Paz Hashash

Arkady Poliakovsky

Approximations in Besov Spaces and Jump Detection of Besov Functions with Bounded Variation

In this paper, we provide a proof that functions belonging to Besov spaces

1), satisfy the following formula under a certain condition: lim →0 + 6 Variations and Besov constants 7 Equivalence Between Gagliardo Constants and Besov Constants 8 Jump Detection in BV ∩ B 1/p,p

8.1 Some observations about jumps of functions in B r q,∞ = B r,q . . . . . . . . . .

1 | ln | [u ] q W r,q (R N ,R d ) = N lim →0 + ˆRN 1 N ˆB (x)
|u(x) -u(y)| q |x -y| rq dydx. (0.1)

Here, [•] W r,q represents the Gagliardo seminorm, and u denotes the convolution of u with a mollifier η ( ) (x) := 1 N η x , η ∈ W 1,1 (R N ), ´RN η(z)dz = 1. Furthermore, we prove that every function

u in BV (R N , R d ) ∩ B 1/p p,∞ (R N , R d ), p ∈ (1, ∞), satisfies lim →0 + 1 | ln | [u ] q W 1/q,q (R N ,R d ) = N lim →0 + ˆRN 1 N ˆB (x) |u(x) -u(y)| q |x -y| dydx = ˆSN-1 |z 1 | dH N -1 (z) ˆJu u + (x) -u -(x) q dH N -1 (x), (0.2)
for every 1 < q < p. Here u + , u -are the one-sided approximate limits of u along the jump set J u . The so-called 'BBM formula', as presented by Bourgain, Brezis, and Mironescu in [START_REF] Bourgain | Another look at Sobolev spaces, Optimal Control and Partial Differential Equations[END_REF], provides a characterization of Sobolev functions W 1,q (Ω) for 1 < q < ∞ and of functions of bounded variation BV (Ω) using double integrals and mollifiers, where Ω ⊂ R N is an open and bounded set with a Lipschitz boundary. The full characterization for BV (Ω) functions is attributed to Dávila [START_REF] Dávila | On an open question about functions of bounded variation[END_REF]. Before describing it, let's recall some definitions.

Definition 1.1. (Decreasing Support Property) Let a ∈ (0, ∞] and ρ ε : (0, ∞) → [0, ∞), ε ∈ (0, a), be a family of L 1 -measurable functions. We say that the family {ρ ε } ε∈(0,a) has the N -dimensional decreasing support property if for every δ ∈ (0, ∞)

lim ε→0 + ˆ∞ δ ρ ε (r)r N -1 dr = 0. (1.1)
Note that by using polar coordinates (see Proposition 10.4), we obtain an alternative form for (1.1):

lim ε→0 + ˆRN \B δ (0) ρ ε (|z|)dz = 0. (1.2)
Definition 1.2. (Kernel) Let a ∈ (0, ∞]. Let ρ ε : (0, ∞) → [0, ∞), ε ∈ (0, a), be a family of L 1 -measurable functions. We say that the family {ρ ε } ε∈(0,a) is a kernel if ´RN ρ ε (|z|)dz = 1, ∀ε ∈ (0, a), and it has the decreasing support property as defined in Definition 1.1.

The BBM formula states that for an open and bounded set Ω ⊂ R N with a Lipschitz boundary, 1 < q < ∞, and u ∈ W 1,q (Ω), for every kernel {ρ ε } ε∈(0,a) (as defined in Definition 1.2), we have lim ε→0 + ˆΩ ˆΩ ρ ε (|x -y|) |u(x) -u(y)| q |x -y| q dy dx = Ĉq,N ∇u q L q (Ω) .

(1.3)

Similarly, for u ∈ BV (Ω), we have lim

ε→0 + ˆΩ ˆΩ ρ ε (|x -y|) |u(x) -u(y)| |x -y| dy dx = Ĉ1,N Du (Ω), (1.4)
where Ĉq,N := ffl S N -1 |z 1 | q dH N -1 (z) for every q ≥ 1. In [START_REF] Poliakovsky | Jump detection in Besov spaces via a new BBM formula. Applications to Aviles-Giga type functionals[END_REF], the following question was investigated: what happens if we replace the left-hand side with q > 1 of Equation (1.3) by the following expression:

lim ε→0 + ˆΩ ˆΩ ρ ε (|x -y|) |u(x) -u(y)| q |x -y| dy dx , (1.5)
where the limit (1.5) is obtained by replacing |u(x)-u(y)| q |x-y| q in (1.3) by |u(x)-u(y)| q |x-y| . Then, the following limit was studied lim

ε→0 + ˆΩ ˆΩ∩Bε(x) 1 L N (B 1 (0))ε N |u(x) -u(y)| q |x -y| dy dx, (1.6)
for 1 < q < ∞, Ω ⊂ R N is an open set with a bounded Lipschitz boundary, and

u ∈ BV (Ω, R d ) ∩ L ∞ (Ω, R d )
. This is a particular case of equation (1.5) with the specific choice of the kernel ρε (r) given by ρε (r) :=

   1 ε N L N (B 1 (0)) if 0 < r < ε 0 if r ≥ ε , ε ∈ (0, ∞). (1.7)
Here, we refer to such a specific kernel as the 'trivial kernel' (see Definition 5.3). The space BV q (Ω, R d ) was also considered in [START_REF] Poliakovsky | Jump detection in Besov spaces via a new BBM formula. Applications to Aviles-Giga type functionals[END_REF]: we define u ∈ BV q (Ω, R d ) if and only if u ∈ L q (Ω, R d ) and lim sup

ε→0 + ˆΩ ˆΩ∩Bε(x) 1 ε N |u(x) -u(y)| q |x -y| dy dx < ∞ (1.8)
holds. In [START_REF] Poliakovsky | Jump detection in Besov spaces via a new BBM formula. Applications to Aviles-Giga type functionals[END_REF], it was proved that the limit in equation (1.6) is determined solely by the jump part of the distributional derivative of u, without considering the absolutely continuous and Cantor parts:

Theorem. (Theorem 1.1 in [START_REF] Poliakovsky | Jump detection in Besov spaces via a new BBM formula. Applications to Aviles-Giga type functionals[END_REF])

Let Ω ⊂ R N be an open set with bounded Lipschitz boundary and let u ∈ BV (Ω, R d )∩L ∞ (Ω, R d ). Then for every 1 < q < ∞ we have u ∈ BV q (Ω, R d ) and

C N ˆJu |u + (x) -u -(x)| q dH N -1 (x) = lim ε→0 + ˆΩ ˆΩ∩Bε(x) 1 ε N |u(x) -u(y)| q |x -y| dy dx, (1.9)
where

C N := 1 N ˆSN-1 |z 1 | dH N -1 (
z), z := (z 1 , ..., z N ).

(1.10)

Here J u is the jump set of the function u, and u + and u -are the one-sided approximate limits of u on J u .

The following proposition gives us a connection between Besov functions in B 1/q q,∞ and BV q -functions.

Proposition. (Proposition 1.1 in [START_REF] Poliakovsky | Jump detection in Besov spaces via a new BBM formula. Applications to Aviles-Giga type functionals[END_REF]) For 1 < q < ∞ we have:

BV q (R N , R d ) = B 1/q q,∞ (R N , R d ).
(1.11)

Moreover, for every open set Ω ⊂ R N we have BV q loc (Ω, R d ) = B 1/q q,∞ loc (Ω, R d ), (1.12) where the local space BV q loc (Ω, R d ) is defined in a usual way.

A more general result than the proposition above was independently obtained by Brasseur in [START_REF] Brasseur | A Bourgain-Brezis-Mironescu characterization of higher order Besov-Nikol'skii spaces[END_REF]. For a comprehensive introduction to Besov spaces, a recommended reference is [START_REF] Leoni | A first course in Sobolev spaces[END_REF].

Recall the definition of Besov space B s q,∞ (R N , R d ):

Definition 1.3. (Besov spaces) Let 1 ≤ q < ∞ and r ∈ (0, 1). Define

B r q,∞ (R N , R d ) := u ∈ L q (R N , R d ) : sup h∈R N \{0} ˆRN |u(x + h) -u(x)| q |h| rq dx < ∞ . (1.13)
For an open set Ω ⊂ R N , the local space B r q,∞ loc (Ω, R d ) is defined to be the set of all functions u ∈ L q loc (Ω, R d ) such that for every compact K ⊂ Ω there exists a function u K ∈ B r q,∞ (R N , R d ) such that u K (x) = u(x) for L N -almost every x ∈ K.

Next, recall the notion of Gagliardo seminorm: Definition 1.4. (Gagliardo Seminorm) Let 1 ≤ q < ∞, E ⊂ R N be an L N -measurable set, u ∈ L q (E, R d ) and r ∈ (0, 1). The Gagliardo seminorm of u in E is defined by

[u] W r,q (E,R d ) := ˆE ˆE |u(x) -u(y)| q |x -y| N +rq dxdy 1 q
.

(1.14)

In [START_REF] Poliakovsky | Asymptotic behavior of W 1/q,q -norm of mollified BV function and its application to singular perturbation problems[END_REF], the following result was proved: for a Lipschitz domain Ω, q ∈ (1, ∞), u ∈ BV (Ω, R d ) ∩ L ∞ (Ω, R d ), and η ∈ W 1,1 (R N ) such that ´RN η(z)dz = 1, if we mollify u by setting for every ε > 0 and x ∈ R N u ε (x) := 1 ε N ˆRN η y -x ε u(y)dy, (1.15) then lim

ε→0 + 1 | ln ε| [u ε ] q W 1/q,q (Ω,R d ) = CN ˆJu u + (x) -u -(x) q dH N -1 (x), (1.16)
with an appropriate dimensional constant CN > 0 (where u in (1.15) is assumed to be continued from Ω to R N such that u ∈ BV (R N , R d ) ∩ L ∞ (R N , R d ) and Du (∂Ω) = 0). It is worth noting that the particular case of (1.16) with η as the Gaussian, q = 2, and Ω = R N was previously proved by Figalli and Jerison in [START_REF] Figalli | How to recognize convexity of a set from its marginals[END_REF] for the characteristic function of a set, and by Hernández in [START_REF] Hernández | Some Properties of a Hilbertian Norm for Perimeter[END_REF] for a general function u. Combining (1.9) and (1.16), we deduce that lim

ε→0 + 1 | ln ε| [u ε ] q W 1/q,q (Ω,R d ) = CN C N lim ε→0 + ˆΩ ˆΩ∩Bε(x) 1 ε N
|u(x) -u(y)| q |x -y| dy dx.

(1.17)

This naturally leads us to pose the following interesting question: does (1.17) hold also for u ∈ BV q \ (BV ∩ L ∞ )? Our first two main results are related to this question:

Theorem 1.1. Let q ∈ [1, ∞) and r ∈ (0, 1). Suppose u ∈ B r q,∞ (R N , R d ), E ⊂ R N be an L Nmeasurable set and η ∈ W 1,1 (R N ). For each ε ∈ (0, ∞) we denote u ε (x) := ˆRN η(z)u(x -εz)dz.

(1.18)

Then, ˆRN η(z)dz

q lim inf ε→0 + ˆSN-1 ˆE χ E (x + εn) |u(x + εn) -u(x)| q ε rq dxdH N -1 (n) ≤ lim inf ε→0 + 1 | ln ε| [u ε ] q W r,q (E,R d ) ≤ lim sup ε→0 + 1 | ln ε| [u ε ] q W r,q (E,R d ) ≤ ˆRN η(z)dz q lim sup ε→0 + ˆSN-1 ˆE χ E (x + εn) |u(x + εn) -u(x)| q ε rq dxdH N -1 (n). (1.19)
Theorem 1.2. Let q ∈ [1, ∞), r ∈ (0, 1). Let u ∈ B r q,∞ (R N , R d ), E ⊂ R N be an L N -measurable set and η ∈ W 1,1 R N . For each ε ∈ (0, ∞) we denote u ε (x) := ´RN η(z)u(x -εz)dz. Assume that the following limit exists:

lim ε→0 + ˆSN-1 ˆE χ E (x + εn) |u(x + εn) -u(x)| q ε rq dxdH N -1 (n).
(1.20)

Then, for every kernel ρ ε we get

lim ε→0 + 1 | ln ε| [u ε ] q W r,q (E,R d ) = ˆRN η(z)dz q H N -1 S N -1 lim ε→0 + ˆE ˆE ρ ε (|x -y|) |u(x) -u(y)| q |x -y| rq dydx = ˆRN η(z)dz q lim ε→0 + ˆSN-1 ˆE χ E (x + εn) |u(x + εn) -u(x)| q ε rq dxdH N -1 (n). (1.21)
Our next result refers to jumps of functions in Besov spaces B 1/p p,∞ , which are also functions of bounded variation. This result generalizes (1.9) (The main improvement is that we don't assume that u ∈ L ∞ ):

Theorem 1.3. Let 1 < p < ∞, u ∈ BV (R N , R d ) ∩ B 1/p
p,∞ (R N , R d ) and 1 < q < p. Then, for every n ∈ R N and every Borel set B ⊂ R N such that H N -1 (∂B ∩ J u ) = 0, we have (1.22) and for every kernel ρ ε , we have

lim ε→0 + ˆB χ B (x + εn) |u(x + εn) -u(x)| q ε dx = ˆB∩Ju u + (x) -u -(x) q |ν u (x) • n|dH N -1 (x),
lim ε→0 + ˆB ˆB ρ ε (|x -y|) |u(x) -u(y)| q |x -y| dydx = lim ε→0 + S N -1 ˆB χ B (x + εn) |u(x + εn) -u(x)| q ε dxdH N -1 (n) = S N -1 |z 1 | dH N -1 (z) ˆJu∩B u + (x) -u -(x) q dH N -1 (x). (1.23)
Here u + , u -are the one-sided approximate limits of u, ν u is a unit normal and J u is the jump set of u (see Definition 10.4).

Corollary 1.1. Let 1 < p < ∞, u ∈ BV (R N , R d ) ∩ B 1/p
p,∞ (R N , R d ) and 1 < q < p. Then, for every n ∈ R N and every Borel set B ⊂ R N such that H N -1 (∂B ∩ J u ) = 0, we obtain for every kernel ρ ε 1 H N -1 (S N -1 ) lim

ε→0 + 1 | ln ε| [u ε ] q W 1/q,q (B,R d ) = ˆRN η(z)dz q lim ε→0 + ˆB ˆB ρ ε (|x -y|) |u(x) -u(y)| q |x -y| dydx = ˆRN η(z)dz q lim ε→0 + S N -1 ˆB χ B (x + εn) |u(x + εn) -u(x)| q ε dxdH N -1 (n) = ˆRN η(z)dz q S N -1 |z 1 | dH N -1 (z) ˆJu∩B u + (x) -u -(x)
q dH N -1 (x). (1.24) Notation 1.1. Throughout the paper, we adopt the following notation: N and d are natural numbers (N, d ∈ N). We denote S N -1 as the (N -1)-dimensional sphere in R N . The N -dimensional Lebesgue measure is denoted as L N , while H N -1 represents the (N -1)-dimensional Hausdorff measure. For an open ball in R N centered at x with a radius of r, we use the notation B r (x). The characteristic function of a set E is denoted as χ E . Furthermore, we use the notation A ⊂⊂ B to indicate that Ā is a compact set and Ā ⊂ B, where Ā represents the topological closure of A.

2 Estimates for Gagliardo Seminorm of Mollified Besov

Functions in Terms of Besov Seminorm

In this section we establish estimates for the Gagliardo seminorm of mollified Besov functions in relation to the Besov seminorm of the functions themselves, without mollification (refer to Corollary 2.1). These estimates will enable us to establish a continuity property for the upper and lower G-functionals in the next section (refer to Definition 3.1 and Lemma 3.1).

Definition 2.1. (Besov Seminorm) Let 1 ≤ q < ∞, r ∈ (0, 1) and E ⊂ R N be an L N -measurable set. Let u : E → R d be an L N -measurable function. The Besov seminorm of u with parameters r, q in E is defined by

[u] B r q,∞ (E,R d ) := sup h∈R N \{0} ˆRN |u(x + h) -u(x)| q |h| rq χ E (x + h)χ E (x)dx 1/q . ( 2
∈ (0, ∞) we denote η (ε) (x) := 1 ε N η x ε , x ∈ R N . The function η (ε) is called an ε-mollifier obtained by η. We call {η (ε) } ε∈(0,∞) a family of mollifiers. For η ∈ L 1 (R N ) and u ∈ L q (R N , R d ) for some 1 ≤ q ≤ ∞, let us define u ε (x) := u * η (ε) (x) = ˆRN η (ε) (x -z)u(z)dz = ˆRN η(z)u(x -εz)dz. (2.2)
The convolution u ε is called mollification of u by the family of mollifiers {η (ε) } ε∈(0,∞) .

Lemma 2.1. (Boundedness of Mollified Functions in Besov and Gagliardo Seminorms)

Let 1 ≤ q < ∞, u ∈ L q (R N , R d ) and η ∈ L 1 (R N ). Then, for every z ∈ R N and ε ∈ (0, ∞) ˆRN |u ε (x) -u ε (x + z)| q dx ≤ ˆRN |η(v)|dv q ˆRN |u(x) -u(x + z)| q dx. (2.3)
In particular, for every r ∈ (0, 1)

sup ε∈(0,∞) [u ε ] B r q,∞ (R N ,R d ) ≤ ˆRN |η(v)|dv [u] B r q,∞ (R N ,R d ) ; (2.4) sup ε∈(0,∞) [u ε ] W r,q (R N ,R d ) ≤ ˆRN |η(v)|dv [u] W r,q (R N ,R d ) . (2.5)
Proof. By (2.2), Hölder's inequality, Fubini's theorem and change of variable formula

ˆRN |u ε (x) -u ε (x + z)| q dx = ˆRN ˆRN η(v) (u(x -εv) -u(x + z -εv)) dv q dx ≤ ˆRN ˆRN |η(v)| |u(x -εv) -u(x + z -εv)| dv q dx = ˆRN ˆRN |η(v)| q-1 q |η(v)| 1 q |u(x -εv) -u(x + z -εv)| dv q dx ≤ ˆRN |η(v)|dv q-1 ˆRN ˆRN |η(v)| |u(x -εv) -u(x + z -εv)| q dvdx = ˆRN |η(v)|dv q-1 ˆRN |η(v)| ˆRN |u(x -εv) -u(x + z -εv)| q dx dv = ˆRN |η(v)|dv q ˆRN |u(y) -u(y + z)| q dy. (2.6)
Let r ∈ (0, 1). Dividing the inequality (2.6) by |z| rq , z = 0, taking the supremum over z ∈ R N \ {0} and then the supremum over ε ∈ (0, ∞), we obtain (2.4). By Definition 1.4 (Gagliardo seminorm), change of variable formula, Fubini's theorem and (2.3) we get

[u ε ] q W r,q (R N ,R d ) = ˆRN ˆRN |u ε (x) -u ε (y)| q |x -y| N +rq dx dy = ˆRN ˆRN |u ε (x + y) -u ε (y)| q |x| N +rq dx dy = ˆRN 1 |x| N +rq ˆRN |u ε (x + y) -u ε (y)| q dy dx ≤ η q L 1 (R N ) ˆRN 1 |x| N +rq ˆRN |u(x + y) -u(y)| q dy dx = η q L 1 (R N ) ˆRN ˆRN |u(x) -u(y)| q |x -y| N +rq dx dy. (2.7)
Inequality (2.5) follows from (2.7).

Lemma 2.2. (Estimates for Gagliardo Seminorm of Mollified Besov Functions -part 1)

Let 1 ≤ q < ∞, r ∈ (0, 1), u ∈ B r q,∞ (R N , R d ) and η ∈ W 1,1 (R N ). For every ε ∈ (0, ∞) and z ∈ R N \ {0} we denote g ε (z) := ˆRN |u ε (x) -u ε (x + z)| q |z| N +rq dx. (2.8)
Then, for every

0 < β < γ < ∞ it follows that ˆRN \Bγ (0) g ε (z)dz ≤ η q L 1 (R N ) 2 q u q L q (R N ,R d ) H N -1 S N -1 rqγ rq ; (2.9) ˆBγ(0)\Bβ(0) g ε (z)dz ≤ η q L 1 (R N ) [u] q B r q,∞ (R N ,R d ) H N -1 S N -1 (ln(γ) -ln(β)) ;
(2.10)

ˆBβ (0) g ε (z)dz ≤ ˆRN |∇η(v)|dv q 2 q u q L q (R N ,R d ) H N -1 S N -1 q -rq β q-rq ε q . (2.

11)

If ε = β, then we have the following alternative to (2.11) estimate:

ˆBε(0) g ε (z)dz ≤ ∇η q-1 L 1 (R N ,R N ) [u] q B r q,∞ (R N ,R d ) ˆRN |∇η(v)|(|v| + 2) rq dv H N -1 S N -1 q -rq . (2.12)
The right hand side of (2.12) can be infinite.

Proof. By Lemma 2.1 and the convexity of the function r -→ r q , r ∈ [0, ∞), we have

g ε (z) ≤ η q L 1 (R N ) |z| N +rq ˆRN |u(x) -u(x + z)| q dx ≤ η q L 1 (R N ) 2 q u q L q (R N ,R d ) 1 |z| N +rq .
(2.13) Thus, by polar coordinates (refer to Proposition 10.4)

ˆRN \Bγ (0) g ε (z)dz ≤ η q L 1 (R N ) 2 q u q L q (R N ,R d ) ˆRN \Bγ (0) 1 |z| N +rq dz = η q L 1 (R N ) 2 q u q L q (R N ,R d ) H N -1 S N -1 rqγ rq . (2.14)
It proves (2.9). By (2.8), (2.4) and polar coordinates

ˆBγ(0)\Bβ(0) g ε (z)dz = ˆBγ(0)\Bβ(0) 1 |z| N ˆRN |u ε (x) -u ε (x + z)| q |z| rq dx dz ≤ ˆRN |η(v)|dv q [u] q B r q,∞ (R N ,R d ) ˆBγ(0)\Bβ(0) 1 |z| N dz = ˆRN |η(v)|dv q [u] q B r q,∞ (R N ,R d ) H N -1 S N -1 (ln(γ) -ln(β)) . (2.15)
It proves (2.10). We now prove (2.11). By (2.8) and Fubini's theorem

ˆBβ (0) g ε (z)dz = ˆRN ˆBβ (0) |u ε (x) -u ε (x + z)| q |z| N +rq dz dx. (2.16) Assume for a moment that η ∈ C 1 (R N ) ∩ W 1,1 (R N ). By (2.
2), change of variable formula, the fundamental theorem of calculus, Fubini's theorem and Jensen's inequality we obtain for every

x ∈ R N ˆBβ (0) |u ε (x) -u ε (x + z)| q |z| N +rq dz = ˆBβ (0) 1 |z| N +rq 1 ε N ˆRN η x -y ε -η x + z -y ε u(y)dy q dz, [z = εw] = 1 ε rq ˆBβ/ε (0) 1 |w| N +rq 1 ε N ˆRN η x -y ε -η x -y ε + w u(y)dy q dw, [y = x -εv] = 1 ε rq ˆBβ/ε (0) 1 |w| N +rq ˆRN (η(v) -η(v + w)) u(x -εv)dv q dw = 1 ε rq ˆBβ/ε (0) 1 |w| N +rq ˆRN (η(v + w) -η(v)) (u(x -εv) -u(x)) dv q dw = 1 ε rq ˆBβ/ε (0) 1 |w| N +rq ˆRN w • ˆ1 0 ∇η(v + tw)dt (u(x -εv) -u(x)) dv q dw ≤ 1 ε rq ˆBβ/ε (0) 1 |w| N +rq-q ˆ1 0 ˆRN |∇η(v + tw)| |u(x -εv) -u(x)| dvdt q dw ≤ 1 ε rq ˆ1 0 ˆBβ/ε (0) 1 |w| N +rq-q ˆRN |∇η(v + tw)| |u(x -εv) -u(x)| dv q dwdt = 1 ε rq ˆ1 0 ˆBβ/ε (0) 1 |w| N +rq-q ˆRN |∇η(v)| |u(x -ε(v -tw)) -u(x)| dv q dwdt. (2.17)
In the forth equality we use ´RN (η(v + w) -η(v)) dv = 0, w ∈ R N . By Hölder's inequality

ˆRN |∇η(v)| |u(x -ε(v -tw)) -u(x)| dv q = ˆRN |∇η(v)| q-1 q |∇η(v)| 1 q |u(x -ε(v -tw)) -u(x)| dv q ≤ ˆRN |∇η(v)|dv q-1 ˆRN |∇η(v)| |u(x -ε(v -tw)) -u(x)| q dv. (2.18)
By (2.18) and Fubini's theorem

ˆRN ˆRN |∇η(v)| |u(x -ε(v -tw)) -u(x)| dv q dx ≤ ˆRN |∇η(v)|dv q-1 ˆRN |∇η(v)| ˆRN |u(x -ε(v -tw)) -u(x)| q dx dv ≤ ˆRN |∇η(v)|dv q 2 q u q L q (R N ,R d ) . (2.
ˆRN ˆBβ (0) |u ε (x) -u ε (x + z)| q |z| N +rq dzdx ≤ ˆRN |∇η(v)|dv q 2 q u q L q (R N ,R d ) 1 ε rq ˆBβ/ε (0) 1 |w| N +rq-q dw = ˆRN |∇η(v)|dv q 2 q u q L q (R N ,R d ) 1 ε rq H N -1 S N -1 q -rq β ε q-rq = ˆRN |∇η(v)|dv q 2 q u q L q (R N ,R d ) H N -1 S N -1 q -rq β q-rq ε q . (2.20) It proves (2.11) in case η ∈ C 1 (R N ) ∩ W 1,1 (R N ). We now prove (2.12) in case η ∈ C 1 (R N ) ∩ W 1,1 (R N )
. By (2.18) and Definition 2.1 (definition of Besov seminorm)

ˆRN ˆRN |∇η(v)| |u(x -ε(v -tw)) -u(x)| dv q dx ≤ ˆRN |∇η(v)|dv q-1 ˆRN |∇η(v)| ˆRN |u(x -ε(v -tw)) -u(x)| q dx dv ≤ ε rq ˆRN |∇η(v)|dv q-1 [u] q B r q,∞ (R N ,R d ) ˆRN |∇η(v)||v -tw| rq dv. (2.21)
By (2.17) with ε = β and (2.21)

ˆRN ˆBε(0) |u ε (x) -u ε (x + z)| q |z| N +rq dzdx ≤ ∇η q-1 L 1 (R N ,R N ) [u] q B r q,∞ (R N ,R d ) ˆ1 0 ˆB1 (0) 1 |w| N +rq-q ˆRN |∇η(v)||v -tw| rq dv dwdt ≤ ∇η q-1 L 1 (R N ,R N ) [u] q B r q,∞ (R N ,R d ) ˆ1 0 ˆB1 (0) 1 |w| N +rq-q ˆRN |∇η(v)|(|v| + 1) rq dv dwdt = ∇η q-1 L 1 (R N ,R N ) [u] q B r q,∞ (R N ,R d ) ˆRN |∇η(v)|(|v| + 1) rq dv ˆB1 (0) 1 |w| N +rq-q dw = ∇η q-1 L 1 (R N ,R N ) [u] q B r q,∞ (R N ,R d ) ˆRN |∇η(v)|(|v| + 1) rq dv H N -1 S N -1 q -rq . (2.22) It proves (2.12) in case η ∈ C 1 (R N ) ∩ W 1,1 (R N )
. We now generalize (2.11) and (2.12) for η ∈

W 1,1 (R N ). For η ∈ W 1,1 (R N ), let η δ := η * γ (δ) , γ (δ) (v) := 1 δ N γ v δ , where γ ∈ C 1 (R N ), supp(γ) ⊂ B 1 (0), γ ≥ 0 and γ L 1 (R N ) = 1.
Here supp(γ) stands for the support of γ. By (2.22) we get for every 0 < δ < 1

ˆRN ˆBε(0) |u * (η δ ) (ε) (x) -u * (η δ ) (ε) (x + z)| q |z| N +rq dzdx ≤ ∇η δ q-1 L 1 (R N ,R N ) [u] q B r q,∞ (R N ,R d ) ˆRN |∇η δ (v)|(|v| + 1) rq dv H N -1 S N -1 q -rq ≤ ∇η q-1 L 1 (R N ,R N ) [u] q B r q,∞ (R N ,R d ) ˆRN |∇η(v)|(|v| + 2) rq dv H N -1 S N -1 q -rq . (2.23)
Let us explain the last inequality of (2.23): Since

|∇η δ (v)| = |∇η * γ (δ) (v)| = ˆRN ∇η(z)γ (δ) (v -z)dz = ˆRN ∇η(v -δy)γ(y)dy ≤ ˆRN |∇η(v -
u * (η δ ) (ε) = u * η * γ (δ) (ε) = u * η (ε) * γ (δ) (ε) = u * η (ε) * γ (ε) (δ) = u * η (ε) * γ (ε) (δ) .
(2.27)

Since u * η (ε) ∈ L q (R N , R d ), γ (ε) ∈ C 1 c (R N ), γ (ε) ≥ 0 and γ (ε) L 1 (R N ) = 1, then the family of functions {u * (η δ ) (ε) } {0<δ<1} converges in L q (R N , R d ) to the function u * η (ε) as δ → 0 + ,
and hence has a subsequence converging almost everywhere. Thus, by (2.23) and Fatou's Lemma we get (2.12) for η ∈ W 1,1 (R N ).

Using the same technique we get also (2.11)

for η ∈ W 1,1 (R N ): Let {u * (η δn ) (ε)
} n∈N be a sequence converging L N -almost everywhere to the function u * η (ε) . By (2.20) and (2.25) we have for every n ∈ N ˆRN ˆBβ (0) |u * (η δn

) (ε) (x) -u * (η δn ) (ε) (x + z)| q |z| N +rq dzdx ≤ ˆRN |∇η δn (v)|dv q 2 q u q L q (R N ,R d ) H N -1 S N -1 q -rq β q-rq ε q ≤ ˆRN |∇η(v)|dv q 2 q u q L q (R N ,R d ) H N -1 S N -1 q -rq β q-rq ε q . (2.28)
Taking the limit as n goes to ∞ and using Fatou's lemma we get (2.11) for η ∈ W 1,1 (R N ).

Corollary 2.1. (Estimates for Gagliardo Seminorm of Mollified Besov Functions -part 2)

Let 1 ≤ q < ∞, r ∈ (0, 1), u ∈ B r q,∞ (R N , R d ) and η ∈ W 1,1 (R N ). For every ε ∈ (0, ∞) and 0 < β < γ < ∞ it follows that [u ε ] q W r,q (R N ,R d ) ≤ η q L 1 (R N ) 2 q u q L q (R N ,R d ) H N -1 S N -1 rqγ rq + η q L 1 (R N ) [u] q B r q,∞ (R N ,R d ) H N -1 S N -1 (ln(γ) -ln(β)) + ˆRN |∇η(v)|dv q 2 q u q L q (R N ,R d ) H N -1 S N -1 q -rq β q-rq ε q . (2.29)
In particular,

sup ε∈(0,1/e) 1 | ln ε| [u ε ] q W r,q (R N ,R d ) ≤ η q L 1 (R N ) 2 q u q L q (R N ,R d ) H N -1 S N -1 rq + η q L 1 (R N ) [u] q B r q,∞ (R N ,R d ) H N -1 S N -1 q q -rq + ˆRN |∇η(v)|dv q 2 q u q L q (R N ,R d ) H N -1 S N -1 q -rq . (2.30)
Proof. By definition of Gagliardo seminorm (Definition 1.4), change of variable formula, Fubini's theorem and additivity of integral we get

[u ε ] q W r,q (R N ,R d ) = ˆRN ˆRN |u ε (x) -u ε (y)| q |x -y| N +rq dy dx = ˆRN ˆRN |u ε (x) -u ε (x + z)| q |z| N +rq dz dx = ˆRN ˆRN |u ε (x) -u ε (x + z)| q |z| N +rq dx dz = ˆRN \Bγ (0) g ε (z)dz+ ˆBγ(0)\Bβ(0) g ε (z)dz+ ˆBβ (0) g ε (z)dz, (2.31)
where we denote choosing γ = 1, β = ε q q-rq , and using that 1 | ln ε| < 1 for every ε ∈ (0, 1/e).

g ε (z) := ˆRN |u ε (x) -u ε (x + z)| q |z| N +rq dx, z ∈ R N \ {0}. ( 2

Continuity of G-Functionals

In this section, we define the upper and lower G-functionals (see Definition 3.1). We prove continuity properties for these functionals (see Lemma 3.1). These continuity properties, in particular, allow us to generalize results involving η ∈ C 1 c (R N ) to cases where η ∈ W 1,1 (R N ) (refer to the proof of Corollary 7.2). Additionally, we introduce the Gagliardo constants, which are specific instances of the G-functionals where the function η is fixed (see Definition 3.2). Let us define for q ∈ [1, ∞), r ∈ (0, 1) and an L N -measurable set E ⊂ R N the upper Gfunctional and the lower G-functional, respectively, to be

G E , G E : B r q,∞ (R N , R d ) × W 1,1 (R N ) → [0, ∞), G E (u, η) := lim sup ε→0 + 1 | ln ε| u * η (ε) q W r,q (E,R d ) , G E (u, η) := lim inf ε→0 + 1 | ln ε| u * η (ε) q W r,q (E,R d ) . (3.1)
Remark 3.1. (Well-definedness of the Upper and Lower G-Functionals)

The well-definedness of the upper and lower G-functionals follows immediately from (2.30). Note that u * η Let q ∈ [1, ∞), r ∈ (0, 1) and E ⊂ R N be an

(ε) W r,q (R N ,R d ) ∈ [0, ∞) for every ε ∈ (0, ∞) assuming only that u ∈ L q (R N , R d ) and η ∈ W 1,1 (R N ): One can show by Hölder's inequality that the convolution u * η lies in L q (R N , R d ), 1 ≤ q ≤ ∞, whenever u ∈ L q (R N , R d ) and η ∈ L 1 (R N ). Therefore, if 1 ≤ q < ∞, u ∈ L q (R N , R d ) and η ∈ W 1,1 (R N ), then u ε ∈ L q (R N , R d ), and it has weak derivatives ∂ ∂x i u ε = u * ∂ ∂x i η (ε) ∈ L q (R N , R d ) for each 1 ≤ i ≤ N . Therefore, u ε ∈ W 1,q (R N , R d ) ⊂ W r,q (R N , R d ), for every r ∈ (0, 1). Thus, u * η (ε) W r,q (R N ,R d ) ∈ [0, ∞) for every ε ∈ (0, ∞).
L N -measurable set. 1. If u ∈ B r q,∞ (R N , R d ) and {η n } ∞ n=1 ⊂ W 1,1 (R N ) is a sequence such that η n converges to η in W 1,1 (R N ), then lim n→∞ G E (u, η n ) = G E (u, η), lim n→∞ G E (u, η n ) = G E (u, η). (3.2) 2. If η ∈ W 1,1 (R N ) and {u n } ∞ n=1 ⊂ B r q,∞ (R N , R d ) is a sequence such that u n converges to u in B r q,∞ (R N , R d ), which means that lim n→∞ u -u n L q (R N ,R d ) + [u -u n ] B r q,∞ (R N ,R d ) = 0, then lim n→∞ G E (u n , η) = G E (u, η), lim n→∞ G E (u n , η) = G E (u, η). (3.3)
Proof. 1. For every n ∈ N we get by (2.30)

sup ε∈(0,1/e) 1 | ln ε| u * (η n ) (ε) q W r,q (E,R d ) < ∞, sup ε∈(0,1/e) 1 | ln ε| u * η (ε) q W r,q (E,R d ) < ∞. (3.4)
Therefore, by Lemma 10.6 we get lim inf

ε→0 + 1 | ln ε| 1/q u * (η n ) (ε) W r,q (E,R d ) -lim inf ε→0 + 1 | ln ε| 1/q u * η (ε) W r,q (E,R d ) ≤ lim sup ε→0 + 1 | ln ε| 1/q u * (η n ) (ε) W r,q (E,R d ) -u * η (ε) W r,q (E,R d ) , (3.5)
and lim sup

ε→0 + 1 | ln ε| 1/q u * (η n ) (ε) W r,q (E,R d ) -lim sup ε→0 + 1 | ln ε| 1/q u * η (ε) W r,q (E,R d ) ≤ lim sup ε→0 + 1 | ln ε| 1/q u * (η n ) (ε) W r,q (E,R d ) -u * η (ε) W r,q (E,R d ) . (3.6)
By the triangle inequality for Gagliardo seminorm we get lim sup

ε→0 + 1 | ln ε| 1/q u * (η n ) (ε) W r,q (E,R d ) -u * η (ε) W r,q (E,R d ) ≤ lim sup ε→0 + 1 | ln ε| 1/q u * (η n ) (ε) -u * η (ε) W r,q (E,R d ) = lim sup ε→0 + 1 | ln ε| 1/q u * (η n ) (ε) -η (ε) W r,q (E,R d ) = lim sup ε→0 + 1 | ln ε| 1/q u * (η n -η) (ε) W r,q (E,R d ) ≤ lim sup ε→0 + 1 | ln ε| 1/q u * (η n -η) (ε) W r,q (R N ,R d ) . (3.7)
Therefore, by (2.30)

lim sup ε→0 + 1 | ln ε| u * (η n ) (ε) W r,q (E,R d ) -u * η (ε) W r,q (E,R d ) q ≤ lim sup ε→0 + 1 | ln ε| u * (η n -η) (ε) q W r,q (R N ,R d ) ≤ sup ε∈(0,1/e) 1 | ln ε| u * (η n -η) (ε) q W r,q (R N ,R d ) ≤ η n -η q L 1 (R N ) 2 q u q L q (R N ,R d ) H N -1 S N -1 rq + η n -η q L 1 (R N ) [u] q B r q,∞ (R N ,R d ) H N -1 S N -1 q q -rq + ˆRN |∇ (η n -η) (v)|dv q 2 q u q L q (R N ,R d ) H N -1 S N -1 q -rq . (3.8)
Taking the limit as n → ∞ in (3.8) we get (3.2) from (3.5) and (3.6).

2. Replacing η n with η and u with u n , we get in the same way lim sup

ε→0 + 1 | ln ε| u n * η (ε) W r,q (E,R d ) -u * η (ε) W r,q (E,R d ) q ≤ η q L 1 (R N ) 2 q u-u n q L q (R N ,R d ) H N -1 S N -1 rq + η q L 1 (R N ) [u -u n ] q B r q,∞ (R N ,R d ) H N -1 S N -1 q q -rq + ˆRN |∇η(v)|dv q 2 q u -u n q L q (R N ,R d )
H N -1 S N -1 q -rq . (3.9)

Taking the limit as n → ∞ we get (3.3).

Definition 3.2. (Gagliardo constants) Let q ∈ [1, ∞), r ∈ (0, 1)
, let E ⊂ R N be an L N -measurable set, and η ∈ W 1,1 (R N ). We define the (r, q) upper Gagliardo constant of u in E with respect to η as the quantity:

lim sup ε→0 + 1 | ln ε| u * η (ε) q W r,q (E,R d ) . (3.10)
Similarly, replacing the lim sup by the lim inf, we define the (r, q) lower Gagliardo constant of u in E with respect to η. If the limit exists, we refer to it as the (r, q) Gagliardo constant of u in E with respect to η.

B r,q -Functions

In this section, we introduce the space B r,q (E, R d ) (see Definition 4.2). We establish several properties of these functions, as detailed in Propositions 4.1 and 4.2, as well as Corollary 4.1. Additionally, we prove the equivalence between the space B r,q (R N , R d ) and the Besov space B r q,∞ (R N , R d ) (refer to Theorem 4.1). Revised Version of the Paper on Approximation in Besov Spaces Definition 4.1. (B r,q -Seminorms)

Let us define for r ∈ (0, 1), q ∈ [1, ∞), an L N -measurable set E ⊂ R N and L N -measurable function u : E → R d the following two quantities: The B r,q -seminorm is defined by

|u| B r,q (E,R d ) := sup ε∈(0,1) ˆE 1 ε N ˆE∩Bε(x) |u(x) -u(y)| q |x -y| rq dydx 1 q ; (4.1)
the upper infinitesimal B r,q -seminorm is defined by Let r ∈ (0, 1), q ∈ [1, ∞) and an L N -measurable set E ⊂ R N . We define a set

[u] B r,q (E,R d ) := lim sup ε→0 + ˆE 1 ε N ˆE∩Bε(x) |u(x) -u(y)| q |x -y| rq dydx 1 q . ( 4
B r,q (E, R d ) := u ∈ L q (E, R d ) : |u| B r,q (E,R d ) < ∞ . (4.3)
We define the local space B r,q loc (E, R d ) as follows: u ∈ B r,q loc (E, R d ) if and only if u ∈ L q loc (E, R d ) and u ∈ B r,q (K, R d ) for every compact set K ⊂ E. Proposition 4.1. (Properties of B r,q -Seminorms)

Let r ∈ (0, 1), q ∈ [1, ∞) and E ⊂ R N be an L N -measurable set. Then, 1. The B r,q -seminorm and the upper infinitesimal B r,q -seminorm are seminorms on B r,q (E, R d );

2. For u ∈ L q (E, R d ), |u| B r,q (E,R d ) < ∞ if and only if [u] B r,q (E,R d ) < ∞; 3. For an open set Ω ⊂ R N , u ∈ B r,q
loc (Ω, R d ) if and only if for every compact set K ⊂ Ω we have

lim sup ε→0 + ˆK 1 ε N ˆBε(x) |u(x) -u(y)| q |x -y| rq dydx < ∞. (4.4)
4. Let us denote:

u 1 := [u] B r,q (E,R d ) + u L q (E,R d ) , u 2 := |u| B r,q (E,R d ) + u L q (E,R d ) . (4.5)
Then, • 1 , • 2 are norms on the space B r,q (E, R d )1 and B r,q (E, R d ), • 2 is a Banach space.

Proof. 1. Let u, v ∈ B r,q (E, R d ) and a ∈ R. It follows immediately from definitions that |u| B r,q (E,R d ) , [u] B r,q (E,R d ) are non-negative and homogeneous, which means that

|au| B r,q (E,R d ) = |a||u| B r,q (E,R d ) and [au] B r,q (E,R d ) = |a|[u] B r,q (E,R d ) .
We have by Minkowski's inequality

ˆE ˆE∩Bε(x) |(u + v)(x) -(u + v)(y)| q |x -y| rq dydx 1 q ≤ ˆE ˆE χ Bε(x) (y) |u(x) -u(y)| |x -y| r + χ Bε(x) (y) |v(x) -v(y)| |x -y| r q dydx 1 q ≤ ˆE ˆE χ Bε(x) (y) |u(x) -u(y)| |x -y| r q dydx 1 q + ˆE ˆE χ Bε(x) (y) |v(x) -v(y)| |x -y| r q dydx 1 q . (4.6)
The triangle inequality for

[•] B r,q (E,R d ) , | • | B r,q (E,R d ) follows from (4.6). 2. Since for every L N -measurable function u : E → R d we have [u] B r,q (E,R d ) ≤ |u| B r,q (E,R d ) , then the finiteness of |u| B r,q (E,R d ) implies the finiteness of [u] B r,q (E,R d ) . Assume [u] B r,q (E,R d ) < ∞.
Then, there exists a number 0 < ε 0 < 1 such that sup

ε∈(0,ε 0 ] ˆE ˆE∩Bε(x) 1 ε N |u(x) -u(y)| q |x -y| rq dy dx < ∞. (4.7)
We have sup

ε∈[ε 0 ,1) ˆE ˆE∩Bε(x) 1 ε N |u(x) -u(y)| q |x -y| rq dy dx ≤ ˆE ˆE∩Bε 0 (x) 1 ε N 0 |u(x) -u(y)| q |x -y| rq dy dx + sup ε∈[ε 0 ,1) ˆE ˆE∩(Bε(x)\Bε 0 (x)) 1 ε N |u(x) -u(y)| q |x -y| rq dy dx, (4.8)
and sup

ε∈[ε 0 ,1) ˆE ˆE∩(Bε(x)\Bε 0 (x)) 1 ε N |u(x) -u(y)| q |x -y| rq dy dx ≤ 2 q-1 1 ε N +rq 0 sup ε∈[ε 0 ,1) ˆE ˆE∩(Bε(x)\Bε 0 (x)) |u(x)| q dy dx + 2 q-1 1 ε N +rq 0 sup ε∈[ε 0 ,1) ˆE ˆE∩(Bε(x)\Bε 0 (x)) |u(y)| q dy dx ≤ 2 q 1 ε N +rq 0 L N (B 1 (0)) u q L q (E,R d ) < ∞. (4.9) Thus, |u| B r,q (E,R d ) < ∞. 3. If for a compact set K ⊂ R N we have (4.4), then [u] B r,q (K,R d ) <
∞ and by item 2 we have also

|u| B r,q (K,R d ) < ∞, hence u ∈ B r,q (K, R d ).
For the opposite implication, let u ∈ B r,q loc (Ω, R d ) and K ⊂ Ω be a compact set. Let Ω 0 ⊂⊂ Ω be an open set containing K. Since Ω 0 is open, then we have for every small enough ε ∈ (0, ∞) that K + B ε (0) ⊂ Ω 0 . Hence, by item 2 we have lim sup

ε→0 + ˆK 1 ε N ˆBε(x) |u(x) -u(y)| q |x -y| rq dydx ≤ lim sup ε→0 + ˆΩ0 1 ε N ˆBε(x)∩Ω0 |u(x) -u(y)| q |x -y| rq dydx < ∞. (4.10) 4. Since by item 1 [•] B r,q (E,R d ) , | • | B r,q (E,R d ) are seminorms and • L q (E,R d ) is a norm, then • 1 , • 2 are norms on the space B r,q (E, R d ). The space B r,q (E, R d ), • 2 is complete: let {u n } ∞ n=1 ⊂ B r,q (E, R d ) be a Cauchy sequence. Then, it is also a Cauchy sequence in L q (E, R d ) so there exists a function u ∈ L q (E, R d ) such that {u n } ∞ n=1 converges to u in L q (E, R d ),
and up to a subsequence the convergence is also L N -almost everywhere. Since {u n } ∞ n=1 is a Cauchy sequence, then it is bounded, so there exists a number M such that |u n | B r,q (E,R d ) ≤ M for every n ∈ N. By Fatou's lemma we get

|u| B r,q (E,R d ) ≤ lim inf n→∞ sup ε∈(0,1) ˆE 1 ε N ˆE∩Bε(x) |u n (x) -u n (y)| q |x -y| rq dydx 1 q ≤ M. (4.11)
Thus, u ∈ B r,q (E, R d ).

Proposition 4.2. (Continuous Embedding of W r,q into B r,q , and Negligibility of the Upper Infinitesimal B r,q -Seminorm for Sobolev Functions)

Let r ∈ (0, 1), q ∈ [1, ∞) and E ⊂ R N be an L N -measurable set. Then, the space W r,q (E,

R d ) with the norm [•] W r,q (E,R d ) + • L q (E,R d ) is continuously embedded in the space B r,q (E, R d ) with the norm • 2 defined in (4.5). Moreover, [u] B r,q (E,R d ) = 0 for every u ∈ W r,q (E, R d ).
Proof. We have for every ε ∈ (0, ∞) and u

∈ W r,q (E, R d ) ∞ > ˆE ˆE |u(x) -u(y)| q |x -y| N +rq dydx ≥ ˆE ˆE∩Bε(x) |u(x) -u(y)| q |x -y| N +rq dy dx ≥ ˆE 1 ε N ˆE∩Bε(x) |u(x) -u(y)| q |x -y| rq dydx. (4.12)
By (4.12) we conclude that W r,q (E,

R d ) is continuously embedded in B r,q (E, R d ). Notice that ˆE sup ε∈(0,∞) ˆE∩Bε(x) |u(x) -u(y)| q |x -y| N +rq dy dx ≤ ˆE ˆE |u(x) -u(y)| q |x -y| N +rq dydx < ∞; lim ε→0 + ˆE∩Bε(x) |u(x) -u(y)| q |x -y| N +rq dy = 0, for L N -almost every x ∈ E. (4.13) Therefore, by Dominated Convergence Theorem [u] B r,q (E,R d ) ≤ lim sup ε→0 + ˆE ˆE∩Bε(x) |u(x) -u(y)| q |x -y| N +rq dydx = ˆE lim ε→0 + ˆE∩Bε(x) |u(x) -u(y)| q |x -y| N +rq dy dx = 0. (4.14) Corollary 4.1. (Non-equivalence of the Seminorms | • | B r,q , [•] B r,q ) Let r ∈ (0, 1), q ∈ [1, ∞)
and Ω ⊂ R N be an open set which is not empty. Let • 1 , • 2 be the norms defined in (4.5). Then, the space B r,q (Ω, R d ), • 1 is not a Banach space. In particular, the seminorms

| • | B r,q , [•] B r,q are not equivalent. Proof. Let u ∈ L q (Ω, R d ) such that [u] B r,q (Ω,R d ) = ∞. Let {u n } ∞ n=1 ⊂ C 1 c (Ω, R d ) be a sequence which converges to u in L q (Ω, R d ), so it is also a Cauchy sequence in L q (Ω, R d ). Therefore, by Proposition 4.2 we have that {u n } ∞ n=1 ⊂ W r,q (Ω, R d ) ⊂ B r,q (Ω, R d
) and this sequence is also a Cauchy sequence with respect to the norm

• 1 because u n -u k 1 = u n -u k L q (Ω,R d ) for every k, n ∈ N. Thus, {u n } ∞
n=1 is a Cauchy sequence in the space B r,q (Ω, R d ), • 1 which does not have a limit in the space. Since by item 4 of Proposition 4.1 B r,q (Ω, R d ), • 2 is a Banach space, then the norms • 1 , • 2 are not equivalent and so as the seminorms

| • | B r,q , [•] B r,q . Theorem 4.1. (Equivalence Between B r,q -Spaces and Besov Spaces B r q,∞ ) Let r ∈ (0, 1), q ∈ [1, ∞). Then, B r q,∞ (R N , R d ) = B r,q (R N , R d ), (4.15)
and for every open set

Ω ⊂ R N B r q,∞ loc (Ω, R d ) = B r,q loc (Ω, R d ). (4.16) Proof. Assume that u ∈ B r q,∞ (R N , R d ). Then, for every ε ∈ (0, ∞) ˆRN 1 ε N ˆBε(x) |u(x) -u(y)| q |x -y| rq dydx = ˆRN ˆB1 (0) |u(x) -u(x + εz)| q |εz| rq dzdx = ˆB1 (0) ˆRN |u(x) -u(x + εz)| q |εz| rq dx dz ≤ [u] q B r q,∞ (R N ,R d ) L N (B 1 (0)) < ∞. (4.17)
Thus, we get

sup ε∈(0,∞) ˆRN 1 ε N ˆBε(x) |u(x) -u(y)| q |x -y| rq dydx ≤ [u] q B r q,∞ (R N ,R d ) L N (B 1 (0)) < ∞. (4.18) Thus, u ∈ B r,q (R N , R d ). Assume that u ∈ B r,q (R N , R d ).
Step 1: for every

h 1 , h 2 ∈ R N such that 0 / ∈ {h 1 , h 2 , h 1 + h 2 } we have ˆRN |u(x + (h 1 + h 2 )) -u(x)| q |h 1 + h 2 | rq dx = ˆRN | (u(x + (h 1 + h 2 )) -u(x + h 1 )) + (u(x + h 1 ) -u(x)) | q |h 1 + h 2 | rq dx ≤ 2 q-1 ˆRN |u(x + (h 1 + h 2 )) -u(x + h 1 )| q |h 1 + h 2 | rq dx + 2 q-1 ˆRN |u(x + h 1 ) -u(x)| q |h 1 + h 2 | rq dx = 2 q-1 |h 2 | rq |h 1 + h 2 | rq ˆRN |u(x + h 2 ) -u(x)| q |h 2 | rq dx + 2 q-1 |h 1 | rq |h 1 + h 2 | rq ˆRN |u(x + h 1 ) -u(x)| q |h 1 | rq dx. (4.19) Step 2: let ν ∈ S N -1 , ε ∈ (0, ∞) and z ∈ R N . Denote h 1 := εz and h 2 := ε(ν -z). Note that B 1/2 1 2 ν ⊂ B 1 (0) ∩ B 1 (ν). For z ∈ B 1/2 1 2 ν , we get by (4.19) ˆRN |u(x + εν) -u(x)| q ε rq dx ≤ 2 q-1 |ν -z| rq ˆRN |u(x + ε(ν -z)) -u(x)| q |ε(ν -z)| rq dx + 2 q-1 |z| rq ˆRN |u(x + εz) -u(x)| q |εz| rq dx ≤ 2 q-1 ˆRN |u(x + ε(ν -z)) -u(x)| q |ε(ν -z)| rq dx + 2 q-1 ˆRN |u(x + εz) -u(x)| q |εz| rq dx. (4.20)
Taking the average with respect to dz on the ball B 1/2 1 2 ν of both sides of the inequality (4.20), we get

ˆRN |u(x + εν) -u(x)| q ε rq dx ≤ 2 q-1 L N B 1/2 1 2 ν ˆB1/2( 1 2 ν) ˆRN |u(x + ε(ν -z)) -u(x)| q |ε(ν -z)| rq dxdz + 2 q-1 L N B 1/2 1 2 ν ˆB1/2( 1 2 ν) ˆRN |u(x + εz) -u(x)| q |εz| rq dxdz ≤ 2 N +q-1 L N (B 1 (0)) ˆB1 (ν) ˆRN |u(x + ε(ν -z)) -u(x)| q |ε(ν -z)| rq dxdz + 2 N +q-1 L N (B 1 (0)) ˆB1 (0) ˆRN |u(x + εz) -u(x)| q |εz| rq dxdz = 2 N +q L N (B 1 (0)) ˆB1 (0) ˆRN |u(x + εz) -u(x)| q |εz| rq dxdz. (4.21) Therefore, since u ∈ B r,q (R N , R d ), then lim sup ε→0 + sup ν∈S N -1 ˆRN |u(x + εν) -u(x)| q ε rq dx ≤ 2 N +q L N (B 1 (0)) lim sup ε→0 + ˆB1 (0) ˆRN |u(x + εz) -u(x)| q |εz| rq dxdz = lim sup ε→0 + ˆRN 1 ε N ˆBε(x) |u(x) -u(y)| q |x -y| rq dydx < ∞. (4.22)
Step 3: notice that

[u] q B r q,∞ (R N ,R d ) = sup h∈R N \{0} ˆRN |u(x + h) -u(x)| q |h| rq dx = sup ε∈(0,∞) sup |h|=ε ˆRN |u(x + h) -u(x)| q ε rq dx = sup ε∈(0,∞) sup h∈S N -1 ˆRN |u(x + εh) -u(x)| q ε rq dx . (4.23) By (4.22) there exists δ ∈ (0, ∞) such that sup ε∈(0,δ) sup h∈S N -1 ˆRN |u(x + εh) -u(x)| q ε rq dx < ∞. (4.24)
Therefore, by (4.23) and ( 4.24) we get

[u] q B r q,∞ (R N ,R d ) ≤ sup ε∈(0,δ) sup h∈S N -1 ˆRN |u(x + εh) -u(x)| q ε rq dx + sup ε∈[δ,∞) sup h∈S N -1 ˆRN |u(x + εh) -u(x)| q ε rq dx ≤ sup ε∈(0,δ) sup h∈S N -1 ˆRN |u(x + εh) -u(x)| q ε rq dx + 2 q δ rq u q L q (R N ,R d ) < ∞. (4.25) Thus, u ∈ B r q,∞ (R N , R d ).
It completes the proof of (4.15). We will derive the local case (4.16) from the global one (4.15). Assume now that u ∈ B r q,∞ loc (Ω, R d ). Let K ⊂ Ω be a compact set and let

Ω 0 ⊂⊂ Ω be an open set such that K ⊂ Ω 0 . Let g ∈ B r q,∞ (R N , R d ) be such that u = g L N -almost everywhere in Ω 0 . We have for ε ∈ (0, ∞) such that K + B ε (0) ⊂ Ω 0 ˆK 1 ε N ˆBε(x) |u(x) -u(y)| q |x -y| rq dydx = ˆK 1 ε N ˆBε(x) |g(x) -g(y)| q |x -y| rq dydx ≤ ˆRN 1 ε N ˆBε(x) |g(x) -g(y)| q |x -y| rq dydx. (4.26)
By (4.15), we get lim sup 2 which is constant 1 on K and constant 0 outside Ω 0 . We have for g := uf and ε

ε→0 + ˆK 1 ε N ˆBε(x) |u(x) -u(y)| q |x -y| rq dydx ≤ lim sup ε→0 + ˆRN 1 ε N ˆBε(x) |g(x) -g(y)| q |x -y| rq dydx < ∞. (4.27) By item 3 of Proposition 4.1 we conclude that u ∈ B r,q loc (Ω, R d ). Assume that u ∈ B r,q loc (Ω, R d ). Let K ⊂ Ω be a compact set and let Ω 0 ⊂⊂ Ω 1 ⊂⊂ Ω be open sets such that K ⊂ Ω 0 . Let f ∈ C 0,r c (R N )
∈ (0, ∞) such that R N \ Ω 1 + B ε (0) ⊂ R N \ Ω 0 ˆRN 1 ε N ˆBε(x) |g(x) -g(y)| q |x -y| rq dydx = ˆΩ1 1 ε N ˆBε(x) |u(x)f (x) -u(x)f (y) + u(x)f (y) -u(y)f (y)| q |x -y| rq dydx ≤ 2 q-1 ˆΩ1 |u(x)| q ε N ˆBε(x) |f (x) -f (y)| q |x -y| rq dydx + 2 q-1 f q L ∞ (R N ) ˆΩ1 1 ε N ˆBε(x) |u(x) -u(y)| q |x -y| rq dydx ≤ 2 q-1 C q L N (B 1 (0)) ˆΩ1 |u(x)| q dx + 2 q-1 f q L ∞ (R N ) ˆΩ1 1 ε N ˆBε(x) |u(x) -u(y)| q |x -y| rq dydx, (4.28)
where

C is a number such that |f (x) -f (y)| ≤ C|x -y| r for x, y ∈ R N . Therefore, lim sup ε→0 + ˆRN 1 ε N ˆBε(x) |g(x) -g(y)| q |x -y| rq dydx < ∞, (4.29)
and by (4.15) we conclude that g

∈ B r q,∞ (R N , R d ). Thus, u ∈ B r q,∞ loc (Ω, R d ).

Kernels

In this section, we analyse the concept of a kernel (see Definition 1.2). Additionally, we discuss specific kernels, namely the logarithmic and trivial kernels (see Definitions 5.2 and 5.3), and establish their properties.

Definition 5.1. (Compact Support Property) Let a ∈ (0, ∞] and ρ ε : (0, ∞) → [0, ∞), ε ∈ (0, a)
, be a family of functions. We say that the family {ρ ε } ε∈(0,a) has the compact support property if for every r > 0 there exists δ r > 0 such that supp(ρ ε ) ⊂ B r (0) for every ε ∈ (0, δ r ).

Note that, if the functions {ρ ε } ε∈(0,a) are L 1 -measurable, the compact support property implies the decreasing support property (see Definition 1.1).

Definition 5.2. (Logarithmic Kernel)

For every ε ∈ (0, 1/e) and ω ∈ (0, 1) let us define a function

ρ ε,ω (r) := 1 H N -1 (S N -1 ) (| ln ε| -| ln R ε,ω |) 1 r N χ [ε,Rε,ω) (r), ρ ε,ω : (0, ∞) → [0, ∞), (5.1)
where R ε,ω := 1 | ln ε| ω , and χ [ε,Rε,ω) is the characteristic function of the interval [ε, R ε,ω ). We call the family of functions {ρ ε,ω } ε∈(0,1/e) the N -dimensional logarithmic kernel, or just logarithmic kernel.

Remark 5.1. (Comments about the Logarithmic Kernel) 1. Note that for every ε, ω ∈ (0, 1) we have ε

< R ε,ω : ε < R ε,ω if and only if ε < 1 ln( 1 ε ) ω if and only if ε 1/ω ln 1 ε < 1.
The last inequality holds since ln(z) < z for every z ∈ (0, ∞). 2. Note that for ε ∈ (0, 1), ln R ε,ω = -ω ln ln 1 ε , and for ε

∈ (0, 1/e), | ln R ε,ω | = ω ln ln 1 ε , so | ln ε| -| ln R ε,ω | = ln 1 ε -ω ln ln 1 ε = ln 1 ε + ln 1 (ln( 1 ε )) ω = ln 1 ε(ln( 1 ε )) ω > 0. The last inequality holds since ε ln 1 ε ω < 1.
3. By L'hopital's rule we have lim x→∞ ln(ln(x)) ln(x) = 0, so we get by definition of R ε,ω For ω ∈ (0, 1), the logarithmic kernel {ρ ε,ω } ε∈(0,1/e) has the following properties: 1. The logarithmic kernel is a kernel that also possesses the compact support property; 2.

lim ε→0 + | ln R ε,ω | | ln ε| = lim ε→0 + ω ln ln 1 ε ln 1 ε = 0. ( 5
lim ε→0 + ε α ´RN ρε,ω(|z|) |z| α dz = 0, ∀α ∈ (0, ∞). Proof. 1. It is easy to see that for every ε ∈ (0, 1/e), ω ∈ (0, 1), the function ρ ε,ω is L 1 -measurable. By polar coordinates ˆRN 1 |z| N χ [ε,Rε,ω) (|z|)dz = ˆBRε,ω (0)\Bε(0) 1 |z| N dz = ˆRε,ω ε ˆ∂Br(0) 1 |z| N dH N -1 (z) dr = ˆRε,ω ε 1 r N r N -1 H N -1 S N -1 dr = H N -1 S N -1 (ln R ε,ω -ln ε) = H N -1 S N -1 (| ln ε| -| ln R ε,ω |) .
(5.3)

Note that, since ε ∈ (0, 1/e), then ε, R ε,ω < 1, and therefore

-ln ε = | ln ε| and ln R ε,ω = -| ln R ε,ω |, so ln R ε,ω -ln ε = | ln ε| -| ln R ε,ω |. Thus, ´RN ρ ε,ω (|z|)dz = 1.
The logarithmic kernel satisfies the compact support property: for every r ∈ (0, ∞) let δ r := e

-1 r 1/ω . Note that if ε ∈ (0, δ r ), then R ε,ω < r, so supp(ρ ε,ω ) ⊂ B Rε,ω (0) ⊂ B r (0), where R ε,ω := 1 | ln ε| ω . 2. By polar coordinates ˆRN ρ ε,ω (|z|) |z| α dz = 1 H N -1 (S N -1 ) (| ln ε| -| ln R ε,ω |) ˆBRε,ω (0)\Bε(0) 1 |z| N +α dz = 1 H N -1 (S N -1 ) (| ln ε| -| ln R ε,ω |) ˆRε,ω ε 1 r N +α r N -1 H N -1 S N -1 dr = 1 (| ln ε| -| ln R ε,ω |) 1 α 1 ε α - 1 R α ε,ω = 1 αε α 1 -ε α R α ε,ω | ln ε| -| ln R ε,ω | = 1 αε α 1 -ε α | ln ε| αω | ln ε| -ω ln | ln ε| .
(5.4)

Hence,

lim ε→0 + ε α ˆRN ρ ε,ω (|z|) |z| α dz = 1 α lim ε→0 + 1 -ε α | ln ε| αω | ln ε| -ω ln | ln ε| = 1 α lim ε→0 + 1 | ln ε| -(ε| ln ε| ω ) α 1 | ln ε| 1 -ω ln | ln ε| | ln ε| = 0 1 = 0.
(5.5)

   1 ε N L N (B 1 (0)) if 0 < r < ε 0 if r ≥ ε , ε ∈ (0, ∞).
(5.6)

Remark 5.2. Notice that the trivial kernel is a kernel. Moreover, it satisfies the compact support property: for every r ∈ (0, ∞), let δ r := r.

Thus, if ε ∈ (0, δ r ), then supp(ρ ε ) ⊂ B ε (0) ⊂ B r (0).
Definition 5.4. (σ-Approximating Kernels) For every number σ ∈ (0, ∞), the N -dimensional σ-approximating kernel is defined to be

ρ σ ε (r) := 1 2σH N -1 (S N -1 )r N -1 χ [ε-σ,ε+σ] (r), ρ σ ε : (0, ∞) → [0, ∞).
(5.7)

Remark 5.3. (σ-Approximating Kernels Give us Kernels) Note that σ-approximating kernels are not kernels because they lack the decreasing support property (see Definition 1.1). However, if we select a number σ ε ∈ (0, ε) for every ε ∈ (0, ∞), then the family {ρ σε ε } ε∈(0,∞) possesses the compact support property, and in particular, it satisfies the decreasing support property. By employing polar coordinates, we find that ´RN ρ σ ε (|z|)dz = 1 for every choice of ε and σ in (0, ∞) with σ < ε. Therefore, {ρ σε ε } ε∈(0,∞) is a kernel, as defined in Definition 1.2.

Variations and Besov constants

In this section, we introduce the notion of (r, q)-Variation (see Definition 6.1). We prove that (r, q)variations control Besov Constants (see Lemma 6.1). Furthermore, we demonstrate that (r, q)variation can be represented as a Besov constant (see Corollary 6.1). Additionally, we establish the continuity of Variations and Besov constants with respect to convergence in Besov Space (see Lemma 6.3). Definition 6.1. ((r, q)-Variation and Directional (r, q)-Variation)

Let r, q ∈ (0, ∞), and u : R N → R d be an L N -measurable function. Suppose E ⊂ R N is an L N -measurable set, and let n ∈ S N -1 be a direction. Then, the (r, q) upper variation of u in E in the direction n is defined by

(r, q) -V (u, E, n) := lim sup ε→0 + ˆE χ E (x + εn) |u(x + εn) -u(x)| q ε rq dx. (6.1)
Similarly, replacing the lim sup by the lim inf, we define the (r, q) lower variation of u in E in the direction n and denote it by (r, q)-V (u, E, n). If the limit exists, we denote it by (r, q)-V (u, E, n), and we call it the (r, q) variation of u in E in the direction n.

The (r, q) upper variation of u in E is defined by

(r, q) -V (u, E) := lim sup ε→0 + ˆSN-1 ˆE χ E (x + εn) |u(x + εn) -u(x)| q ε rq dxdH N -1 (n). (6.2)
Similarly, replacing the lim sup by the lim inf, we define the (r, q) lower variation of u in E and denote it by (r, q) -V (u, E). If the limit exists, we denote it by (r, q) -V (u, E), and we call it the (r, q) variation of u in E. Definition 6.2. (Besov Constants) Let r, q ∈ (0, ∞), and u : R N → R d be an L N -measurable function. Suppose E ⊂ R N is an L N -measurable set, and let {ρ ε } ε∈(0,a) be a kernel for some a ∈ (0, ∞]. The upper infinitesimal (r, q) Besov constant of u in E with respect to the kernel ρ ε is defined as the quantity:

lim sup ε→0 + ˆE ˆE ρ ε (|x -y|) |u(x) -u(y)| q |x -y| rq dydx. (6.3)
Similarly, replacing the lim sup by the lim inf, we define the lower infinitesimal (r, q) Besov constant of u in E with respect to the kernel ρ ε . If the limit exists, we refer to it as the infinitesimal (r, q) Besov constant of u in E with respect to the kernel ρ ε .

Remark 6.1. (The Upper Infinitesimal B r,q -seminorm is a Besov Constant) Note that if we select the trivial kernel as defined in Definition 5.3 in (6.3), multiply the result by L N (B 1 (0)), and then take the result to the power of 1 q , we obtain the upper infinitesimal B r,q -seminorm as defined in 4.1. Remark 6.2. (Variations of W 1,q , BV and B r q,∞ ) From the BBM formula, for an open and bounded set Ω ⊂ R N with a Lipschitz boundary, where 1 < q < ∞ and u ∈ W 1,q (Ω), we have

(1, q) -V (u, Ω) = C q,N ∇u q L q (Ω) ; (6.4) for u ∈ BV (Ω), we have (1, 1) -V (u, Ω) = C 1,N Du (Ω), (6.5)
where C q,N := ´SN-1 |z 1 | q dH N -1 (n) for every q ≥ 1. For proof of this result see [START_REF] Poliakovsky | Some remarks on a formula for Sobolev norms due to Brezis, Van Schaftingen and Yung[END_REF].

For r ∈ (0, 1) and q ∈ [1, ∞), we observe from Sandwich Lemma 6.1 and Theorem 4.1 that the finiteness of the upper variation (r, q)

-V (u, R N ) of u together with u ∈ L q (R N , R d ) is equivalent to u ∈ B r q,∞ (R N , R d ).
Lemma 6.1. (The Sandwich Lemma with Variations and Besov Constants Included) Let E ⊂ R N be an L N -measurable set and u : E → R d be an L N -measurable function. Let a ∈ (0, ∞] and let ρ ε : (0, ∞) → [0, ∞), ε ∈ (0, a), be a kernel, and α, q ∈ (0, ∞). Assume that at least one of the following three assumptions holds:

1. ess sup ε∈(0,∞) ´SN-1 ´E χ E (x + εn) |u(x+εn)-u(x)| q ε α dxdH N -1 (n) < ∞; 2. u ∈ L q (R N , R d );
3. The kernel {ρ ε } ε∈(0,a) has the compact support property as defined in Definition 5.1.

Then,

lim inf ε→0 + S N -1 ˆE χ E (x + εn) |u(x + εn) -u(x)| q ε α dxdH N -1 (n) ≤ ess liminf ε→0 + S N -1 ˆE χ E (x + εn) |u(x + εn) -u(x)| q ε α dxdH N -1 (n) ≤ lim inf ε→0 + ˆE ˆE ρ ε (|x -y|) |u(x) -u(y)| q |x -y| α dydx ≤ lim sup ε→0 + ˆE ˆE ρ ε (|x -y|) |u(x) -u(y)| q |x -y| α dydx ≤ ess limsup ε→0 + S N -1 ˆE χ E (x + εn) |u(x + εn) -u(x)| q ε α dxdH N -1 (n) ≤ lim sup ε→0 + S N -1 ˆE χ E (x + εn) |u(x + εn) -u(x)| q ε α dxdH N -1 (n). (6.6)
In particular, for r ∈ (0, ∞) and α = rq, we get (6.6) for (r, q) variations and Besov constants.

Proof. Using polar coordinates we get

ˆE ˆE ρ ε (|x -y|) |u(x) -u(y)| q |x -y| α dydx = ˆE ˆRN χ E (y)ρ ε (|x -y|) |u(x) -u(y)| q |x -y| α dy dx = ˆE ˆRN χ E (x + z)ρ ε (|z|) |u(x) -u(x + z)| q |z| α dz dx = ˆRN ˆE χ E (x + z)ρ ε (|z|) |u(x) -u(x + z)| q |z| α dx dz = ˆ∞ 0 ˆ∂Bt(0) ˆE χ E (x + z)ρ ε (|z|) |u(x) -u(x + z)| q |z| α dx dH N -1 (z)dt = ˆ∞ 0 ˆSN-1 t N -1 ˆE χ E (x + tn)ρ ε (t) |u(x) -u(x + tn)| q t α dx dH N -1 (n)dt = ˆ∞ 0 t N -1 ρ ε (t) ˆSN-1 ˆE χ E (x + tn) |u(x) -u(x + tn)| q t α dxdH N -1 (n) dt = ˆδ 0 t N -1 ρ ε (t)V (t)dt + ˆ∞ δ t N -1 ρ ε (t)V (t)dt. (6.7)
In formula (6.7) we denote

V (t) := ˆSN-1 ˆE χ E (x + tn) |u(x) -u(x + tn)| q t α dxdH N -1 (n). (6.8)
By polar coordinates we see that

ˆRN ρ ε (|z|)dz = 1 =⇒ 1 H N -1 (S N -1 ) = ˆ∞ 0 t N -1 ρ ε (t)dt. (6.9)
Since {ρ ε } ε∈(0,a) is a kernel, then it has the decreasing support property (see Definition 1.1). Therefore, for every δ > 0, we get lim ε→0 + ´∞ δ t N -1 ρ ε (t)dt = 0, and by (6.9) we obtain

1 H N -1 (S N -1 ) = lim ε→0 + ˆδ 0 t N -1 ρ ε (t)dt + ˆ∞ δ t N -1 ρ ε (t)dt = lim ε→0 + ˆδ 0 t N -1 ρ ε (t)dt. (6.10)
By equation (6.7) we obtain ess sup t∈(0,δ)

V (t) ˆδ 0 t N -1 ρ ε (t)dt + ˆ∞ δ t N -1 ρ ε (t)V (t)dt ≥ ˆE ˆE ρ ε (|x -y|) |u(x) -u(y)| q |x -y| α dydx ≥ ess inf t∈(0,δ) V (t) ˆδ 0 t N -1 ρ ε (t)dt + ˆ∞ δ t N -1 ρ ε (t)V (t)dt. (6.11) If ess sup t∈(0,∞) V (t) < ∞, then we get lim ε→0 + ´∞ δ t N -1 ρ ε (t)V (t)dt = 0. If u ∈ L q (R N , R d ), then sup t∈(δ,∞) V (t) ≤ 2 q H N -1 (S N -1 ) δ α u q L q (R N ,R d ) < ∞. (6.12)
So we get again that lim ε→0 + ´∞ δ t N -1 ρ ε (t)V (t)dt = 0. Therefore, in both cases, we obtain (6.6) by first taking the liminf(limsup) as ε → 0 + and then the limit as δ → 0 + in inequality (6.11).

In case {ρ ε } ε∈(0,a) has the compact support property, for r > 0 there exists δ r such that for every ε ∈ (0, δ r ) we obtain supp(ρ ε ) ⊂ (0, r), and by (6.7) we get

ˆE ˆE ρ ε (|x -y|) |u(x) -u(y)| q |x -y| α dydx = ˆr 0 t N -1 ρ ε (t)V (t)dt ≤ ˆr 0 t N -1 ρ ε (t)dt ess sup t∈(0,r) V (t) = 1 H N -1 (S N -1 )
ess sup t∈(0,r) V (t), (6.13)

ˆE ˆE ρ ε (|x -y|) |u(x) -u(y)| q |x -y| α dydx = ˆr 0 t N -1 ρ ε (t)V (t)dt ≥ ˆr 0 t N -1 ρ ε (t)dt ess inf t∈(0,r) V (t) = 1 H N -1 (S N -1 )
ess inf t∈(0,r)

V (t). (6.14)

Taking the upper limit as ε → 0 + and then the limit as r → 0 + in (6.13), we obtain the third inequality in (6.6). Similarly, taking the lower limit as ε → 0 + and then the limit as r → 0 + in (6.14), we obtain the second inequality of (6.6).

Lemma 6.2. (Variations and Essential Variations) Let q, α ∈ (0, ∞), and let

u ∈ L q (R N , R d ). Assume that E ⊂ R N is a Lebesgue measurable set such that for every v ∈ R N we have L N (E ∩ (∂E + v)) = 0. Then, lim inf ε→0 + S N -1 ˆE χ E (x + εn) |u(x + εn) -u(x)| q ε α dxdH N -1 (n) = ess liminf ε→0 + S N -1 ˆE χ E (x + εn) |u(x + εn) -u(x)| q ε α dxdH N -1 (n) (6.15) and lim sup ε→0 + S N -1 ˆE χ E (x + εn) |u(x + εn) -u(x)| q ε α dxdH N -1 (n) = ess limsup ε→0 + S N -1 ˆE χ E (x + εn) |u(x + εn) -u(x)| q ε α dxdH N -1 (n). (6.16)
In particular, for r ∈ (0, ∞) and α = rq, we get the result for (r, q) essential variations and Besov constants.

Proof. Let us denote

V (t) := ˆSN-1 ˆE χ E (x + tn) |u(x) -u(x + tn)| q t α dxdH N -1 (n), (6.17)
and

F (t) := ˆSN-1 ˆE χ E (x + tn)|u(x + tn) -u(x)| q dxdH N -1 (n), F : R → R. (6.18)
Note that F (t) = t α V (t). We prove the continuity of F in R, and consequently establish the continuity of V in (0, ∞). Thus, every point in (0, ∞) is a Lebesgue point of V . Therefore, by Proposition 10.1 and Corollary 10.1, we conclude that

lim inf ε→0 + V (ε) = ess liminf ε→0 + V (ε), lim sup ε→0 + V (ε) = ess limsup ε→0 + V (ε). (6.19)
Let t 0 ∈ R be any number, and let us show that F is continuous at t 0 . Note first that

χ E (x + tn)|u(x + tn) -u(x)| q -χ E (x + t 0 n)|u(x + t 0 n) -u(x)| q ≤ χ E (x + tn) |u(x + tn) -u(x)| q -|u(x + t 0 n) -u(x)| q + |χ E (x + tn) -χ E (x + t 0 n)| |u(x + t 0 n) -u(x)| q ≤ |u(x + tn) -u(x)| q -|u(x + t 0 n) -u(x)| q + χ (E-tn)∆(E-t 0 n) (x)|u(x + t 0 n) -u(x)| q . (6.20)
Therefore, by (6.20)

|F (t) -F (t 0 )| = ˆSN-1 ˆE χ E (x + tn)|u(x + tn) -u(x)| q dxdH N -1 (n) - ˆSN-1 ˆE χ E (x + t 0 n)|u(x + t 0 n) -u(x)| q dxdH N -1 (n) ≤ ˆSN-1 ˆE χ E (x + tn)|u(x + tn) -u(x)| q -χ E (x + t 0 n)|u(x + t 0 n) -u(x)| q dxdH N -1 (n) ≤ ˆSN-1 ˆE |u(x + tn) -u(x)| q -|u(x + t 0 n) -u(x)| q dxdH N -1 (n) + ˆSN-1 ˆE χ (E-tn)∆(E-t 0 n) (x)|u(x + t 0 n) -u(x)| q dxdH N -1 (n). (6.21)
By Dominated Convergence Theorem and continuity of translations in L q we obtain lim

t→t 0 ˆSN-1 ˆE |u(x + tn) -u(x)| q -|u(x + t 0 n) -u(x)| q dxdH N -1 (n) = ˆSN-1 lim t→t 0 ˆE |u(x + tn) -u(x)| q -|u(x + t 0 n) -u(x)| q dxdH N -1 (n) = 0. (6.22)
We utilized the continuity of translations as follows: since u(• + tn) converges to u(• + t 0 n) in L q as t tends to t 0 , then u(• + tn) -u converges to u(• + t 0 n) -u in L q as t tends to t 0 . Consequently, |u(• + tn) -u| converges to |u(• + t 0 n) -u| in L q as t tends to t 0 , and thus |u(• + tn) -u| q converges to |u(• + t 0 n) -u| q in L 1 as t tends to t 0 Let us define for every ε ∈ (0, ∞) the ε-neighbourhood of ∂E -t 0 n by

E ε := x ∈ R N dist(x, ∂E -t 0 n) ≤ ε . (6.23) Note that ∩ ε∈(0,∞) E ε = ∂E -t 0 n. Therefore, for every ε ∈ (0, ∞), there exists R(ε) ∈ (0, ∞) such that for every t ∈ R with |t -t 0 | < R(ε), we have (E -tn)∆(E -t 0 n) ⊂ E ε and so χ (E-tn)∆(E-t 0 n) (x) ≤ χ Eε (x) for every x ∈ R N . Therefore, lim sup t→t 0 ˆSN-1 ˆE χ (E-tn)∆(E-t 0 n) (x)|u(x + t 0 n) -u(x)| q dxdH N -1 (n) ≤ ˆSN-1 ˆE χ Eε (x)|u(x + t 0 n) -u(x)| q dxdH N -1 (n). (6.24)
Therefore, taking the limit as ε → 0 + in (6.24), we get by Dominated Convergence Theorem and the assumption about

E lim sup t→t 0 ˆSN-1 ˆE χ (E-tn)∆(E-t 0 n) (x)|u(x + t 0 n) -u(x)| q dxdH N -1 (n) ≤ ˆSN-1 ˆE∩(∩ ε>0 Eε) |u(x + t 0 n) -u(x)| q dxdH N -1 (n) = ˆSN-1 ˆE∩(∂E-t 0 n) |u(x + t 0 n) -u(x)| q dxdH N -1 (n) = 0. (6.25)
Using (6.21), (6.22), and (6.25), we conclude the continuity of F at t 0 ∈ R. It completes the proof.

Proposition 6.1. (Besov Constants and Essential Variations) Let q, α ∈ (0, ∞), and let u ∈ L q (R N , R d ). Assume that E ⊂ R N is a Lebesgue measurable set. Then, there exists a kernel {ρ ε } ε∈(0,∞) such that (6.26)

lim inf ε→0 + ˆE ˆE ρ ε (|x -y|) |u(x) -u(y)| q |x -y| α dydx = ess liminf ε→0 + ˆE ˆE ρ ε (|x -y|) |u(x) -u(y)| q |x -y| α dydx = ess liminf ε→0 + S N -1 ˆE χ E (x + εn) |u(x + εn) -u(x)| q ε α dxdH N -1 (n),
and lim sup ε→0 + ˆE ˆE ρ ε (|x -y|) |u(x) -u(y)| q |x -y| α dydx = ess limsup ε→0 + ˆE ˆE ρ ε (|x -y|) |u(x) -u(y)| q |x -y| α dydx = ess limsup ε→0 + S N -1 ˆE χ E (x + εn) |u(x + εn) -u(x)| q ε α dxdH N -1 (n). (6.27)
In particular, for r ∈ (0, ∞) and α = rq, we get the result for (r, q) variations and Besov constants.

Proof. For every ε ∈ (0, ∞) and σ ∈ (0, ε) let ρ σ ε as in (5.7). By (6.7), we get

ˆE ˆE ρ σ ε (|x -y|) |u(x) -u(y)| q |x -y| α dydx = ˆ∞ 0 t N -1 ρ σ ε (t) ˆSN-1 ˆE χ E (x + tn) |u(x) -u(x + tn)| q t α dxdH N -1 (n) dt = 1 2σH N -1 (S N -1 ) ˆε+σ ε-σ V (t)dt, (6.28) where V (t) := ˆSN-1 ˆE χ E (x + tn) |u(x) -u(x + tn)| q t α dxdH N -1 (n). (6.29) Since u ∈ L q (R N , R d ), the function V is locally integrable in (0, ∞), so almost every point in (0, ∞) is a Lebesgue point of V . Let ε ∈ (0, ∞) be a Lebesgue point of V . There exists 0 < σ ε < ε such that 1 2σε ´ε+σε ε-σε V (t)dt -V (ε) < ε. Therefore, ˆE ˆE ρ σε ε (|x -y|) |u(x) -u(y)| q |x -y| α dydx = 1 2σ ε H N -1 (S N -1 ) ˆε+σε ε-σε V (t)dt = 1 H N -1 (S N -1 ) V (ε) + 1 H N -1 (S N -1 ) 1 2σ ε ˆε+σε ε-σε V (t)dt -V (ε) . (6.30)
By taking the lower limit in (6.30) as ε → 0 + , with ε being a Lebesgue point of V , we derive the second equation in (6.26) using Proposition 10.1 and Corollary 10.1. Similarly, by taking the upper limit in (6.30) as ε → 0 + , with ε being a Lebesgue point of V , we obtain the second equation in (6.27). Note that {ρ σε ε } ε∈(0,∞) is a kernel as was explained in Remark 5.3. By the definition of ess liminf and the second equation of (6.26) we obtain lim inf

ε→0 + ˆE ˆE ρ σε ε (|x -y|) |u(x) -u(y)| q |x -y| α dydx ≤ ess liminf ε→0 + 1 H N -1 (S N -1 )
V (ε). (6.31) By Lemma 6.1, we get lim inf

ε→0 + ˆE ˆE ρ σε ε (|x -y|) |u(x) -u(y)| q |x -y| α dydx ≥ ess liminf ε→0 + 1 H N -1 (S N -1 ) V (ε). (6.32)
We get the first equation of (6.26) by (6.31) and (6.32). We get the first equation of (6.27) in a similar way.

Corollary 6.1. (Representability of Variations as Besov Constants) Let q, α ∈ (0, ∞), and let u ∈ L q (R N , R d ). Assume that E ⊂ R N is a Lebesgue measurable set such that for every v ∈ R N we have L N (E ∩ (∂E + v)) = 0. Then, there exists a kernel {ρ ε } ε∈(0,∞) such that (6.33)

lim inf ε→0 + ˆE ˆE ρ ε (|x -y|) |u(x) -u(y)| q |x -y| α dydx = lim inf ε→0 + S N -1 ˆE χ E (x + εn) |u(x + εn) -u(x)| q ε α dxdH N -1 (n),
and lim sup ε→0 + ˆE ˆE ρ ε (|x -y|) |u(x) -u(y)| q |x -y| α dydx = lim sup ε→0 + S N -1 ˆE χ E (x + εn) |u(x + εn) -u(x)| q ε α dxdH N -1 (n). (6.34)
In particular, for r ∈ (0, ∞) and α = rq, we get the result for (r, q) variations and Besov constants.

Proof. Formulas (6.33) and (6.34) follow immediately from Lemma 6.2 and Proposition 6.1.

Lemma 6.3. (Continuity of Variations and Besov Constants in Besov Spaces

B r q,∞ ) Let q ∈ [1, ∞), r ∈ (0, 1), and E ⊂ R N be an L N -measurable set. Consider a sequence {u k } ∞ k=1 ⊂ B r q,∞ (R N , R d ) such that u k converges to u in B r q,∞ (R N , R d ).
Then, 1. for every n ∈ R N , we have

lim k→∞ lim sup ε→0 + ˆE χ E (x + εn) |u k (x + εn) -u k (x)| q ε rq dx = lim sup ε→0 + ˆE χ E (x + εn) |u(x + εn) -u(x)| q ε rq dx, (6.35)
and a similar result also holds when replacing the lim sup with the lim inf.

It follows that

lim k→∞ lim sup ε→0 + ˆSN-1 ˆE χ E (x + εn) |u k (x + εn) -u k (x)| q ε rq dxdH N -1 (n) = lim sup ε→0 + ˆSN-1 ˆE χ E (x + εn) |u(x + εn) -u(x)| q ε rq dxdH N -1 (n), (6.36)
and a similar result also holds when replacing the lim sup with the lim inf.

3. Let a ∈ (0, ∞]. For every kernel ρ ε : (0, ∞) → [0, ∞), ε ∈ (0, a), we get

lim k→∞ lim sup ε→0 + ˆE ˆE ρ ε (|x -y|) |u k (x) -u k (y)| q |x -y| rq dydx = lim sup ε→0 + ˆE ˆE ρ ε (|x -y|) |u(x) -u(y)| q |x -y| rq dydx, (6.37)
and a similar result also holds when replacing the lim sup with the lim inf.

Proof. Let us prove assertion 1. Note that if n = 0, then equation (6.35) in both lim inf and lim sup cases trivially holds. Assume n = 0. Let us denote

I ε (u k , n)(x) := χ E (x + εn) |u k (x + εn) -u k (x)| q ε rq , (6.38)
and

I ε (u, n)(x) := χ E (x + εn) |u(x + εn) -u(x)| q ε rq . (6.39)
By Lemma 10.6, Minkowski's inequality, and the definition of the Besov seminorm [•] B r q,∞ , we obtain lim sup

ε→0 + ˆE I ε (u k , n)(x)dx 1 q -lim sup ε→0 + ˆE I ε (u, n)(x)dx 1 q ≤ lim sup ε→0 + ˆE I ε (u k , n)(x)dx 1 q -ˆE I ε (u, n)(x)dx 1 q = lim sup ε→0 + ˆE (I ε (u k , n)(x)) 1/q q dx 1 q - ˆE (I ε (u, n)(x)) 1/q q dx 1 q ≤ lim sup ε→0 + ˆE (I ε (u k , n)(x)) 1/q -(I ε (u, n)(x)) 1/q q dx 1 q = lim sup ε→0 + ˆE χ E (x + εn) |u k (x + εn) -u k (x)| ε r -χ E (x + εn) |u(x + εn) -u(x)| ε r q dx 1 q = lim sup ε→0 + ˆE χ E (x + εn) ||u k (x + εn) -u k (x)| -|u(x + εn) -u(x)|| q ε rq dx 1 q ≤ lim sup ε→0 + ˆE χ E (x + εn) |(u k -u)(x + εn) -(u k -u)(x)| q ε rq dx 1 q ≤ |n| r [u k -u] B r q,∞ (R N ,R d ) .
(6.40)

We take the limit as k → ∞ on both sides of (6.40) to obtain (6.35). Similarly, we get lim inf

ε→0 + ˆE I ε (u k , n)(x)dx 1 q -lim inf ε→0 + ˆE I ε (u, n)(x)dxdx 1 q ≤ |n| r [u k -u] B r q,∞ (R N ,R d ) . (6.41)
Assertion 2 of the Lemma is proven in the same way. By replacing the integral ´E(•)dx with the integral ´SN-1 ´E(•)dxdH N -1 (n) in (6.40) throughout, we obtain lim sup

ε→0 + ˆSN-1 ˆE I ε (u k , n)(x)dxdH N -1 (n) 1 q -lim sup ε→0 + ˆSN-1 ˆE I ε (u, n)(x)dxdH N -1 (n) 1 q ≤ H N -1 (S N -1 ) 1/q |n| r [u k -u] B r q,∞ (R N ,R d ) , (6.42) and lim inf ε→0 + ˆSN-1 ˆE I ε (u k , n)(x)dxdH N -1 (n) 1 q -lim inf ε→0 + ˆSN-1 ˆE I ε (u, n)(x)dxdH N -1 (n) 1 q ≤ H N -1 (S N -1 ) 1/q |n| r [u k -u] B r q,∞ (R N ,R d ) . (6.43)
Taking the limit as k tends to infinity in inequalities (6.42) and (6.43), we get formula (6.36) in both cases lim inf and lim sup. We prove assertion 3. Let us denote

B ε,u k (x, y) := ρ ε (|x -y|) |u k (x) -u k (y)| q |x -y| rq , B ε,u (x, y) := ρ ε (|x -y|) |u(x) -u(y)| q |x -y| rq . (6.44)
As in (6.40), for n = 0, by Lemma 10.6, Minkowski's inequality, Sandwich Lemma with α = rq (Lemma 6.1) and the definition of the Besov seminorm [•] B r q,∞ , we obtain lim sup

ε→0 + ˆE ˆE B ε,u k (x, y)dydx 1 q -lim sup ε→0 + ˆE ˆE B ε,u (x, y)dydx 1 q ≤ lim sup ε→0 + ˆE ˆE (B ε,u k (x, y)) 1 q q dydx 1 q - ˆE ˆE (B ε,u (x, y)) 1 q q dydx 1 q ≤ lim sup ε→0 + ˆE ˆE (B ε,u k (x, y)) 1 q -(B ε,u (x, y)) 1 q q dydx 1 q = lim sup ε→0 + ˆE ˆE (ρ ε (|x -y|)) 1 q |u k (x) -u k (y)| |x -y| r -(ρ ε (|x -y|)) 1 q |u(x) -u(y)| |x -y| r q dydx 1 q = lim sup ε→0 + ˆE ˆE ρ ε (|x -y|) |(u k -u)(x) -(u k -u)(y)| q |x -y| rq dydx 1 q ≤ lim sup ε→0 + S N -1 ˆE χ E (x + εn) |(u k -u)(x + εn) -(u k -u)(x)| q ε rq dxdH N -1 (n) 1 q . ≤ |n| r [u k -u] B r q,∞ (R N ,R d
) . (6.45) Taking the limit as k → ∞ in the inequality (6.45), we obtain (6.37). We get this result for lim inf in a similar way. Remark 6.3. In Lemma 6.3, we can utilize Corollary 6.1 to derive assertion 2 from assertion 3 in Lemma 6.3, provided that we limit ourselves to sets E satisfying the conditions of Corollary 6.1.

Equivalence Between Gagliardo Constants and Besov

Constants

In this section and the next one, we will frequently use the term 'equivalence' to denote the relationship between different quantities. For instance, we may refer to the equivalence between Gagliardo constants and Besov constants of a function u to indicate that they are equal up to a scalar multiplication that does not depend on u.

In this section, we demonstrate that the upper and lower variations control Gagliardo constants (refer to Theorem 7.1). Furthermore, we establish that, under certain conditions, Gagliardo constants and Besov constants are equivalent (see Theorem 7.2). As a special case, we derive the equivalence between Gagliardo constants and infinitesimal B r,q -seminorms (refer to Corollary 7.4).

q ∈ [1, ∞), r ∈ (0, 1) and u ∈ B r q,∞ (R N , R d ). Let ρ ε : (0, ∞) → [0, ∞), ε ∈ (0, a)
, be a kernel for some a ∈ (0, ∞). Then

lim sup ε→0 + ˆRN ˆRN ρ ε (|x -y|) |u(x) -u(y)| q |x -y| rq dydx ≤ [u] q B r q,∞ (R N ,R d ) < ∞. (7.1)
Proof. By Lemma 6.1 with α = rq and E = R N , Definition 2.1 (definition of Besov seminorm) and the assumption u

∈ B r q,∞ (R N , R d ) we get lim sup ε→0 + ˆRN ˆRN ρ ε (|x -y|) |u(x) -u(y)| q |x -y| rq dydx ≤ 1 H N -1 (S N -1 ) lim sup ε→0 + ˆSN-1 ˆRN |u(x + εn) -u(x)| q ε rq dxdH N -1 (n) ≤ [u] q B r q,∞ (R N ,R d ) < ∞. (7.2)
Lemma 7.1. (Approximation of Gagliardo Constants by Besov Constants through the Logarithmic Kernel) Let 1 ≤ q < ∞, r ∈ (0, 1). Let ω ∈ (0, 1) be such that rq < 1/ω. Let u ∈ B r q,∞ (R N , R d ), E ⊂ R N be an L N -measurable set, and let η be such that

η ∈ W 1,1 R N , ˆRN |∇η(v)||v| rq dv < ∞. (7.3)
Then for every ε ∈ (0, 1/e) it follows that

1 | ln ε| [u ε ] q W r,q (E,R d ) = H N -1 S N -1 ˆE ˆE ρ ε,ω (|x -y|) |u ε (x) -u ε (y)| q |x -y| rq dydx + o ε (1), (7.4)
where u ε (x) := ´RN η(z)u(x -εz)dz, lim ε→0 + o ε (1) = 0 and ρ ε,ω is the logarithmic kernel defined in Definition 5.2.

Proof. Let ε ∈ (0, 1/e) be fixed. By definition of Gagliardo seminorm [•] W r,q (Definition 1.4), change of variable formula, Fubini's theorem and additivity of integral

1 | ln ε| [u ε ] q W r,q (E,R d ) = 1 | ln ε| ˆE ˆE |u ε (x) -u ε (y)| q |x -y| N +rq dy dx = 1 | ln ε| ˆRN ˆRN |u ε (x) -u ε (x + z)| q |z| N +rq χ E (x + z)χ E (x)dz dx = 1 | ln ε| ˆRN ˆRN |u ε (x) -u ε (x + z)| q |z| N +rq χ E (x + z)χ E (x)dx dz = 1 | ln ε| ˆRN \B Rε,ω (0) g ε (z)dz + ˆBRε,ω (0)\Bε(0) g ε (z)dz + ˆBε(0) g ε (z)dz , (7.5)
where we set

g ε (z) : = ˆRN |u ε (x) -u ε (x + z)| q |z| N +rq χ E (x + z)χ E (x)dx. (7.6) Using (2.9) with γ = R ε,ω := | ln ε| -ω we get 1 | ln ε| ˆRN \B Rε,ω (0) g ε (z)dz ≤ η q L 1 (R N ) 2 q u q L q (R N ,R d ) H N -1 S N -1 | ln ε| 1-rqω rq = o ε (1). (7.7)
Using (2.12) we get

1 | ln ε| ˆBε(0) g ε (z)dz ≤ ∇η q-1 L 1 (R N ,R N ) ˆRN |∇η(v)|(|v| + 2) rq dv [u] q B r q,∞ (R N ,R d ) H N -1 S N -1 q -rq 1 | ln ε| = o ε (1). (7.8)
Therefore, we obtain by (7.5),(7.7),(7.8) and the definition of the logarithmic kernel ρ ε,ω

1 | ln ε| [u ε ] q W r,q (E,R d ) = 1 | ln ε| ˆBRε,ω (0)\Bε(0) g ε (z)dz + o ε (1) = 1 | ln ε| ˆBRε,ω (0)\Bε(0) ˆRN |u ε (x) -u ε (x + z)| q |z| N +rq χ E (x + z)χ E (x)dx dz + o ε (1) = 1 | ln ε| ˆRN χ [ε,Rε,ω) (|z|) |z| N ˆRN |u ε (x) -u ε (x + z)| q |z| rq χ E (x + z)χ E (x)dx dz + o ε (1) = 1 - | ln R ε,ω | | ln ε| H N -1 S N -1 ˆRN ρ ε,ω (|z|) ˆRN |u ε (x) -u ε (x + z)| q |z| rq χ E (x + z)χ E (x)dx dz+o ε (1) = H N -1 S N -1 ˆE ˆE ρ ε,ω (|x -y|) |u ε (x) -u ε (y)| q |x -y| rq dxdy - | ln R ε,ω | | ln ε| H N -1 S N -1 ˆRN ρ ε,ω (|z|) ˆRN |u ε (x) -u ε (x + z)| q |z| rq χ E (x + z)χ E (x)dx dz + o ε (1) = H N -1 S N -1 ˆE ˆE ρ ε,ω (|x -y|) |u ε (x) -u ε (y)| q |x -y| rq dxdy + o ε (1). (7.9)
In the last equality we used (5.2), item 1 of Proposition 5.1, (2.4) and u ∈ B r q,∞ (R N , R d ) in order to get

| ln R ε,ω | | ln ε| H N -1 S N -1 ˆRN ρ ε,ω (|z|) ˆRN |u ε (x) -u ε (x + z)| q |z| rq χ E (x + z)χ E (x)dx dz ≤ | ln R ε,ω | | ln ε| H N -1 S N -1 ˆRN |η(v)|dv q [u] q B r q,∞ (R N ,R d ) = o ε (1). (7.10) It completes the proof. Lemma 7.2. (The η-Separating Lemma) Assume q ∈ [1, ∞), r ∈ (0, 1) and u ∈ B r q,∞ (R N , R d ). Let η ∈ L 1 (R N ) be such that ˆRN |η(z)||z| rq dz < ∞. (7.11)
Let {ρ ε } ε∈(0,a) , a ∈ (0, ∞], ρ ε : (0, ∞) → [0, ∞), be a kernel such that

lim ε→0 + ε rq ˆRN ρ ε (|z|)
|z| rq dz = 0. (7.12)

Then for every L N -measurable set E ⊂ R N we have

lim inf ε→0 + ˆE ˆE ρ ε (|x -y|) |u ε (x) -u ε (y)| q |x -y| rq dydx = ˆRN η(z)dz q lim inf ε→0 + ˆE ˆE ρ ε (|x -y|) |u(x) -u(y)| q |x -y| rq dydx, (7.13)
and lim sup

ε→0 + ˆE ˆE ρ ε (|x -y|) |u ε (x) -u ε (y)| q |x -y| rq dydx = ˆRN η(z)dz q lim sup ε→0 + ˆE ˆE ρ ε (|x -y|) |u(x) -u(y)| q |x -y| rq dydx. (7.14) Proof. Let 0 < α < 1. It follows for L N -almost every x, z ∈ R N that |u(x) -u(x + z)| q = |(u(x) -u ε (x)) + (u ε (x) -u ε (x + z)) + (u ε (x + z) -u(x + z))| q ≤ (|u(x) -u ε (x)| + |u ε (x) -u ε (x + z)| + |u ε (x + z) -u(x + z)|) q ≤ 1 α q-1 |u ε (x) -u ε (x + z)| q + 1 (1 -α) q-1 (|u(x) -u ε (x)| + |u ε (x + z) -u(x + z)|) q . (7.15)
In the last inequality we use the following convex inequalityj: for numbers A, B ≥ 0 and convex function Ψ : [0, ∞) → R it follows that

Ψ(A + B) = Ψ α A α + (1 -α) B 1 -α ≤ αΨ A α + (1 -α)Ψ B 1 -α . (7.16)
In the inequality (7.15) we choose

A = |u ε (x) -u ε (x + z)|, B = |u(x) -u ε (x)| + |u ε (x + z) -u(x + z)|, Ψ(r) = r q . (7.17)
Therefore, by (7.15)

ˆE ˆE ρ ε (|x -y|) |u(x) -u(y)| q |x -y| rq dydx = ˆRN ˆRN ρ ε (|x -y|) |u(x) -u(y)| q |x -y| rq χ E (y)χ E (x)dydx = ˆRN ˆRN ρ ε (|z|) |u(x) -u(x + z)| q |z| rq χ E (x + z)χ E (x)dzdx ≤ 1 α q-1 ˆRN ˆRN ρ ε (|z|) |u ε (x) -u ε (x + z)| q |z| rq χ E (x + z)χ E (x)dzdx + 1 (1 -α) q-1 ˆRN ˆRN ρ ε (|z|) (|u(x) -u ε (x)| + |u ε (x + z) -u(x + z)|) q |z| rq dzdx = 1 α q-1 ˆE ˆE ρ ε (|x -y|) |u ε (x) -u ε (y)| q |x -y| rq dydx + 1 (1 -α) q-1 ˆRN ˆRN ρ ε (|z|) (|u(x) -u ε (x)| + |u ε (x + z) -u(x + z)|) q |z| rq dzdx. (7.18) Notice that ˆRN ˆRN ρ ε (|z|) (|u(x) -u ε (x)| + |u(x + z) -u ε (x + z)|) q |z| rq dzdx ≤ 2 q-1 ˆRN ˆRN ρ ε (|z|) |u(x) -u ε (x)| q + |u(x + z) -u ε (x + z)| q |z| rq dzdx = 2 q ˆRN ˆRN ρ ε (|z|) |u(x) -u ε (x)| q |z| rq dzdx = 2 q ˆRN ρ ε (|z|) |z| rq dz u -u ε q L q (R N ,R d ) . (7.19)
Assume for a moment that ´RN η(z)dz = 1. Then, by Hlder's inequality

u -u ε q L q (R N ,R d ) = ˆRN |u(x) -u ε (x)| q dx = ˆRN ˆRN η(v) (u(x) -u(x -εv)) dv q dx ≤ ˆRN ˆRN |η(v)| q-1 q |η(v)| 1 q |u(x) -u(x -εv)| dv q dx ≤ η q-1 L 1 (R N ) ˆRN ˆRN |η(v)| |u(x) -u(x -εv)| q dv dx = η q-1 L 1 (R N ) ˆRN |η(v)| ˆRN |u(x) -u(x -εv)| q dx dv ≤ ε rq η q-1 L 1 (R N ) [u] q B r q,∞ (R N ,R d ) ˆRN |η(v)||v| rq dv. (7.20)
Hence, by (7.18), (7.19) and (7.20)

ˆE ˆE ρ ε (|x -y|) |u(x) -u(y)| q |x -y| rq dydx ≤ 1 α q-1 ˆE ˆE ρ ε (|x -y|) |u ε (x) -u ε (y)| q |x -y| rq dydx + 2 q (1 -α) q-1 ε rq ˆRN ρ ε (|z|) |z| rq dz u -u ε q L q (R N ,R d ) ε rq ≤ 1 α q-1 ˆE ˆE ρ ε (|x -y|) |u ε (x) -u ε (y)| q |x -y| rq dydx + 2 q (1 -α) q-1 ε rq ˆRN ρ ε (|z|) |z| rq dz η q-1 L 1 (R N ) [u] q B r q,∞ (R N ,R d ) ˆRN |η(v)||v| rq dv. (7.21)
By (7.11), (7.12), u ∈ B r q,∞ (R N , R d ) and (7.21) we obtain lim inf

ε→0 + ˆE ˆE ρ ε (|x -y|) |u(x) -u(y)| q |x -y| rq dydx ≤ 1 α q-1 lim inf ε→0 + ˆE ˆE ρ ε (|x -y|) |u ε (x) -u ε (y)| q |x -y| rq dydx, (7.22)
and lim sup

ε→0 + ˆE ˆE ρ ε (|x -y|) |u(x) -u(y)| q |x -y| rq dydx ≤ 1 α q-1 lim sup ε→0 + ˆE ˆE ρ ε (|x -y|) |u ε (x) -u ε (y)| q |x -y| rq dydx.
(7.23)

Taking the limit as α → 1 -we get lim inf

ε→0 + ˆE ˆE ρ ε (|x -y|) |u(x) -u(y)| q |x -y| rq dydx ≤ lim inf ε→0 + ˆE ˆE ρ ε (|x -y|) |u ε (x) -u ε (y)| q |x -y| rq dydx, (7.24)
and lim sup

ε→0 + ˆE ˆE ρ ε (|x -y|) |u(x) -u(y)| q |x -y| rq dydx ≤ lim sup ε→0 + ˆE ˆE ρ ε (|x -y|) |u ε (x) -u ε (y)| q |x -y| rq dydx. (7.25)
Replacing the roles of u ε (x), u ε (y) with u(x), u(y), respectively, one can prove similarly (to the inequality (7.21)) the inequality

ˆE ˆE ρ ε (|x -y|) |u ε (x) -u ε (y)| q |x -y| rq dydx ≤ 1 α q-1 ˆE ˆE ρ ε (|x -y|) |u(x) -u(y)| q |x -y| rq dydx + 2 q (1 -α) q-1 ε rq ˆRN ρ ε (|z|) |z| rq dz η q-1 L 1 (R N ) [u] q B r q,∞ (R N ,R d ) ˆRN |η(v)||v| rq dv (7.26)
in order to obtain lim inf

ε→0 + ˆE ˆE ρ ε (|x -y|) |u ε (x) -u ε (y)| q |x -y| rq dydx ≤ lim inf ε→0 + ˆE ˆE ρ ε (|x -y|) |u(x) -u(y)| q |x -y| rq dydx, (7.27)
and lim sup

ε→0 + ˆE ˆE ρ ε (|x -y|) |u ε (x) -u ε (y)| q |x -y| rq dydx ≤ lim sup ε→0 + ˆE ˆE ρ ε (|x -y|) |u(x) -u(y)| q |x -y| rq dydx. (7.28)
Assume now that ´RN η(z)dz = 0. Replacing η with cη, where c := 1 ´RN η(z)dz , and using the homogeneity of the convolution u * (cη) = c (u * η), one can get (7.13) and (7.14). In case ´RN η(z)dz = 0, let us choose any η 0 ∈ C c (R N ) such that ´RN η 0 (z)dz = 1, and for each n ∈ N

define η n := η -1 n η 0 . It follows that u * η (ε) (x) -u * η (ε) (y) = u * η n + 1 n η 0 (ε) (x) -u * η n + 1 n η 0 (ε) (y) = u * (η n ) (ε) + 1 n η 0 (ε) (x) -u * (η n ) (ε) + 1 n η 0 (ε) (y) = u * (η n ) (ε) (x) -u * (η n ) (ε) (y) + u * 1 n η 0 (ε) (x) -u * 1 n η 0 (ε) (y) . (7.29) Therefore, u * η (ε) (x) -u * η (ε) (y) q ≤ 2 q-1 u * (η n ) (ε) (x) -u * (η n ) (ε) (y) q + 1 n q u * (η 0 ) (ε) (x) -u * (η 0 ) (ε) (y) q . (7.30) Thus, since ´RN η n (v)dv = -1 n = 0, ´RN η 0 (v)dv = 1 = 0, then lim sup ε→0 + ˆE ˆE ρ ε (|x -y|) |u ε (x) -u ε (y)| q |x -y| rq dydx ≤ 2 q-1 lim sup ε→0 + ˆE ˆE ρ ε (|x -y|) u * (η n ) (ε) (x) -u * (η n ) (ε) (y) q |x -y| rq dydx + 2 q-1 n q lim sup ε→0 + ˆE ˆE ρ ε (|x -y|) u * (η 0 ) (ε) (x) -u * (η 0 ) (ε) (y) q |x -y| rq dydx = 2 q-1 n q lim sup ε→0 + ˆE ˆE ρ ε (|x-y|) |u(x) -u(y)| q |x -y| rq dydx+ 2 q-1 n q lim sup ε→0 + ˆE ˆE ρ ε (|x-y|) |u(x) -u(y)| q |x -y| rq dydx = 2 q n q lim sup ε→0 + ˆE ˆE ρ ε (|x -y|) |u(x) -u(y)| q |x -y| rq dydx. (7.31)
Taking the limit as n → ∞ and using Corollary 7.1 we get lim Let q ∈ [1, ∞), r ∈ (0, 1). Let ω ∈ (0, 1) be such that rq < 1/ω. Let u ∈ B r q,∞ (R N , R d ), E ⊂ R N be an L N -measurable set and η ∈ W 1,1 R N . Then,

ε→0 + ˆE ˆE ρ ε (|x -y|) |u ε (x) -u ε (y)| q |x -y| rq dydx = 0. ( 7
lim inf ε→0 + 1 | ln ε| u * η (ε) q W r,q (E,R d ) = ˆRN η(z)dz q H N -1 S N -1 lim inf ε→0 + ˆE ˆE ρ ε,ω (|x -y|) |u(x) -u(y)| q |x -y| rq dydx, (7.33) lim sup ε→0 + 1 | ln ε| u * η (ε) q W r,q (E,R d ) = ˆRN η(z)dz q H N -1 S N -1 lim sup ε→0 + ˆE ˆE ρ ε,ω (|x -y|) |u(x) -u(y)| q |x -y| rq dydx, (7.34)
where ρ ε,ω is the logarithmic kernel.

Proof. Assume for a moment that η ∈ C 1 c (R N ). By Lemma 7.1 we have lim inf

ε→0 + 1 | ln ε| [u ε ] q W r,q (E,R d ) = H N -1 S N -1 lim inf ε→0 + ˆE ˆE ρ ε,ω (|x -y|) |u ε (x) -u ε (y)| q |x -y| rq dydx, (7.35) lim sup ε→0 + 1 | ln ε| [u ε ] q W r,q (E,R d ) = H N -1 S N -1 lim sup ε→0 + ˆE ˆE ρ ε,ω (|x -y|) |u ε (x) -u ε (y)| q |x -y| rq dydx. (7.36)
Note that by item 2 of Proposition 5.1 with α = rq, the logarithmic kernel satisfies condition (7.12) of Lemma 7.2. Therefore, by Lemma 7.2 lim inf

ε→0 + ˆE ˆE ρ ε,ω (|x -y|) |u ε (x) -u ε (y)| q |x -y| rq dydx = ˆRN η(z)dz q lim inf ε→0 + ˆE ˆE ρ ε,ω (|x -y|) |u(x) -u(y)| q
|x -y| rq dydx, (7.37) and lim sup

ε→0 + ˆE ˆE ρ ε,ω (|x -y|) |u ε (x) -u ε (y)| q |x -y| rq dydx = ˆRN η(z)dz q lim sup ε→0 + ˆE ˆE ρ ε,ω (|x -y|) |u(x) -u(y)| q |x -y| rq dydx. (7.38)
Now, (7.33) and ( 7.34) follow from (7.35),(7.36),(7.37), (7.38).

For η ∈ W 1,1 (R N ) choose any sequence {η n } ∞ n=1 ⊂ C 1 c (R N ) which converges to η in W 1,1 (R N
). So we have (7.33) and ( 7.34) for η n , for every n ∈ N. Taking the limit as n goes to ∞ and using item 1 of Lemma 3.1, we obtain (7.33) and (7.34) for η ∈ W 1,1 (R N ).

Corollary 7.3. (Gagliardo Constants are Controlled by Besov Seminorms

) Let 1 ≤ q < ∞, r ∈ (0, 1), u ∈ B r q,∞ (R N , R d ) and E ⊂ R N be an L N -measurable set. Let η ∈ W 1,1 (R N ). Then, lim sup ε→0 + 1 | ln ε| [u ε ] q W r,q (E,R d ) ≤ ˆRN η(z)dz q H N -1 (S N -1 )[u] q B r q,∞ (E,R d ) < ∞. (7.39)
Proof. Let ω ∈ (0, 1) be such that rq < 1/ω and ρ ε,ω be the logarithmic kernel as defined in Definition 5.2. By (7.34), (6.6) and Definition 2.1 (Definition of Besov seminorm) we get lim sup

ε→0 + 1 | ln ε| [u ε ] q W r,q (E,R d ) = ˆRN η(z)dz q H N -1 S N -1 lim sup ε→0 + ˆE ˆE ρ ε,ω (|x-y|) |u(x) -u(y)| q |x -y| rq dydx ≤ ˆRN η(z)dz q lim sup ε→0 + ˆSN-1 ˆE χ E (x + εn) |u(x + εn) -u(x)| q ε rq dxdH N -1 (n) ≤ ˆRN η(z)dz q H N -1 (S N -1 )[u] q B r q,∞ (E,R d ) < ∞. (7.40) Theorem 7.1. (Variations Control Gagliardo Constants) Let q ∈ [1, ∞) and r ∈ (0, 1). Suppose u ∈ B r q,∞ (R N , R d ), E ⊂ R N be an L N -measurable set and η ∈ W 1,1 (R N ). Then, ˆRN η(z)dz q lim inf ε→0 + ˆSN-1 ˆE χ E (x + εn) |u(x + εn) -u(x)| q ε rq dxdH N -1 (n) ≤ lim inf ε→0 + 1 | ln ε| [u ε ] q W r,q (E,R d ) ≤ lim sup ε→0 + 1 | ln ε| [u ε ] q W r,q (E,R d ) ≤ ˆRN η(z)dz q lim sup ε→0 + ˆSN-1 ˆE χ E (x + εn) |u(x + εn) -u(x)| q ε rq dxdH N -1 (n). (7.41)
Proof. Let ω ∈ (0, 1) be such that rq < 1/ω and ρ ε,ω be the logarithmic kernel as defined in Definition 5.2. By Sandwich Lemma (Lemma 6.1) we get for α = rq and ρ ε = ρ ε,ω lim inf

ε→0 + S N -1 ˆE χ E (x + εn) |u(x + εn) -u(x)| q ε rq dxdH N -1 (n) ≤ lim inf ε→0 + ˆE ˆE ρ ε,ω (|x -y|) |u(x) -u(y)| q |x -y| rq dydx ≤ lim sup ε→0 + ˆE ˆE ρ ε,ω (|x -y|) |u(x) -u(y)| q |x -y| rq dydx ≤ lim sup ε→0 + S N -1 ˆE χ E (x + εn) |u(x + εn) -u(x)| q ε rq dxdH N -1 (n). ( 7
ε→0 + S N -1 ˆE χ E (x + εn) |u(x + εn) -u(x)| q ε rq dxdH N -1 (n) ≤ lim inf ε→0 + 1 | ln ε| u * η (ε) q W r,q (E,R d ) ´RN η(z)dz q H N -1 (S N -1 ) ≤ lim sup ε→0 + 1 | ln ε| u * η (ε) q W r,q (E,R d ) ´RN η(z)dz q H N -1 (S N -1 ) ≤ lim sup ε→0 + S N -1 ˆE χ E (x + εn) |u(x + εn) -u(x)| q ε rq dxdH N -1 (n). (7.43)
Multiplying both sides of inequality (7.43) by ´RN η(z)dz q H N -1 S N -1 we obtain (7.41).

Theorem 7.2. (Equivalence Between Gagliardo and Beosv Constants) Let q ∈ [1, ∞), r ∈ (0, 1). Let u ∈ B r q,∞ (R N , R d ), E ⊂ R N be an L N -measurable set and η ∈ W 1,1 R N . If the following limit exists:

lim ε→0 + ˆSN-1 ˆE χ E (x + εn) |u(x + εn) -u(x)| q ε rq dxdH N -1 (n), (7.44)
then, for every kernel ρ ε we get

lim ε→0 + 1 | ln ε| [u ε ] q W r,q (E,R d ) = ˆRN η(z)dz q lim ε→0 + ˆSN-1 ˆE χ E (x+εn) |u(x + εn) -u(x)| q ε rq dxdH N -1 (n) = ˆRN η(z)dz q H N -1 S N -1 lim ε→0 + ˆE ˆE ρ ε (|x -y|) |u(x) -u(y)| q |x -y| rq dydx. (7.45)
Proof. Formulas (7.45) follows from assumption (7.44), Theorem 7.1 and Sandwich Lemma with α = rq (Lemma 6.1).

Corollary 7.4. (Equivalence Between Gagliardo Constants and B r,q -Seminorms) Let 1 ≤ q < ∞, r ∈ (0, 1), u ∈ B r q,∞ (R N , R d ), E ⊂ R N be an L N -measurable set and η ∈ W 1,1 R N . If the following limit exists:

lim ε→0 + ˆSN-1 ˆE χ E (x + εn) |u(x + εn) -u(x)| q ε rq dxdH N -1 (n), (7.46) then lim ε→0 + 1 | ln ε| [u ε ] q W r,q (E,R d ) = N ˆRN η(z)dz q lim ε→0 + ˆE 1 ε N ˆE∩Bε(x) |u(x) -u(y)| q |x -y| rq dydx = N ˆRN η(z)dz q [u] q B r,q (E,R d ) , (7.47)
where [u] B r,q (E,R d ) is the upper infinitesimal B r,q -seminorm defined in 4.1.

Proof. By Remark 5.2 the trivial kernel, ρε , is a kernel. Therefore, by assumption (7.46) and Theorem 7.2 we obtain lim

ε→0 + 1 | ln ε| [u ε ] q W r,q (E,R d ) = ˆRN η(z)dz q H N -1 S N -1 lim ε→0 + ˆE ˆE ρε (|x -y|) |u(x) -u(y)| q |x -y| rq dydx = ˆRN η(z)dz q H N -1 S N -1 L N (B 1 (0)) lim ε→0 + ˆE 1 ε N ˆE∩Bε(x) |u(x) -u(y)| q |x -y| rq dydx = N ˆRN η(z)dz q lim ε→0 + ˆE 1 ε N ˆE∩Bε(x) |u(x) -u(y)| q |x -y| rq dydx. (7.48)
The equation

L N (B 1 (0)) = H N -1 (S N -1 )
N follows from polar coordinates.

Remark 7.1. (Consistency with Previous Results) Equation ( 7.47) can be derived for functions

u ∈ BV (R N , R d ) ∩ L ∞ (R N , R d ), 1 < q < ∞, r = 1 q , η ∈ W 1,1 (R N )
, and an open set Ω ⊂ R N with a bounded Lipschitz boundary such that Du (∂Ω) = 0, by combining Theorem 1.2 in [START_REF] Poliakovsky | Asymptotic behavior of W 1/q,q -norm of mollified BV function and its application to singular perturbation problems[END_REF] and Theorem 1.1 in [START_REF] Poliakovsky | Jump detection in Besov spaces via a new BBM formula. Applications to Aviles-Giga type functionals[END_REF], which provide:

lim ε→0 + 1 | ln ε| [u ε ] q W 1 q ,q (Ω,R d ) = ´RN-1 2 (1 + |v| 2 ) -N +1 2 dv 1 N ´SN-1 |z 1 |dH N -1 (z) ˆRN η(z)dz q lim ε→0 + ˆΩ 1 ε N ˆΩ∩Bε(x) |u(x) -u(y)| q |x -y| dydx. (7.49)
According to Proposition 10.3, we obtain ´RN-1 2 (1 + |v| 2 )

-N +1 2 dv = ´SN-1 |z 1 |dH N -1 (z).
8 Jump Detection in BV ∩ B 1/p,p

In this section we prove formulas for lim ε→0

+ 1 | ln ε| [u ε ] q W 1/q,q (B,R d )
, where u ∈ BV ∩ B 1/p,p and B ⊂ R N is a Borel set (refer to Corollary 8.2).

Remark 8.1. (BV ∩ L ∞ is a Subset of B r q,∞ , rq ≤ 1) Let u ∈ BV R N , R d ∩ L ∞ R N , R d . Let 1 ≤ q < ∞
and r ∈ (0, 1) be such that rq ≤ 1. By Lemma 10.9 we get

sup h∈R N \{0} ˆRN |u(x + h) -u(x)| q |h| rq dx ≤ sup h∈R N \B 1 (0) ˆRN |u(x + h) -u(x)| q |h| rq dx + sup h∈B 1 (0)\{0} ˆRN |u(x + h) -u(x)| q |h| rq dx ≤ 2 q u q L q (R N ,R d ) + 2 q-1 u q-1 L ∞ (R N ,R d ) sup h∈B 1 (0)\{0} ˆRN |u(x + h) -u(x)| |h| dx ≤ 2 q u q L q (R N ,R d ) + 2 q-1 u q-1 L ∞ (R N ,R d ) Du (R N ) < ∞. (8.1) Note that since u ∈ L 1 R N , R d ∩ L ∞ R N , R d , then u ∈ L q R N , R d . Thus, by Definition 1.3 (definition of Besov space) we get u ∈ B r q,∞ (R N , R d ). Lemma 8.1. (Interpolation for Besov Seminorms) Let p ∈ (1, ∞) and u ∈ BV (R N , R d ) ∩ B 1/p,p (R N , R d ).
Then for every q ∈ (1, p) we have u ∈ B 1/q,q (R N , R d ) and

[u] q B 1/q q,∞ (R N ,R d ) ≤ Du (R N ) α [u] p B 1/p p,∞ (R N ,R d ) 1-α , (8.2)
where α := p-q p-1 .

Proof. Since α = p-q p-1 , then q = α + (1 -α)p. By Hölder's inequality and Lemma 10.9 we get

[u] q B 1/q q,∞ (R N ,R d ) = sup h∈R N \{0} ˆRN |u(x + h) -u(x)| q |h| dx = sup h∈R N \{0} ˆRN |u(x + h) -u(x)| |h| α |u(x + h) -u(x)| p |h| 1-α dx ≤ sup h∈R N \{0} ˆRN |u(x + h) -u(x)| |h| dx α sup h∈R N \{0} ˆRN |u(x + h) -u(x)| p |h| dx 1-α ≤ Du (R N ) α [u] p B 1/p p,∞ (R N ,R d ) 1-α . (8.3) Corollary 8.1. (Convergence of the Truncated Family in Besov Seminorm) Let p ∈ (1, ∞) and u ∈ BV (R N , R d ) ∩ B 1/p p,∞ (R N , R d ).
Then, for every q ∈ (1, p) we have

lim l→∞ [u -u l ] B 1/q q,∞ (R N ,R d ) = 0, (8.4)
where {u l } l∈[0,∞) is the truncated family obtained by u as defined in Definition 10.2. In particular, the truncated family u l converges to u in the norm of the space B

1/q q,∞ (R N , R d ), which means that lim l→∞ [u -u l ] B 1/q q,∞ (R N ,R d ) + u -u l L q (R N ,R d ) = 0. (8.5)
Proof. Let q ∈ (1, p) and denote α := p-q p-1 . From Lemma 8.1 we get

[u -u l ] q B 1/q q,∞ (R N ,R d ) ≤ D(u -u l ) (R N ) α [u -u l ] p B 1/p p,∞ (R N ,R d ) 1-α . (8.6) Note that [u -u l ] p B 1/p p,∞ (R N ,R d ) = sup h∈R N \{0} ˆRN |(u -u l )(x + h) -(u -u l )(x)| p |h| dx ≤ 2 p-1 sup h∈R N \{0} ˆRN |u(x + h) -u(x)| p |h| dx + 2 p-1 sup h∈R N \{0} ˆRN |u l (x + h) -u l (x)| p |h| dx ≤ 2 p [u] p B 1/p p,∞ (R N ,R d ) < ∞. (8.7)
Since by Lemma 10.11 we have lim l→∞ D(u -u l ) (Ω) = 0, then (8.4) follows. The convergence of u l to u as l → ∞ in the norm of the space B 1/q q,∞ (R N , R d ) follows from (8.4) and Lemma 10.10.

Recall Definition 10.4 for u + , u -, J u , ν u .

Theorem 8.1. (Proposition 2.4 in [START_REF] Poliakovsky | Jump detection in Besov spaces via a new BBM formula. Applications to Aviles-Giga type functionals[END_REF])

Let Ω ⊂ R N be an open set, 1 < q < ∞, u ∈ BV loc (Ω, R d ) ∩ L ∞ loc (Ω, R d ).
Then, for every h ∈ R N and every compact set K ⊂ Ω such that Du (∂K) = 0 we have For an open set Ω ⊂ R N and a function u ∈ BV (Ω, R d ), it follows that Du ≥ |u + -u -|H N -1 J u (refer to Lemma 3.76 in [START_REF] Ambrosio | Functions of Bounded Variation and Free Discontinuity Problems[END_REF]). It is important to note that according to Definition 10.4, |u + (x)u -(x)| > 0 for x ∈ J u . Therefore, for a set E ⊂ Ω, the assumption Du (∂E) = 0 indicates that H N -1 (∂E ∩ J u ) = 0, implying that the portion of the jump set J u within the topological boundary of E is negligible with respect to H N -1 .

lim ε→0 + ˆK |u(x + εh) -u(x)| q ε dx = ˆK∩Ju u + (x) -u -(x) q |ν u (x) • h|dH N -1 (x). ( 8

Lemma 8.2. (Equivalence Between Variation and Jump Variation in the BV

Case) Let 1 < p < ∞, u ∈ BV (R N , R d ) ∩ B 1 p ,p (R N , R d ) and 1 < q < p. Then, for every n ∈ R N and every Borel set B ⊂ R N such that H N -1 (∂B ∩ J u ) = 0 we have lim ε→0 + ˆB χ B (x + εn) |u(x + εn) -u(x)| q ε dx = ˆB∩Ju u + (x) -u -(x) q |ν u (x) • n|dH N -1 (x). (8.9)
In particular, the following limit exists: [START_REF] Hashash | Jumps in Besov spaces and fine properties of Besov and fractional Sobolev functions[END_REF]. Here, we will mention that bi-Hölder functions can be used to demonstrate that the jump variation of such a function is zero (the right-hand side of (8.10)), but the variation of u (the left-hand side of (8.10)) is positive.

lim ε→0 + ˆSN-1 ˆB χ B (x + εn) |u(x + εn) -u(x)| q ε dxdH N -1 (n) = ˆSN-1 |z 1 | dH N -1 (z) ˆB∩Ju u + (x) -u -(x) q dH N -1 (x)
Proof. Step 1: u ∈ L ∞ (R N , R d ). Let n ∈ R N and B ⊂ R N be a Borel set. Let K ⊂ B o
lim inf ε→0 + ˆB χ B (x + εn) |u(x + εn) -u(x)| q ε dx ≥ lim inf ε→0 + ˆBo χ B o (x + εn) |u(x + εn) -u(x)| q ε dx ≥ lim inf ε→0 + ˆK |u(x + εn) -u(x)| q ε dx = ˆK∩Ju u + (x) -u -(x) q |ν u (x) • n|dH N -1 (x). (8.11)
Taking the supremum over compact sets K ⊂ B o such that Du (∂K) = 0 we get by Lemma 10.2

lim inf ε→0 + ˆB χ B (x + εn) |u(x + εn) -u(x)| q ε dx ≥ ˆBo ∩Ju u + (x) -u -(x) q |ν u (x) • n|dH N -1 (x). (8.12)
Let Ω be an open set such that B ⊂ Ω and Du (∂Ω) = 0. By Lemma 10.1, there exists a sequence of numbers {R k } ∞ k=1 such that for every k ∈ N:

R k > 0, R k < R k+1 , Du (∂B R k (0)) = 0, and lim k→∞ R k = ∞. Note that since ∂ Ω ∩ B R k (0) ⊂ ∂Ω ∪ ∂B R k (0), then Du ∂ Ω ∩ B R k (0) = 0.
Note that if n = 0, then equation (8.9) holds trivially. Assume n = 0. It follows from Theorem 8.1, Lemma 10.9 and Remark 8.2 lim sup

ε→0 + ˆB χ B (x + εn) |u(x + εn) -u(x)| q ε dx ≤ lim sup ε→0 + ˆΩ |u(x + εn) -u(x)| q ε dx ≤ lim sup ε→0 + ˆΩ |u(x + εn) -u(x)| q ε dx ≤ lim sup ε→0 + ˆΩ∩B R k+1 (0) |u(x + εn) -u(x)| q ε dx + lim sup ε→0 + ˆΩ\B R k+1 (0) |u(x + εn) -u(x)| q ε dx ≤ ˆ(Ω∩B R k+1 (0))∩Ju u + (x) -u -(x) q |ν u (x) • n|dH N -1 (x) + 2 q-1 u q-1 L ∞ (R N ,R d ) lim sup ε→0 + ˆRN \B R k+1 (0) |u(x + εn) -u(x)| ε dx ≤ ˆΩ∩Ju u + (x) -u -(x) q |ν u (x) • n|dH N -1 (x) + 2 q-1 u q-1 L ∞ (R N ,R d ) |n| Du R N \ B R k (0) . (8.13)
Taking the limit as k → ∞ in (8.13), we get lim sup

ε→0 + ˆB χ B (x + εn) |u(x + εn) -u(x)| q ε dx ≤ ˆΩ∩Ju u + (x) -u -(x) q |ν u (x) • n|dH N -1 (x). (8.14)
Therefore, by the Lemma 10.3 we get lim sup

ε→0 + ˆB χ B (x + εn) |u(x + εn) -u(x)| q ε dx ≤ ˆB∩Ju u + (x) -u -(x) q |ν u (x) • n|dH N -1 (x). (8.15)
By (8.12) and (8.15) we get (8.9) for every Borel set

B ⊂ R N such that H N -1 (∂B ∩ J u ) = 0.
Since by Lemma 10.9

sup ε∈(0,∞) ˆB χ B (x + εn) |u(x + εn) -u(x)| q ε dx ≤ sup ε∈(0,∞) ˆB |u(x + εn) -u(x)| q ε dx ≤ 2 q-1 u q-1 L ∞ (R N ,R d ) sup ε∈(0,∞) ˆB |u(x + εn) -u(x)| ε dx ≤ 2 q-1 u q-1 L ∞ (R N ,R d ) |n| Du (R N ) < ∞, (8.16
) then we get by Dominated Convergence Theorem, equation (8.9), Fubini's Theorem and Proposition 10.2

lim ε→0 + ˆSN-1 ˆB χ B (x + εn) |u(x + εn) -u(x)| q ε dxdH N -1 (n) = ˆSN-1 lim ε→0 + ˆB χ B (x + εn) |u(x + εn) -u(x)| q ε dxdH N -1 (n) = ˆSN-1 ˆB∩Ju u + (x) -u -(x) q |ν u (x) • n|dH N -1 (x) dH N -1 (n) = ˆB∩Ju u + (x) -u -(x) q ˆSN-1 |ν u (x) • n|dH N -1 (n) dH N -1 (x) = ˆSN-1 |e 1 • n|dH N -1 (n) ˆB∩Ju u + (x) -u -(x)
q dH N -1 (x). (8.17)

In particular, the limit in (8.10) exists.

Step 2: u is not necessarily bounded. For every

l ∈ [0, ∞) we have u l ∈ L ∞ (R N , R d ), where {u l } l∈[0,∞)
is the truncated family defined in Definition 10.2. So we get for every l ∈ [0, ∞) by the previous step the formulas

lim ε→0 + ˆB χ B (x + εn) |u l (x + εn) -u l (x)| q ε dx = ˆB∩Ju l (u l ) + (x) -(u l ) -(x) q |ν u l (x) • n|dH N -1 (x), (8.18
) and

lim ε→0 + ˆSN-1 ˆB χ B (x + εn) |u l (x + εn) -u l (x)| q ε dxdH N -1 (n) = ˆSN-1 |z 1 | dH N -1 (z) ˆB∩Ju l (u l ) + (x) -(u l ) -(x) q dH N -1 (x
(u l ) + (x) -(u l ) -(x) q |ν u l (x)•n|dH N -1 (x) = ˆB∩Ju u + (x) -u -(x) q |ν u (x)•n|dH N -1 (x), (8.20) and lim l→∞ ˆB∩Ju l (u l ) + (x) -(u l ) -(x) q dH N -1 (x) = ˆB∩Ju u + (x) -u -(x) q dH N -1 (x). (8.21)
By Corollary 8.1, we know that the truncated family u l converges to u in Besov space B

1/q q,∞ . Let us denote F ε (u) := ˆB χ B (x + εn) |u(x + εn) -u(x)| q ε dx. (8.22)
By Lemma 6.3, we get

lim l→∞ lim sup ε→0 + F ε (u l ) = lim sup ε→0 + F ε (u), lim l→∞ lim inf ε→0 + F ε (u l ) = lim inf ε→0 + F ε (u). (8.23)
By (8.18) the limit lim ε→0 + F ε (u l ) exists for every l ∈ [0, ∞). Thus, by (8.23), we conclude the existence of the limit lim ε→0 + F ε (u), and

lim l→∞ lim ε→0 + F ε (u l ) = lim ε→0 + F ε (u). (8.24)
Taking the limit in (8.18) as l → ∞, and using (8.20) and (8.24), we obtain (8.9). By the Dominated Convergence Theorem, we deduce (8.10) from (8.9), as shown in calculation (8.17).

Definition 8.1. (q-Jump Variation)

Let Ω ⊂ R N be an open set, u ∈ L 1 loc (Ω, R d ), q ∈ R, and S ⊂ Ω is an H N -1 -measurable set. We define the q-jump variation of u in S by JV u,q (S) :=

ˆS∩Ju |u + (x) -u -(x)| q dH N -1 (x). (8.25)
Let n ∈ S N -1 . We define the q-jump variation of u in S in direction n by JV u,q,n (S) :=

ˆS∩Ju |u + (x) -u -(x)| q |ν u (x) • n|dH N -1 (x). (8.26)
Corollary 8.2. (Equivalence Between Gagliardo Constants and the q-Jump Variations)

Let p ∈ (1, ∞), q ∈ (1, p), u ∈ BV (R N , R d ) ∩ B 1 p ,p (R N , R d ), η ∈ W 1,1 R N and B ⊂ R N be a Borel set such that H N -1 (∂B ∩ J u ) = 0. Then, lim ε→0 + 1 | ln ε| [u ε ] q W 1/q,q (B,R d ) = ˆRN η(z)dz q H N -1 S N -1 lim ε→0 + ˆB ˆB ρ ε (|x -y|) |u(x) -u(y)| q |x -y| dydx = ˆRN η(z)dz q lim ε→0 + ˆSN-1 ˆB χ B (x + εn) |u(x + εn) -u(x)| q ε dxdH N -1 (n) = ˆRN η(z)dz q ˆSN-1 |z 1 | dH N -1 (z) ˆJu∩B u + (x) -u -(x) q dH N -1 (x). (8.27)
Proof. By Lemma 8.2 the limit

lim ε→0 + ˆSN-1 ˆB χ B (x + εn) |u(x + εn) -u(x)| q ε dxdH N -1 (n) (8.28)
exists, and

lim ε→0 + ˆSN-1 ˆB χ B (x + εn) |u(x + εn) -u(x)| q ε dxdH N -1 (n) = ˆSN-1 |z 1 | dH N -1 (z) ˆB∩Ju u + (x) -u -(x) q dH N -1 (x). (8.29)
Since the limit in (8.28) exists, we get by Theorem 7.2 with r = 1 q and E = B that lim

ε→0 + 1 | ln ε| [u ε ] q W 1/q,q (B,R d ) = ˆRN η(z)dz q H N -1 S N -1 lim ε→0 + ˆB ˆB ρ ε (|x -y|) |u(x) -u(y)| q |x -y| dydx = ˆRN η(z)dz q lim ε→0 + ˆSN-1 ˆB χ B (x + εn) |u(x + εn) -u(x)| q ε dxdH N -1 (n). (8.30)
We get (8.27) by equations (8.29) and (8.30).

Some observations about jumps of functions in

B r q,∞ = B r,q Lemma 8.3. (Besov Spaces embed in Fractional Sobolev Spaces) Let 0 < r < s < 1, q ∈ [1, ∞). Then, B s q,∞ (R N , R d ) ⊂ W r,q loc (R N , R d ). (8.31)
Proof. Let u ∈ B s q,∞ (R N , R d ) and K ⊂ R N be a compact set. We have by additivity of integral

ˆK ˆK |u(x) -u(y)| q |x -y| rq+N dydx = ˆK ˆK∩B 1 (x) |u(x) -u(y)| q |x -y| rq+N dy dx + ˆK ˆK\B 1 (x) |u(x) -u(y)| q |x -y| rq+N dy dx. (8.32)
By Change of variable formula, Fubini's theorem, definition of the Besov seminorm, polar coordinates and the assumption u

∈ B s q,∞ (R N , R d ) we have that ˆK ˆK∩B 1 (x) |u(x) -u(y)| q |x -y| rq+N dy dx = ˆK ˆB1 (x) χ K (y) |u(x) -u(y)| q |x -y| rq+N dy dx = ˆK ˆB1 (0) χ K (x + z) |u(x) -u(x + z)| q |z| rq+N dz dx = ˆB1 (0) ˆK χ K (x + z) |u(x) -u(x + z)| q |z| rq+N dx dz = ˆB1 (0) |z| sq-rq-N ˆK χ K (x + z) |u(x) -u(x + z)| q |z| sq dx dz ≤ [u] q B s q,∞ (R N ,R d ) ˆB1 (0) |z| sq-rq-N dz = [u] q B s q,∞ (R N ,R d ) H N -1 (S N -1 ) (s -r)q < ∞. (8.33)
By monotonicity of integral, the convexity of the function r -→ r q , r ∈ [0, ∞), and

u ∈ L q (K, R d ) we obtain that ˆK ˆK\B 1 (x) |u(x) -u(y)| q |x -y| rq+N dy dx ≤ ˆK ˆK\B 1 (x) |u(x) -u(y)| q dy dx ≤ ˆK ˆK |u(x) -u(y)| q dy dx ≤ ˆK 2 q-1 ˆK |u(x)| q + |u(y)| q dy dx = ˆK 2 q-1 |u(x)| q L N (K) + 2 q-1 u q L q (K,R d ) dx = 2 q u q L q (K,R d ) L N (K) < ∞. (8.34)
Thus, we derive from (8.32), (8.33) and (8.34) that u ∈ W r,q loc (R N , R d ). Theorem 8.2. (H N -1 -negligibility of the jump set of fractional Sobolev functions, Theorem 1.7 in [START_REF] Hashash | Jumps in Besov spaces and fine properties of Besov and fractional Sobolev functions[END_REF])

Let

Ω ⊂ R N be an open set, q ∈ (1, ∞) and u ∈ W 1/q,q loc (Ω, R d ). Then H N -1 (J u ) = 0. Corollary 8.3. (H N -1 -Negligibility of the Jump Set of u ∈ B r q,∞ , rq > 1) Let r ∈ (0, 1) and q ∈ [1, ∞) be such that rq > 1 and u ∈ B r q,∞ (R N , R d ). Then H N -1 (J u ) = 0. Proof. By Lemma 8.3 we have B r q,∞ (R N , R d ) ⊂ W 1/q,q loc (R N , R d ), so by Theorem 8.2 we get H N -1 (J u ) = 0. Remark 8.4. (Functions in B r q,∞ , rq ≤ 1, Have Jumps) If r ∈ (0, 1), q ∈ [1, ∞) are such that rq ≤ 1, then, as was proved in Remark 8.1, BV (R N , R d ) ∩ L ∞ (R N , R d ) ⊂ B r q,∞ (R N , R d ).
Therefore, for functions u ∈ B r q,∞ (R N , R d ), the measure of the jump set with respect to Hausdorff measure, H N -1 (J u ), can be any value in the interval [0, ∞]. 9 Open questions Question 9.1. Let 1 < q < ∞ and u ∈ B 1/q q,∞ (R N , R d ). Does the following limit exist?

lim ε→0 + ˆSN-1 ˆRN |u(x + εn) -u(x)| q ε dxdH N -1 (n). (9.1)
Note that if the limit lim

ε→0 + ˆRN |u(x + εn) -u(x)| q ε dx (9.2)
exists for H N -1 -almost every n ∈ S N -1 , then the limit in (9.1) exists by Dominated Convergence Theorem: Since u ∈ B 1/q q,∞ (R N , R d ), then we get by Definition 2.1 that

sup n∈S N -1 sup ε∈(0,∞) ˆRN |u(x + εn) -u(x)| q ε dx ≤ [u] q B 1/q q,∞ (R N ,R d ) < ∞, (9.3)
so by Dominated Convergence Theorem we have the existence of the limit in (9.1) and

lim ε→0 + ˆSN-1 ˆRN |u(x + εn) -u(x)| q ε dxdH N -1 (n) = ˆSN-1 lim ε→0 + ˆRN |u(x + εn) -u(x)| q ε dxdH N -1 (n). (9.4) Question 9.2. Let 1 < q < ∞, u ∈ B 1/q q,∞ (R N , R d ), η ∈ W 1,1 (R N ). Does the following inequality hold? lim inf ε→0 + 1 | ln ε| [u ε ] q W 1/q,q (R N ,R d ) ≥ ˆRN η(z)dz q ˆSN-1 |z 1 | dH N -1 (z) ˆJu u + (x) -u -(x) q dH N -1 (x). (9.5) Question 9.3. Let 1 < q < ∞, u ∈ L q (R N , R d ).
Does the following implication hold?

∀η ∈ W 1,1 (R N ), lim sup ε→0 + 1 | ln ε| [u ε ] q W 1/q,q (R N ,R d ) < ∞ =⇒ u ∈ B 1/q q,∞ (R N , R d ). (9.6) Theorem 9.1. (Theorem 1.3 in [9]) Let 1 ≤ q < ∞, Ω ⊂ R N be an open set and u ∈ L 1 loc (Ω, R d ). Then, 1 N ˆSN-1 |z 1 | dH N -1 (z) ˆJu |u + (x) -u -(x)| q dH N -1 (x) ≤ lim inf ε→0 + ˆΩ ˆΩ∩Bε(x) 1 ε N |u(x) -u(y)| q |x -y| dy dx. (9.7)
Remark 9.1. If the limit in (9.1) exists, then the answer on the other questions is yes: If the limit in (9.1) exists, then we get (9.5) from equation (7.47), Theorem 9.1 and Proposition 10.3; and we get (9.6) from equation (7.47) and Theorem 4.1.

Question 9.4. Assume r ∈ (0, 1), q ∈ [1, ∞) and u ∈ B r q,∞ (R N , R d ). Does the following limit hold? Let (X, E, σ) be a measure space, which means that X is a set, E is a sigma-algebra on X and σ : E → [0, ∞] is a measure. Assume that E ∈ E is such that σ(E) < ∞. Assume {E α } α∈I is a family of sets, where I is a set of indexes, such that for every α ∈ I, E α ⊂ E,E α ∈ E, and E α ∩ E α = ∅ for every different α, α ∈ I. Define the set

F := α ∈ I : σ(E α ) > 0 . (10.1)
Then, F is at most countable.

Proof. Let us decompose

F = ∪ k∈N F k , F k := α ∈ I : σ(E α ) > 1 k . For each k ∈ N the set F k is finite. Otherwise, there exists a sequence {α j } j∈N ⊂ F k of different elements such that ∞ > σ(E) ≥ σ j∈N E α j = j∈N σ(E α j ) ≥ j∈N 1 k = ∞. (10.2)
This contradiction shows that each F k is a finite set and hence F is at most countable set as a countable union of finite sets.

Lemma 10.2. (The Compact Negligible Boundary Property) Let (X, d) be a locally compact metric space and let µ be a positive Borel measure on X which is finite on compact sets. Then for every compact set K ⊂ X there exists a compact set E ⊂ X such that K ⊂ E and µ(∂E) = 0.

Proof. Since K is compact and X is locally compact, then there exists an open set W such that K ⊂ W and W is compact, where W is the topological closure of W . Note that since ∂W ⊂ X \W , then d (∂W, K) ≥ d (X \ W, K). Since K is compact and X \ W is closed and K ∩ (X \ W ) = ∅, then d (X \ W, K) > 0. Therefore, D := d (∂W, K) > 0. For each ε ∈ (0, ∞) we define a set

W ε := x ∈ W : d(x, ∂W ) ≥ ε . (10.3)
∂W ε = W ε \ W o ε ⊂ W ε \ x ∈ W : d(x, ∂W ) > ε = x ∈ W : d(x, ∂W ) = ε , (10.4
∂Ω ε = Ω ε \ Ω ε ⊂ x ∈ X : d(x, C) ≤ ε \ Ω ε = x ∈ X : d(x, C) = ε . (10.6)
Therefore, for every different ε 1 , ε 2 ∈ (0, ∞) we get ∂Ω ε 1 ∩ ∂Ω ε 2 = ∅. Thus, we get by Lemma 10.1 for the family {∂Ω ε } ε∈(0,∞) the existence of an infinitesimal sequence ε k ∈ (0, ∞) such that µ(∂Ω ε k ) = 0. Since C is closed we have C = k∈N Ω ε k .

Proposition 10.1. (Extremal Sets for Essential Infimum and Supremum) Let X be a set and µ be a positive measure on X. Let f : X → R be a µ-measurable function. Assume that K ⊂ X is a set with the following two properties:

1. µ(X \ K) = 0; 2. For every σ ∈ (0, ∞) and x 0 ∈ K, µ ({x ∈ X | |f (x) -f (x 0 )| < σ}) > 0.
f (x) ≥ inf x∈K f (x). Suppose, by contra- diction, that inf x∈K\Θ f (x) > inf x∈K f (x). This implies that inf x∈K f (x) = inf x∈Θ f (x). Otherwise, if inf x∈K f (x) < inf x∈Θ f (x), then inf x∈K f (x) = min inf x∈Θ f (x), inf x∈K\Θ f (x) > inf x∈K f (x),
which leads to a contradiction.

Therefore, for any ε ∈ (0, ∞), there exists

x 0 ∈ Θ such that f (x 0 ) -inf x∈K f (x) < ε 2 . By property 2 of K, there exists y ∈ K \ Θ such that |f (y) -f (x 0 )| < ε 2 . Hence, 0 < inf x∈K\Θ f (x) -inf x∈K f (x) = f (x 0 ) -inf x∈K f (x) + (f (y) -f (x 0 )) + inf x∈K\Θ f (x) -f (y) < ε.
(10.10) Since is arbitrarily small, we arrive at a contradiction, which proves that inf x∈K\Θ f (x) = inf x∈K f (x). The proof of of formula ess sup x∈X f (x) = sup x∈K f (x) is similar. Let X be a metric space, and let µ be a Borel measure on X such that 0 < µ(B r (x)) < ∞ for every r ∈ (0, ∞) and every x ∈ X. Suppose p ∈ [1, ∞) and f ∈ L p (X). Then, there exists a set K ⊂ X with properties 1 and 2 as outlined in Proposition 10.1. More precisely, the set of Lebesgue points of f possesses these properties.

Proof. Since f ∈ L p (X), by the Lebesgue Differentiation Theorem, we know that almost every point in X is a Lebesgue point of f with respect to µ. Let us denote this set by K. Therefore, we have property 1: µ(X \ K) = 0. To establish property 2, let x 0 ∈ K and α ∈ (0, 1). Note that for an arbitrary positive number σ, there exists R such that

ˆBR (x 0 ) |f (x) -f (x 0 )| p dµ(x) < ασ p µ (B R (x 0 )) . (10.11)
By Chebyshev's inequality

µ ({x ∈ B R (x 0 ) | |f (x) -f (x 0 )| > σ}) µ(B R (x 0 )) ≤ 1 σ p B R (x 0 ) |f (x) -f (x 0 )| p dµ(x) < α. (10.12)
Since f is µ-measurable, we obtain

µ ({x ∈ B R (x 0 ) | |f (x) -f (x 0 )| > σ}) µ(B R (x 0 )) + µ ({x ∈ B R (x 0 ) | |f (x) -f (x 0 )| ≤ σ}) µ(B R (x 0 )) = 1. (10.13) Therefore, µ ({x ∈ B R (x 0 ) | |f (x) -f (x 0 )| ≤ σ}) µ(B R (x 0 )) ≥ 1 -α, (10.14)
and hence, Let µ be a positive measure on the measurable space (X, E), X is a set and E is a σ-algebra on X. Let f ∈ L 1 (X, R N ). Then, the variation of the R N -valued measure f µ(B) := ˆB f dµ, B ∈ E (10.17 Let X be a set, E be a σ-algebra on X and µ : E → R N be a measure. Let f : X → R be such that f ∈ L 1 (X, µ ). Then,

µ ({x ∈ X | |f (x) -f (x 0 )| ≤ σ}) ≥ (1 -α)µ(B R (x 0 )) > 0. ( 10
f µ (E) ≤ N 1/2 |f | µ (E), E ∈ E. (10.19)
Proof. Let us denote µ := (µ 1 , ..., µ N ). For every Proposition 10.2. For every

E ∈ E |f µ(E)| = |(f µ 1 (E), ..., f µ N (E))| = N j=1 (f µ i (E)) 2 1/2 ≤ N j=1 (|f | µ i (E)) 2 1/2 ≤ N 1/2 |f | µ (E). (10.20) Therefore, f µ (E) = sup j∈N |f µ(E j )| : E j ∈ E pairwise disjoint, E = ∪ j∈N E j ≤ N 1/2 sup j∈N |f | µ (E j ) : E j ∈ E pairwise disjoint, E = ∪ j∈N E j = N 1/2 |f | µ (E). ( 10
v 1 , v 2 ∈ S N -1 we have ˆSN-1 |v 1 • n| dH N -1 (n) = ˆSN-1 |v 2 • n| dH N -1 (n). (10.22)
Proof. Take an isometry A :

R N → R N such that A(v 2 ) = v 1 . Then, ˆSN-1 |v 1 • n| dH N -1 (n) = ˆA-1 (S N -1 ) |A(v 2 ) • A(w)| dH N -1 (w) = ˆSN-1 |v 2 • w| dH N -1 (w).
(10.23) Therefore, we get from (10.29) for every natural m > 1

A 2m = 1 2m -1 A 2 and A 2m+1 = 2 2m A 3 . (10.31)
Let us calculate A 2 , A 3 separately. Note that

A 3 := ˆ∞ 0 2rdr 2 1 + r 2 2 = - 1 2 1 + r 2 r=∞ r=0 = 1 2 . (10.32)
Let us prove that

A 2 := ˆ∞ 0 1 √ 1 + r 2 3 dr = 1.
(10.33)

Changing variables r = z 2 -1 2z in the last integral gives: Thus, for every natural N > 1

A 2 = ˆ∞ 1 1 1 + z 2 -1 2z 2 3 1 2 + 1 2z 2 dz = ˆ∞ 1 1 1 + 1 4z 2 z 2 -1 2 3 1 2z 2 z 2 + 1 dz = ˆ∞ 1 1 4z 2 + z 2 -1 2 3 4z z 2 + 1 dz = ˆ∞ 1 1 z 2 + 1 2 3 4z z 2 + 1 dz = ˆ∞ 1 4z (z 2 + 1) 2 dz = - 2 z 2 + 1
A N = 1 N -1 . (10.36)
Therefore, by (10.27), (10.28), (10.36) and polar coordinates we get for every N > 1 ˆRN-1 dv

1 + |v| 2 N +1 = H N -2 (S N -2 ) 1 N -1 = L N -1 B N -1 1 (0) . ( 10
≤ i ≤ d, i ∈ N, l ∈ [0, ∞) and x ∈ E we define u i l (x) := l ∧ (-l ∨ u i (x))
, where a ∧ b := min{a, b}, a ∨ b := max{a, b}, for a, b ∈ R; and we define u l (x) := (u 1 l (x), ..., u d l (x)). We call the family of functions {u l } l∈[0,∞) the truncated family obtained by u. Proof. For x ∈ R, l ∈ [0, ∞) we define x l := l ∧ (-l ∨ x). Notice that for every x, y ∈ R and l, m ∈ [0, ∞), l < m, we have |x l -y l | ≤ |x m -y m |. For a point x = (x 1 , ..., x d ) ∈ R d , and l ∈ [0, ∞) we define x l := (x 1 l , ..., x d l ). Similarly, we have for every x, y ∈ R d and l, m ∈ [0, ∞), l < m, the inequality |x l -y l | ≤ |x m -y m |. Notice also that for x ∈ R d , the family {x l } l∈[0,∞) has the property lim l→∞ x l = x. In particular, for every x, y ∈ R d , the family {|x l -y l |} l∈[0,∞) is monotone increasing to |x -y|. Therefore, we get items 1,2 and 3 by choosing the points u(x), u(y) in place of the points x, y. The set S u of points where this property does not hold is called the approximate discontinuity set.

For any x ∈ Ω the point z, uniquely determined by (10.46), is called the approximate limit of u at x and denoted by ũ(x).

∈ L 1 loc (Ω, R d ). (a) S u is a Borel set, L N (S u ) = 0 and ũ : Ω \ S u → R d is a Borel function, coinciding L N -almost everywhere in Ω \ S u with u; (b) if f : R d → R p is a Lipschitz map and v = f • u, then S v ⊂ S u and ṽ(x) = f (ũ(x)) for any x ∈ Ω \ S u .
Proposition 10.9. (Properties of One-Sided Approximate Limits, Proposition 3.69 in [START_REF] Ambrosio | Functions of Bounded Variation and Free Discontinuity Problems[END_REF])

Let Ω ⊂ R N be an open set and u ∈ L 1 loc (Ω, R d ). (a) The set J u is a Borel subset of S u and there exist Borel functions

u + , u -, ν u : J u → R d × R d × S N -1 (10.50)
such that for every x ∈ J u we have

lim ρ→0 + B + ρ (x,νu(x))
|u(y) -u + (x)|dy = 0, lim

ρ→0 + B - ρ (x,νu(x)) |u(y) -u -(x)|dy = 0. (10.51) (b) if f : R d → R p is a Lipschitz map, v = f • u and x ∈ J u , then x ∈ J v if and only if f (u + (x)) = f (u -(x))
, and in this case For each l ∈ [0, ∞), let us define the l-truncated function by T l : R d → R d , T l (x) := x l , where x l is defined as in the proof of Proposition 10.5. Then we have the following assertions: 1. T l is a Lipschitz map; 2. The jumps set of u can be decomposed in terms of the jump sets of T l • u through the formula:

v + (x), v -(x), ν v (x) = f (u + (x)), f (u -(x)), ν u (x) . (10.52) Otherwise, x / ∈ S v and ṽ(x) = f (u + (x)) = f (u -(x)).
J u = l∈[0,∞) J T l •u ∩ J u ;
(10.53)

3. For every l, m ∈ [0, ∞) such that l ≤ m we have the following monotonicity property:

J T l •u ∩ J u ⊂ J Tm•u ∩ J u . ( 10
→ R d , T l (x) := x l , is Lipschitz. 2. For u ∈ L 1 loc (Ω, R d ),
where Ω ⊂ R N is an open set, and x ∈ J u , we know by Proposition 10.9 that x ∈ J T l •u if and only if T l (u + (x)) = T l (u -(x)), and in this case

(T l • u) + (x), (T l • u) -(x), ν T l •u (x) = T l (u + (x)), T l (u -(x)), ν u (x) ;
(10.55) and if T l (u + (x)) = T l (u -(x)), then x / ∈ S T l •u . Thus, since for every x ∈ J u there exists a big enough

l ∈ [0, ∞) such that T l (u + (x)) = u + (x) = u -(x) = T l (u -(x)), we have J u = l∈[0,∞) J T l •u ∩ J u .
(10.56)

we have for every

l, m ∈ [0, ∞), l ≤ m, that J T l •u ∩ J u ⊂ J Tm•u ∩ J u : If x ∈ J T l •u ∩ J u , then T l (u + (x)) = T l (u -(x)) and so T m (u + (x)) = T m (u -(x)). If not, then T m (u + (x)) = T m (u - (
x / ∈ S T l •u . It is a contradiction since x ∈ J T l •u ⊂ S T l •u . From T m (u + (x)) = T m (u -(x)) and x ∈ J u we get x ∈ J Tm•u ∩ J u .
Lemma 10.7. (Lower Semi-Continuity for Jump-Integral with respect to the Truncated Family) Let Ω ⊂ R N be an open set, u ∈ L 1 loc (Ω, R d ), h : R N → R be a non-negative, H N -1 -measurable function and F : R → R be a non-negative continuous function. Then,

lim inf l→∞ ˆJu l F |(u l ) + (x) -(u l ) -(x)| h(x)dH N -1 (x) ≥ ˆJu F |u + (x) -u -(x)| h(x)dH N -1 (x), (10.57)
where {u l } l∈[0,∞) is the truncated family obtained by u.

Proof. By Proposition 10.10 we obtain

ˆJu l F |(u l ) + (x) -(u l ) -(x)| h(x)dH N -1 (x) ≥ ˆJu l ∩Ju F |(u l ) + (x) -(u l ) -(x)| h(x)dH N -1 (x) = ˆJu l ∩Ju F |(u + (x)) l -(u -(x)) l | h(x)dH N -1 (x) = ˆJu χ Ju l ∩Ju (x)F |(u + (x)) l -(u -(x)) l | h(x)dH N -1 (x). ( 10
|(u + (x)) l -(u -(x)) l | = |u + (x) -u -(x)|, ∀x ∈ J u .
Taking the lower limit as l → ∞ on both sides of (10.58) and using Fatou's lemma we obtain (10.57).

lim l→∞ ˆJu l F |(u l ) + (x) -(u l ) -(x)| h(x)dH N -1 (x) = ˆJu F |u + (x) -u -(x)| h(x)dH N -1 (x). (10.60) 2. For every n ∈ R N lim l→∞ ˆJu l F |(u l ) + (x) -(u l ) -(x)| |ν u l (x) • n|h(x)dH N -1 (x) = ˆJu F |u + (x) -u -(x)| |ν u (x) • n|h(x)dH N -1 (x). (10.61)
Proof. Let us prove assertion 1. Since u ∈ BV loc (Ω, R d ), then for every l ∈ [0, ∞) we have by chain rule for BV -functions (refer to Theorem 10.5) that u l ∈ BV loc (Ω, R d ), and by Federer-Vol'pert theorem (refer to Theorem 10.2) we have

H N -1 (S u \ J u ) = H N -1 (S u l \ J u l ) = 0. Therefore, H N -1 (J u l \ J u ) = H N -1 (S u l \ S u ) = 0, because S u l ⊂ S u . Therefore, by item (b) of Proposition 10.9 we get ˆJu l F |(u l ) + (x) -(u l ) -(x)| h(x)dH N -1 (x) = ˆJu l ∩Ju F |(u l ) + (x) -(u l ) -(x)| h(x)dH N -1 (x) + ˆJu l \Ju F |(u l ) + (x) -(u l ) -(x)| h(x)dH N -1 (x) = ˆJu l ∩Ju F |(u + (x)) l -(u -(x)) l | h(x)dH N -1 (x) = ˆJu χ Ju l ∩Ju (x)F |(u + (x)) l -(u -(x)) l | h(x)dH N -1 (x). ( 10
.62) By Proposition 10.5, Proposition 10.10 and monotone convergence theorem we get (10.60) by taking the limit as l → ∞ on both sides of (10.62).

For assertion 2, note that, by item (b) of Proposition 10.9 we get ˆJu Theorem 10.2. (Federer-Vol'pert Theorem, Theorem 3.78 in [START_REF] Ambrosio | Functions of Bounded Variation and Free Discontinuity Problems[END_REF])

l F |(u l ) + (x) -(u l ) -(x)| |ν u l (x) • n|h(x)dH N -1 (x) = ˆJu l ∩Ju F |(u l ) + (x) -(u l ) -(x)| |ν u l (x) • n|h(x)dH N -1 (x) + ˆJu l \Ju F |(u l ) + (x) -(u l ) -(x)| |ν u l (x) • n|h(x)dH N -1 (x) = ˆJu l F |(u l ) + (x) -(u l ) -(x)| |ν u (x) • n|χ Ju (x)h(x)dH N -1 (x). ( 10
Let Ω ⊂ R N be an open set, and u ∈ BV loc (Ω, R d ). Then, the jump set J u is countably (N -1)-rectifiable set, oriented with the jump vector ν u (x), and moreover, we have H N -1 S u \ J u = 0. In particular, S u is σ-finite with respect to H N -1 . Therefore, since for every natural 1 ≤ i ≤ d we have that u i ∈ L 1 (Ω, R), then we get by (10.105) that u i ∈ BV (Ω, R). Therefore, we obtain (10.85) from (10.106) taking the limit as l goes to infinity. The convergence of u l to u as l → ∞ in the norm of the space BV (Ω, R d ) follows from Lemma 10.10 with p = 1 and (10.85).

2 6 3

 6 Estimates for Gagliardo Seminorm of Mollified Besov Functions in Terms of Besov Seminorm Continuity of G-Functionals 13 4 B r,q -

. 1 )

 1 Definition 2.2. (Mollification and Mollifier)Let η : R N → R be a function. For each ε

Definition 3 . 1 .

 31 (The Upper and Lower G-Functionals)

Lemma 3 . 1 .

 31 (Continuity of the Upper and Lower G-Functionals)

. 2 )

 2 Definition 4.2. (The Space B r,q )

. 2 )

 2 Proposition 5.1. (Properties of the Logarithmic Kernel)

Definition 5 . 3 .

 53 (The Trivial Kernel) Let us define the N -dimensional trivial kernel, or just trivial kernel, to be ρε (r) :=

Corollary 7 . 1 .

 71 (Besov Constants Bounded by Besov Seminorms) Let

. 8 )

 8 Remark 8.2. (Variation Negligibility of Boundaries of Sets)The purpose of this remark is to explain the condition Du (∂K) = 0 in Theorem 8.1.

  be a compact set such that Du (∂K) = 0, where B o is the topological interior of B. By Theorem 8.1 we obtain

1 .

 1 lim l→∞ sup h∈R N \{0} ˆ{x∈R N | |u(x)|>l} |u(x + h) -u(x)| q|h| rq dx = (Countability of Measurable Sets with Finite Measure)

  Note that K ⊂ W ε for every ε ∈ (0, D). If ∂W = ∅, then we can choose E = W , because E is compact and since ∂E ⊂ ∂W = ∅, then µ(∂E) = 0. So we can assume that ∂W = ∅. Notice that for a general non-empty set S ⊂ X, the map f (x) := d(x, S), f : X → [0, ∞) is Lipschitz and so continuous. Thus, the set W ε is a closed set. Since W ε is a subset of the compact set W , then it is compact. For every different ε, ε ∈ (0, D) we have ∂W ε ∩ ∂W ε = ∅: since W is open and the distance function f is continuous, then W ∩ x ∈ X : d(x, ∂W ) > ε is an open set, and it is a subset of W ε . Therefore, x ∈ W : d(x, ∂W ) > ε ⊂ W o ε , where W o ε is the topological interior of W ε . Hence,

  ) and the sets x ∈ W : d(x, ∂W ) = ε are disjoint for different numbers ε. Using Lemma 10.1 with the family of sets {∂W ε } ε∈(0,D) ⊂ W , µ(W ) < ∞, we derive the existence of ε ∈ (0, D) such that µ(∂W ε ) = 0. We choose E := W ε .Lemma 10.3. (The Open Negligible Boundary Property)Let (X, d) be a metric space and let µ be a finite positive Borel measure on X. Let C ⊂ X be a closed set. Then, there exists a monotone decreasing sequence of open sets Ω k ⊂ X such that for every k ∈ N µ(∂Ω k ) = 0, and C = k∈N Ω k .Proof. Define for every ε ∈ (0, ∞)Ω ε := x ∈ X : d(x, C) < ε . (10.5) Assume that C = ∅; if C = ∅, then we can choose Ω k = ∅. Since the function x -→ d(x, C) is continuous, then Ω ε is an open set. We have

7 ) 9 )

 79 We call K an extremal set for the function f . Let us consider a set Θ ⊂ K such that µ(Θ) = 0. We aim to show that inf x∈K\Θ f (x) = inf x∈K f (x). By taking the supremum over all such Θ, we obtain ess inf x∈K f (x) = inf x∈K f (x), and hence ess inf x∈X f (x) = inf x∈K f (x).It follows from the definition of infimum that inf x∈K\Θ

Corollary 10 . 1 .

 101 (Existence of Extremal Sets for Lebesgue Functions)

.15) 10 . 2

 102 Vector Valued Measures and Variation Definition 10.1. (Vector Valued Measures and Variation)Let X be a set and E be a σ-algebra on X. Let µ : E → R d be a measure, which means that µ(∅) = 0 and for any sequence {E j } j∈N ⊂ E of pairwise disjoint sets we have µ j∈N E j = j∈N µ (E j ). The variation of µ is defined to beµ (E) := sup j∈N |µ(E j )| : E j ∈ E pairwise disjoint, E = j∈N E j , E ∈ E. (10.16) Lemma 10.4. (Variation of Multiplication of a Vector Valued Function with Positive Measure, Proposition 1.23 in [1])

  ) satisfies f µ (B) = ˆB |f |dµ, B ∈ E. (10.18) Lemma 10.5. (Variation of Multiplication of Scalar Function with Vector Valued Measure)

.21) 10 . 3

 103 Aspects of Integration on S N -1 with respect to H N -1

  .37) Thus, by(10.26) and (10.37) we get(10.24). Proposition 10.4. (Polar coordinates, see 3.4.4 in[START_REF] Evans | Measure theory and fine properties of functions[END_REF])Let g ∈ L 1 R N , R d . Then ˆRN g(x)dx = ˆ∞ 0 ˆ∂Br(0) g(z)dH N -1 (z) dr = ˆ∞ 0 r N -1 ˆSN-1 g(rz)dHN -1 (z) dr. (10.38) 10.4 Sequences of Real Numbers Lemma 10.6. (Liminfsup Lemma) Let {a k} ∞ k=1 , {b k } ∞ k=1 ⊂ R be bounded sequences. Then, max | lim inf k→∞ a k -lim inf k→∞ b k |, | lim sup k→∞ a k -lim sup k→∞ b k | ≤ lim sup k→∞ |a k -b k |. (10.39) Proof. Recall the general inequalities: lim sup k→∞ (a k + b k ) ≤ lim sup k→∞ a k + lim sup k→∞ b k , (10.40) lim inf k→∞ (a k + b k ) ≤ lim sup k→∞ a k + lim inf k→∞ b k . (10.41) By (10.40) we get lim sup k→∞ a k = lim sup k→∞ (a k -b k + b k ) ≤ lim sup k→∞ (a k -b k ) + lim sup k→∞ b k . (10.42) Changing the roles of a k and b k , we get lim sup k→∞ a k -lim sup k→∞ b k ≤ lim sup k→∞ |a k -b k |. (10.43) By (10.41) we get lim inf k→∞ a k = lim inf k→∞ (a k -b k + b k ) ≤ lim sup k→∞ (a k -b k ) + lim inf k→∞ b k . (10.44) Changing the roles of a k and b k , we get lim inf k→∞ a k -lim inf k→∞ b k ≤ lim sup k→∞ |a k -b k |. (10.45) 10.5 The Truncated Family Definition 10.2. (Truncated Family) Let E ⊂ R N be a set and let u : E → R d , u = (u 1 , ..., u d ) be a function. For every 1

Proposition 10 . 5 .

 105 (Properties of the Truncated Family) Let E ⊂ R N be a set and let u : E → R d , u = (u 1 , ..., u d ) be a function. Let {u l } l∈[0,∞) be the truncated family obtained by u. Then, 1. lim l→∞ u l (x) = u(x), ∀x ∈ E; 2. For every x, y ∈ E and l, m ∈ [0, ∞), l ≤ m, we have |u l (x) -u l (y)| ≤ |u m (x) -u m (y)| ≤ |u(x) -u(y)|; 3. For every x, y ∈ E, the family {|u l (x) -u l (y)|} l∈[0,∞) is monotone increasing to |u(x) -u(y)|.

Definition 10 . 4 .

 104 (Approximate Jump Points) Let Ω ⊂ R N be an open set, u ∈ L 1 loc (Ω, R d ) and x ∈ Ω. We say that x is an approximate jump point of u if and only if there exist different a, b ∈ R d and ν ∈ S N -1 such that limρ→0 + 1 ρ N ˆB+ ρ (x,ν) |u(z) -a|dz + ˆBρ (x,ν) |u(z) -b|dz = 0, (10.47)whereB + ρ (x, ν) := y ∈ B ρ (x) : (y -x) • ν > 0 , B - ρ (x, ν) := y ∈ B ρ (x) : (y -x) • ν < 0 . (10.48)The triple (a, b, ν), uniquely determined by (10.47) up to a permutation of (a, b) and the change of sign of ν, is denoted by (u+ (x), u -(x), ν u (x)). The set of approximate jump points is denoted by J u . Note that J u ⊂ S u . Definition 10.5. (Approximate Differentiability, definition 3.70 in [1]) Let Ω ⊂ R N be an open set and let u ∈ L 1 loc (Ω, R d ). Let x ∈ Ω \ S u . We say that u is approximately differentiable at x if there exists a d × N matrix L such that lim ρ→0 + Bρ(x) |u(y) -ũ(x) -L(y -x)| ρ dy = 0. (10.49) If u is approximately differentiable at x, the matrix L, uniquely determined by (10.49), is called the approximate differential of u at x and denoted by ∇u(x). The set of approximate differentiability points of u is denoted by D u . Proposition 10.6. (Properties of Approximate Differential, Proposition 3.71 in [1]) Let Ω ⊂ R N be an open set and let u ∈ L 1 loc (Ω, R d ). Then, D u is a Borel set and the map ∇u : D u → R dN is a Borel map. Proposition 10.7. (Locality Properties of Approximate Differential, Proposition 3.73 in [1]) Let Ω ⊂ R N be an open set, u, v ∈ L 1 loc (Ω, R d ). If x ∈ D u ∩ D v and the set {u = v} has density 1 at x, then ∇u(x) = ∇v(x). In particular, ∇u(x) = ∇v(x) for L N -almost every x ∈ {u = v} ∩ D u ∩ D v . Proposition 10.8. (Properties of Approximate Limits, Proposition 3.64 in [1]) Let Ω ⊂ R N be an open set and u

Proposition 10 . 10 .

 1010 (Truncation and Jumps) Let Ω ⊂ R N be an open set, and let u ∈ L 1 loc (Ω, R d ).

.54) Proof. 1 .

 1 For each l ∈ [0, ∞), by Proposition 10.5 we get that the map T l : R d

  x)) and then x / ∈ S Tm•u , and since T l • (T m • u) = T l • u, then, by part (b) of Proposition 10.8 with T l in place of f and T m • u in place of u, we obtain S T l •u ⊂ S Tm•u and so

10. 7

 7 Aspects of BV -Functions Definition 10.6. (Definition of BV Functions) Let Ω ⊂ R N be an open set. We say that u ∈ BV (Ω, R d ) if and only if u ∈ L 1 (Ω, R d ) and there exists an d × N matrix valued measure µ :B(Ω) 3 → R d×N such that for every ϕ ∈ C ∞ c (Ω) it follows that ˆΩ u(x)∇ϕ(x)dx = -ˆΩ ϕ(x)dµ(x).(10.59)In this case we denote µ := Du. In formula (10.59) we think about u as a column vector u = (u 1 , ..., u d ) T and ∇ϕ = (∂ 1 ϕ, ..., ∂ N ϕ). Lemma 10.8. (Continuity for Jump-Integral with respect to the Truncated Family) Let Ω ⊂ R N be an open set, u ∈ BV loc (Ω, R d ). Let h : R N → R be a non-negative H N -1measurable function, and F : R → R be a non-negative, monotone increasing function. Let {u l } l∈[0,∞) be the truncated family obtained by u. Then, 1.

  .63) Using item 1 with |ν u (x) • n|χ Ju (x)h(x) in place of h(x), we conclude (10.61).Theorem 10.1. (Calderón-Zygmund, Theorem 3.83 in[START_REF] Ambrosio | Functions of Bounded Variation and Free Discontinuity Problems[END_REF])Let Ω ⊂ R N be an open set. Any function u ∈ BV (Ω, R d ) is approximately differentiable at L Nalmost every point of Ω. Moreover, the approximate differential ∇u is the density of the absolutely continuous part of Du with respect to L N , in particular ∇u ∈ L 1 (Ω, R d×N ).

Lemma 10 . 9 .

 109 (Variation Inequality) Let Ω ⊂ R N be an open set and u ∈ BV (Ω, R d ). Let E ⊂ Ω be an L N -measurable set and let h ∈ R N \ {0}. Assume that dist(E, ∂Ω) > |h|. Then, ˆE |u(x + h) -u(x)| |h| dx ≤ Du (Ω). (10.64)In particular, if Ω = R N , thensup h∈R N \{0} ˆRN |u(x + h) -u(x)| |h| dx ≤ Du (R N ). (10.65) Proof. Let {u k } ∞ k=1 ⊂ C 1 (Ω, R d) be a sequence of functions which converges to u L N -almost everywhere and lim k→∞ Du k (Ω) = Du (Ω). Then, for every k ∈ N, by the fundamental theorem of calculus and Fubini's theorem we getˆE |u k (x + h) -u k (x)| |h| dx = ˆE | ´1 0 ∇u k (x + th) • hdt| |h| dx ≤ ˆ1 0 ˆE |∇u k (x + th)|dxdt = ˆ1 0 ˆE+th |∇u k (y)|dydt ≤ Du k (Ω). (10.66)Taking the lower limit as k → ∞ and using Fatou's Lemma we get (10.64). To get (10.65) note that for every h ∈ R N , dist(R N , ∅) = ∞ > |h|.

10. 8  1 ,

 81 Negligibility of Sets with respect to Du Definition 10.7. (Measure-theoretic Boundary)Let E ⊂ R N be a set. We write x ∈ ∂ * E if and only if the following two inequalities hold:lim sup ε→0 + L N (B ε (x) ∩ E) L N (B ε (x)) > 0, lim sup ε→0 + L N B ε (x) ∩ R N \ E L N (B ε (x)) > 0.(10.67)Equivalently, x ∈ ∂ * E if and only if E and its complement R N \ E do not have density 0 at x; if and only if the set E does not have density neither 0 nor 1. In other words, if we denote by E 0 the set of points at which E has density 0 and by E 1 the set of points at which E has density 1, namelyE 0 := x ∈ R N : lim ε→0 + L N (B ε (x) ∩ E) L N (B ε (x)) = 0 , E 1 := x ∈ R N : lim ε→0 + L N (B ε (x) ∩ E) L N (B ε (x)) = 1 , (10.68) then x ∈ ∂ * E if and only if x / ∈ E 0 ∪ E 1 .We call ∂ * E the measure-theoretic boundary of the set E.Theorem 10.3. (The Co-Area Formula for BV -Functions, see equation(3.63) in[START_REF] Ambrosio | Functions of Bounded Variation and Free Discontinuity Problems[END_REF])Let Ω ⊂ R N be an open set, and let u ∈ BV (Ω). Then, for every Borel set B ⊂ ΩDu (B) = ˆR H N -1 (B ∩ ∂ * {u > t}) dH 1 (t). (10.69) Proposition 10.11. (Variation-Negligibility of Sets with H 1 -Negligible Images) Let Ω ⊂ R N be an open set and u ∈ BV (Ω, R d ). Let B ⊂ Ω \ S u be a Borel set such that H 1 (ũ(B)) = 0. Then, Du (B) = 0.Proof. Assume first that d = 1. Let us first prove that for every t ∈ R we haveũ (Ω \ S u ) ∩ ∂ * z ∈ Ω : u(z) > t ⊂ {t}. (10.70)It means that the approximate limit ũ takes the measure-theoretic boundaries of super-level sets ∂ * z ∈ Ω : u(z) > t , which are outside S u , to the corresponding points t. We use the short notation {u > t} := z ∈ Ω : u(z) > t , as well as for similar sets. Assume that z 0 ∈ (Ω \ S u ) ∩ ∂ * {u > t}. Therefore, if ũ(z 0 ) < t, then for every ε ∈ (0, ∞) we have by Chebyshev's inequalityL N (B ε (z 0 ) ∩ {u > t}) L N (B ε (z 0 )) = L N (B ε (z 0 ) ∩ {u -ũ(z 0 ) > t -ũ(z 0 )}) L N (B ε (z 0 )) ≤ 1 t -ũ(z 0 ) Bε(z 0 ) |u(x) -ũ(z 0 )|dx. (10.71)Since z 0 ∈ Ω \ S u , then we get from (10.71) that the density of {u > t} at z 0 is zero, which contradicts the assumption that z 0 ∈ ∂ * {u > t}. Similarly, if ũ(z 0 ) > t, then for every ε ∈ (0, ∞) respect to L N . We define the jump part and the Cantor part of Du, respectively, to be the following measures:D j u := D s u J u , D c u := D s u (Ω \ S u ) . (10.76) Theorem 10.4. (Decomposition of Du into the Absolutely Continuous, Jump and Cantor Parts) Let Ω ⊂ R N be an open set, and let u ∈ BV (Ω, R d ). Then, Du = D a u + D j u + D c u, (10.77) where D a u, D j u, D c u are defined in Definition 10.8. They have the following properties: 1. D a u, D j u, D c u are finite Radon measures in Ω (it means that they are measures from B(Ω), the Borel σ-algebra, into R d×N , the set of all matrices of size d × N with entries from R); 2. They are orthogonal to each other; 3. It follows that:D a u = ∇uL N , D j u = u + -u -⊗ ν u H N -1 J u ,(10.78)where for points a = (a 1 , ...,a d ) ∈ R d , b = (b 1 , ..., b N ) ∈ R N we define a ⊗ b to be the d × N matrix given by (a ⊗ b) ij := a i b j . 4. We have Du = |∇u|L N + u + -u -H N -1 J u + D c u .(10.79)One can find proofs for the assertions of Theorem 10.4 in section 3.9 in[START_REF] Ambrosio | Functions of Bounded Variation and Free Discontinuity Problems[END_REF].Theorem 10.5. (Chain Rule in BV , Theorem 3.99 in[START_REF] Ambrosio | Functions of Bounded Variation and Free Discontinuity Problems[END_REF])Let Ω ⊂ R N be an open set. Let u ∈ BV (Ω, R) and let f : R → R be a Lipschitz function satisfyingf (0) = 0 if L N (Ω) = ∞. Then, v := f • u belongs to BV (Ω, R) and Dv = f (u)∇uL N + f (u + ) -f (u -) ν u H N -1 J u + f (ũ)D c u. (10.80) Remark 10.2. (Well-Definedness of Compositions in Chain Rule for BV -Functions) In this remark we would like to explain why f • u ∈ L 1 (Ω, |∇u|L N ) and f • ũ ∈ L 1 (Ω, D c u ). Let Ω ⊂ R N be an open set, and let u ∈ BV (Ω, R). Let f : R → R be a Lipschitz function. 1. By Rademacher's theorem there exists a Borel set Θ ⊂ R such that f is differentiable at every x ∈ R \ Θ and H 1 (Θ) = 0. 2. Since the approximate limit ũ : Ω \ S u → R is a Borel function, then ũ-1 (Θ) ⊂ Ω \ S u is a Borel set. 3. Therefore, we get by Proposition 10.11 that Du (ũ -1 (Θ)) = 0. 4. Since f is Lipschitz, then its L 1 -almost everywhere derivative f : R \ Θ → R is a Borel function. Therefore, the composition f • ũ : Ω \ (S u ∪ ũ-1 (Θ)) → R is a Borel function. 5. Since by Remark 10.1 we have D c u (S u ) = 0, then f • ũ is defined almost everywhere in Ω with respect to the measure D c u . Since D c u is a Borel measure, then f • ũ is a measurable function with respect to the measure D c u . Since f is Lipschitz and u∈ BV (Ω, R), then ˆΩ |f (ũ(x))|d D c u (x) ≤ f L ∞ (R) D c u (Ω) ≤ f L ∞ (R) Du (Ω) < ∞.(10.81) 10.2) we have u l = f l • u. By the chain rule for BV -functions (refer to Theorem 10.5) we have u l ∈ BV (Ω, R) andDu l = f l (u)∇uL N + (f l (u + ) -f l (u -))ν u H N -1 J u + f l (ũ)D c u. (10.87) By (10.86), (10.87) and Remark 10.2 we haveD(u -u l ) = (1 -f l (u)) ∇uL N + (u + -u -) -(f l (u + ) -f l (u -)) ν u H N -1 J u + (1 -f l (ũ))D c u. (10.88)By Lemma 10.4 we get(1 -f l (u)) ∇uL N (Ω) = ˆΩ |1 -f l (u(x))| |∇u(x)| dL N (x) (10.89) and (u + -u -) -(f l (u + ) -f l (u -)) ν u H N -1 J u (Ω) = ˆJu (u + (x) -u -(x)) -(f l (u + (x)) -f l (u -(x))) dH N -1 (x). (10.90) Note that for getting (10.90) we use that |ν u | = 1 (refer to Definition 10.4). By Lemma 10.5 we get (1 -f l (ũ))D c u (Ω) ≤ N 1/2 ˆΩ |1 -f l (ũ(x))| d D c u (x), (10.91) where • stands for the variation (refer to Definition 10.1). Therefore, we get by (10.88), the triangle inequality of the variation, (10.89),(10.90) and (10.91) thatD(u -u l ) (Ω) ≤ ˆΩ |1 -f l (u(x))| |∇u(x)| dL N (x) + ˆJu (u + (x) -u -(x)) -(f l (u + (x)) -f l (u -(x))) dH N -1 (x) + N 1/2 ˆΩ |1 -f l (ũ(x))| d D c u (x). (10.92)For every l ∈ [0, ∞), from item (a) of Proposition 10.8 we get|∇u|L N x ∈ Ω : |u(x)| = l = |∇u|L N x ∈ Ω \ S u : |ũ(x)| = l ,(10.93)and from Proposition 10.11 we haveD c u x ∈ Ω \ S u : |ũ(x)| = l = |∇u|L N x ∈ Ω \ S u : |ũ(x)| = l = 0. (10.94)Note that for getting (10.94) we use the assumption that u is a scalar function in order to get thatH 1 (ũ (E l )) ≤ H 1 ({l, -l}) = 0, E l := x ∈ Ω \ S u : |ũ(x)| = l . (10.95)For every l ∈ (0, ∞) we havef l (z) :=   if |z| < l 0, if |z| > l .(10.96) By (10.93) and (10.96) we get for every l ∈ (0, ∞) thatˆΩ |1 -f l (u(x))| |∇u(x)| dL N (x) = ˆΩ |1 -f l (u(x))| d |∇u| L N (x) = ˆ x∈Ω:|u(x)|>l |1 -f l (u(x))| d |∇u| L N (x) + ˆ x∈Ω:|u(x)|=l |1 -f l (u(x))| d |∇u| L N (x) + ˆ x∈Ω:|u(x)|<l |1 -f l (u(x))| d |∇u| L N (x) =ˆ x∈Ω:|u(x)|>l d |∇u| L N (x). (10.97) By Calderón-Zygmund theorem (refer to Theorem 10.1), we have ∇u ∈ L 1 (Ω, R N ). Therefore, we get by (10.97) and the decreasing monotonicity of the measure |∇u| L N that lim l→∞ ˆΩ |1 -f l (u(x))| |∇u(x)| dL N (x) = 0. (10.98) Since u ∈ BV (Ω, R), then we get from Federer-Vol'pert Theorem (refer to Theorem 10.2) that S u is σ-finite with respect to H N -1 . Thus, by Proposition 10.12 and Remark 10.1 we have D c u (S u ) = 0. Therefore, by (10.94) and (10.96) we obtain ˆΩ |1 -f l (ũ(x))| d D c u (x) = ˆΩ\Su |1 -f l (ũ(x))| d D c u (x) = ˆ x∈Ω\Su:|ũ(x)|>l |1 -f l (ũ(x))| d D c u (x) + ˆ x∈Ω\Su:|ũ(x)|=l |1 -f l (ũ(x))| d D c u (x) + ˆ x∈Ω\Su:|ũ(x)|<l |1 -f l (ũ(x))| d D c u (x) = ˆ x∈Ω\Su:|ũ(x)|>l 1d D c u (x). (10.99) Note that since S u is a Borel set in Ω and ũ : Ω \ S u → R is a Borel function (refer to Proposition 10.8), then the sets x ∈ Ω\S u : |ũ(x)| > l , x ∈ Ω\S u : |ũ(x)| < l and x ∈ Ω\S u : |ũ(x)| = l are Borel sets in Ω, so they are measurable with respect to the measure D c u , because D c u is a Borel measure (refer to Theorem 10.4). Since D c u is a finite Borel measure in Ω, then we get by (10.99) and the decreasing monotonicity of the measure D c u that lim l→∞ ˆΩ |1 -f l (ũ(x))| d D c u (x) = lim l→∞ D c u x ∈ Ω \ S u : |ũ(x)| > l = D c u l∈N x ∈ Ω \ S u : |ũ(x)| > l = D c u (∅) = 0. (10.100) At last, by Proposition 10.5 we get lim l→∞(u + (x) -u -(x)) -(f l (u + (x)) -f l (u -(x))) = 0, x ∈ J u ; (10.101) (u + (x) -u -(x)) -(f l (u + (x)) -f l (u -(x))) ≤ 2 u + (x) -u -(x) , x ∈ J u . (10.102) By Theorem 10.4 we get u + -u -∈ L 1 (J u , H N -1 ). (10.103) Therefore, Dominated Convergence Theorem gives lim l→∞ ˆJu (u + (x) -u -(x)) -(f l (u + (x)) -f l (u -(x))) dH N -1 (x) = 0. (10.104) Equation (10.85) follows from (10.92), (10.98), (10.100) and (10.104) in case u ∈ BV (Ω, R). The general case, u ∈ BV (Ω, R d ), follows from the inequality max 1≤i≤d,i∈N Du i (Ω) ≤ Du (Ω) ≤ d i=1 Du i (Ω), (10.105) where u = (u 1 , ..., u d ). Indeed, note that by the definition of the truncated family, Definition 10.2, it follows that (u l ) i = (u i ) l for every natural 1 ≤ i ≤ d and l ∈ [0, ∞). Therefore, we get D(u -u l ) (Ω) ≤ d i=1 D(u -u l ) i (Ω) = d i=1 D(u i -(u i ) l ) (Ω). (10.106)

  Remark 8.3. (The Assumption of Bounded Variation in Lemma 8.2) In Lemma 8.2, the assumption that u has bounded variation cannot be dropped in general to obtain inequality (8.10). There are examples of functions in B 1 p ,p (R N , R d ) for which equation (8.10) does not hold. Examples can be found in

,

(8.10)

  10.6 Approximate Continuity and Differentiability of L 1 loc -functions Definition 10.3. (Approximate Limit) Let Ω ⊂ R N be an open set and u ∈ L 1 loc (Ω, R d ). We say that u has approximate limit at x ∈ Ω if and only if there exists z ∈ R d such that

	lim ρ→0 + Bρ(x)	|u(y) -z|dy = 0.	(10.46)

  .58) By Proposition 10.10 we have lim l→∞ χ Ju l ∩Ju (x) = χ Ju (x), ∀x ∈ J u , and by Proposition 10.5 we have lim l→∞

As usual, on the space of equivalent classes obtained by equality L N -almost everywhere.

The space of Hlder continuous functions with exponent r and compact support.

Borel sigma algebra

Proof. Note that (10.24) holds for N = 1. So we can assume that N > 1. Let B N -1 1 (0) be the ball of radius 1 around the origin in R N -1 . Define g : B N -1 1 (0) → R N , g(z 2 , ..., z N ) := (f (z 2 , ..., z N ), z 2 , ..., z N ) , f (z 2 , ..., z N ) := 1 -N j=2 z 2 j .

(10.25)

The image of g is S + := z = (z 1 , ..., z N ) ∈ S N -1 : z 1 > 0 . Denote z = (z 1 , z ), z := (z 2 , ..., z N ).

By the area formula

In addition, by polar coordinates we obtain for N ≥ 2 ˆRN-1 dv

(10.27)

Let us denote

(10.28)

Assume for the moment that N > 3. Integration by parts gives

We got a recursive sequence. Note for example

(10.30)

we have by Chebyshev's inequality

Since z 0 ∈ Ω \ S u , then we get from (10.72) that the density of {u ≤ t} at z 0 is zero, which contradicts the assumption that z 0 ∈ ∂ * {u > t}. We conclude that ũ(z 0 ) = t, which proves (10.70). By (10.70) we get that, if t / ∈ ũ(B), then B ∩ ∂ * {u > t} = ∅. We get from the co-area formula (Theorem 10.3) and the assumption H 1 (ũ(B)) = 0 that

In the general case, d ∈ N, let us denote u = (u 1 , ..., u d ). Notice that for every natural 1 ≤ j ≤ d we have S u j ⊂ S u , and for x ∈ S u we have by uniqueness of approximate limit (u j )(x) = (ũ) j (x). Therefore, Since D c u vanishes on sets which are σ-finite with respect to H N -1 , and any subset of such a set is also σ-finite with respect to H N -1 , then the variation D c u vanishes on sets which are σ-finite with respect to H N -1 (recall that a variation of a vector valued measure µ vanishes on a set if and only if µ vanishes on every subset of the set). Therefore, f • ũ ∈ L 1 (Ω, D c u ). 6. Without loss of generality assume that the L N -almost everywhere defined function u is defined on all of Ω and it is a Borel function. Let us denote by