Eduard Curcȃ

Some Bourgain-Brézis type solutions via complex interpolation

Keywords: Bourgain-Brézis solutions, Fourier multipliers, Divergence equation. MSC 2020 classification: 42B15, 42B35, 46B70 1

In 2002 Bourgain and Brézis proved that given a vector field

We prove several results of a similar nature in which we take into consideration the Fourier support of the solutions. For instance, in the case d ≥ 3 we prove the following: for any vector field

and u L ∞ ∩ Ḃd/p,p 2 v Ḃd/p,p q .

Our arguments rely on a version of the complex interpolation method combined with some ideas of Bourgain and Brézis.

∈ S (R d ) ∩ Ẇ 1,d R d such that div u = f, (1)
in the distributions sense on R d . Indeed, it sufices to set

u := ∇|∇| -2 f, (2)
and to use the fact that the components of ∇u are of the form R i R j u where R 1 , ..., R d are the Riesz transforms on R d ( R j ϕ(ξ) = (ξ j / |ξ|) ϕ(ξ), for any Schwartz ϕ). Since each R j is a Calderón-Zygmund operator, we easily get that each component of ∇u belongs to L d (R d ).

Note that the space Ẇ 1,d does not embed in L ∞ and hence, the solution in Ẇ 1,d provided by the expression (2) may fall outside L ∞ (see for instance the example given by L. Nirenberg in [START_REF] Bourgain | On the equation div Y = f and application to control of phases[END_REF]Remark 7,p. 400]). However, as it was shown 1 by Bourgain and Brézis (2002), the fact that (1) admits a (possibly another) solution u ∈ L ∞ (R d ) is a direct consequence of the Gagliardo embedding (W 1,1 (R d ) → L d (R d ), where d := d/(d -1)) (see [5, Proposition 1]).

Even more, Bourgain and Brézis have proved in [5, Theorem 1] the following striking fact: there exists a solution u to (1) that is simultaneously bounded and in the "right" Sobolev space Ẇ 1,d (R d ). In other words, there exists a vector field u ∈ L ∞ (R d ) ∩ Ẇ 1,d (R d ) which is a solution to [START_REF] Bahouri | Fourier Analysis and Nonlinear Partial Differential Equations[END_REF]. In the general case where d ≥ 2, this result was proved by an involved approximation argument using the Littlewood-Paley square function. We mention that the complicated construction used in [START_REF] Bourgain | On the equation div Y = f and application to control of phases[END_REF] can also be used in more general situations. By similar constructive methods, Bourgain and Brézis proved an analogue existence result for more general underdeterminated Hodge systems. Following the ideas in [START_REF] Bourgain | On the equation div Y = f and application to control of phases[END_REF] and [START_REF] Bourgain | New estimates for eliptic equations and Hodge type systems[END_REF], Bousquet, Mironescu and Russ ( [START_REF] Bousquet | A limiting case for the divergence equation[END_REF], 2014) and later Bousquet, Russ, Wang and Yung ( [START_REF] Bousquet | Approximation in fractional Sobolev spaces and Hodge systems[END_REF], 2017) provided generalizations of the Bourgain-Brézis results in the scale of Triebel-Lizorkin spaces. For instance, adapted to the case of the divergence equation, Theorem 2 in [START_REF] Bousquet | Approximation in fractional Sobolev spaces and Hodge systems[END_REF] gives us: Theorem 1. Suppose that 1 < p, q < ∞ and consider some vector field v ∈ S (R d ) ∩ Ḟ d/p,p q (R d ). Then, there exists a vector field u ∈ L ∞ (R d ) ∩ Ḟ d/p,p q (R d ) such that div u = div v, and u L ∞ ∩ Ḟ d/p,p q v Ḟ d/p,p q .

Remark 2. Note that by Calderón-Zygmund theory any (compactly supported) f ∈ Ḟ d/p-1,p q (R d ) can be written as the divergence of the vector field v = ∇ |∇| -2 f ∈ Ḟ d/p,p q (R d ). Hence, since Ḟ 1,d 2 = Ẇ 1,d , when p = d and q = 2, from Theorem 1 above we recover the result of Bourgain and Brézis.

A similar existence result holds for the scale of Besov spaces, the proof being technically the same as for Theorem 1. Since in this paper we are concerned more with the Besov version, we explicitely state it below: Theorem 3. Suppose that 1 < p, q < ∞ and consider some vector field v ∈ S (R d ) ∩ Ḃd/p,p q (R d ). Then, there exists a vector field u ∈ L ∞ (R d ) ∩ Ḃd/p,p q (R d ) that satisfies

div u = div v,
and such that u L ∞ ∩ Ḃd/p,p q v Ḃd/p,p q .

Remark 4. Throughout the paper we will call the given vector v the source and u will be called solution. A similar convention will also be applied to more general equations.

It is worth noticing that in the special case where d = 2 (and hence, p = 2) Bourgain and Brézis have found a much simpler proof of their existence result (see [START_REF] Bourgain | On the equation div Y = f and application to control of phases[END_REF]Section 4,p. 403]). In this case the proof is by duality and it is nonconstructive. Also, by similar methods, a proof was found by [START_REF] Mazya | Bourgain-Brezis type inequality with explicit constants[END_REF] for the case p = q = 2 of Theorem 1 (or equivalently of Theorem 3) (see [START_REF] Mazya | Bourgain-Brezis type inequality with explicit constants[END_REF]). Again, the proof is by duality and strikingly simple. (See also [START_REF] Mironescu | On some inequalities of Bourgain, Brézis, Maz'ya, and Shaposhnikova related to L 1 vector fields[END_REF] for some related discussions.) However, both approaches, namely that of Bourgain and Brézis in the case d = 2 and that of Mazya are based on L 2 -Fourier analysis arguments that are unlikely to be extended to the case where p = 2.

There is yet another situation of a different nature: Proposition 5. Let d ≥ 2 be an integer and consider some r ∈ (1, ∞). Then, for any vector field v ∈ Ḃd/r,r Indeed, since Ḃd/r,r

1 (R d ) → L ∞ (R d ) ∩ Ḃd/r,r 1 (R d
), it suffices to set u := v (see subsection 2.1 for the definition of Ḃd/r,r 1 (R d ) we use in this paper). Since Proposition 5 and Theorem 3 have much easier proofs than the constructive proof used in [START_REF] Bourgain | On the equation div Y = f and application to control of phases[END_REF] and in [START_REF] Bousquet | A limiting case for the divergence equation[END_REF], it would be interesting to find a way to "interpolate" between Proposition 5 and Theorem 3.

A naive interpolation strategy

One may try to interpolate in the following way. By the closed range theorem and the lifting property of the Besov spaces we can reformulate Proposition 5 and Theorem 3

g Ḃ-d/r+1,r ∞ ∼ ∇g Ḃ-d/r,r ∞ ∇g L 1 + Ḃ-d/r,r ∞ , (3)
and respectively,

g Ẇ -d/2,2 ∼ ∇g Ẇ -d/2,2 ∇g L 1 + Ẇ -d/2,2 , (4)
for any Schwartz function g with g vanishing in a neighborhood of 0.

For each Banach function space Y on R d denote by G(Y ) the space of all vector fields in Y that are gradients, i.e., G(Y

) := {g ∈ Y | curl g = 0} .
With this notation one can view ( 3) and ( 4) as the embeddings

G(L 1 + Ḃ-d/r,r ∞ ) → Ḃ-d/r+1,r ∞ ,
and respectively,

G(L 1 + Ẇ -d/2,2 ) → Ẇ -d/2+1,2 .
By complex interpolation (we may consider the real interpolation as well), we conclude that, for any θ ∈ (0, 1),

(G(L 1 + Ḃ-d/r,r ∞ ), G(L 1 + Ẇ -d/2,2 )) θ → ( Ḃ-d/r+1,r ∞ , Ẇ -d/2+1,2 ) θ . (5)
The right hand side of (5) can be easily computed explicitly (see for instance [START_REF] Bergh | Interpolation spaces. An introduction[END_REF]Chapter 6]):

( Ḃ-d/r+1,r ∞ , Ẇ -d/2+1,2 ) θ = Ḃ-d/p+1,p q ,
where 1/p = (1 -θ)/r + θ/2 and 1/q = (1 -θ)/1 + θ/2. Now we would like to have

G(L 1 + Ḃ-d/p,p q ) → (G(L 1 + Ḃ-d/r,r ∞ ), G(L 1 + Ẇ -d/2,2 )) θ , (6)
and combining this with (5), we would get via the closed range theorem the fact that for any vector field v ∈ Ḃd/p,p q (R d ) there exists another vector field u ∈ L ∞ (R d ) ∩ Ḃd/p,p q (R d ) of the same divergence as v. However, computing explicitly the left hand side of (5) or proving [START_REF] Bourgain | New estimates for eliptic equations and Hodge type systems[END_REF] only by interpolation theory is a quite difficult task. Naively we may have the following strategy for proving [START_REF] Bourgain | New estimates for eliptic equations and Hodge type systems[END_REF]. We can observe that, since we have the embeddings

L 1 , Ḃ-d/r,r ∞ → L 1 + Ḃ-d/r,r ∞ and L 1 , Ẇ -d/2,2 → L 1 + Ẇ -d/2,2 , we can conclude by interpolation that L 1 , Ḃ-d/p,p q → (L 1 + Ḃ-d/r,r ∞ , L 1 + Ẇ -d/2,2 ) θ or equivalently L 1 + Ḃ-d/p,p q → (L 1 + Ḃ-d/r,r ∞ , L 1 + Ẇ -d/2,2 ) θ .
Consequently,

G(L 1 + Ḃ-d/p,p q ) → G(L 1 + Ḃ-d/r,r ∞ , L 1 + Ẇ -d/2,2 θ ).
Hence, in order to obtain [START_REF] Bourgain | New estimates for eliptic equations and Hodge type systems[END_REF] it would be sufficient to have

G((L 1 + Ḃ-d/r,r ∞ , L 1 + Ẇ -d/2,2 ) θ ) → (G(L 1 + Ḃ-d/r,r ∞ ), G(L 1 + Ẇ -d/2,2 )) θ ,
or, equivalently, since the other embedding is trivial,

G((L 1 + Ḃ-d/r,r ∞ , L 1 + Ẇ -d/2,2 ) θ ) = (G(L 1 + Ḃ-d/r,r ∞ ), G(L 1 + Ẇ -d/2,2 )) θ . (7)
We can further reformulate this fact as

N ∩ (Y 0 , Y 1 ) θ = (N ∩ Y 0 , N ∩ Y 1 ) θ , (8)
where

Y 0 = L 1 + Ḃ-d/r,r ∞ , Y 1 = L 1 + Ẇ -d/2,2
and N is the spaces of the fields in Y 0 + Y 1 that are gradients, i.e, N := G(Y 0 + Y 1 ). The difficulty of proving [START_REF] Bousquet | A limiting case for the divergence equation[END_REF] consists in the fact that, for the general situation when Y 0 , Y 1 , N are Banach spaces the question whether or not (8) holds does not have yet a satisfactory answer. If we replace the complex interpolation method in [START_REF] Bousquet | Approximation in fractional Sobolev spaces and Hodge systems[END_REF] with the real K-method of interpolation, then, (8) may be false for some particular choice of the spaces Y 0 , Y 1 , N (see for instance [START_REF] Krugliak | The failure of the Hardy inequality and interpolation of intersections[END_REF]). In the case of the complex interpolation method it seems that even less is known when [START_REF] Bousquet | Approximation in fractional Sobolev spaces and Hodge systems[END_REF] is valid. This naive interpolation strategy seems inappropriate to prove [START_REF] Bourgain | New estimates for eliptic equations and Hodge type systems[END_REF] or even a weaker statement like

G(L 1 + Ḃ-d/p,p 1 ) → (G(L 1 + Ḃ-d/r,r ∞ ), G(L 1 + Ẇ -d/2,2 )) θ ,
which corresponds to the following existence result: Proposition 6. Let d ≥ 2 be an integer and consider some parameters p ∈ (2, ∞) and q ∈ (1, 2). Then, for any vector field

v ∈ S (R d )∩ Ḃd/p,p q (R d ) there exists a vector field u ∈ L ∞ (R d )∩ Ḃd/p,p ∞ (R d ) such that div u = div v, and
u L ∞ ∩ Ḃd/p,p ∞ v Ḃd/p,p q .

The main results

In this paper we take into consideration the spectrum of solutions. In what follows, the spectrum of a tempered distribution v is its Fourier support, i.e., spec(v) := supp v (see Remark 14 for a more general definition). Adapted to the case of the divergence equation, our main result reads:

Theorem 7. Let d ≥ 3 be an integer and consider the set

∆ := R d \(-∞, 0) d . Consider some parameters p ∈ [2, ∞) and q ∈ (1, 2). Then, for any vector field v ∈ S (R d ) ∩ Ḃd/p,p q (R d ), with spec(v) ⊆ ∆ there exists a vector field u ∈ L ∞ (R d ) ∩ Ḃd/p,p 2 (R d ), with spec(u) ⊆ ∆ such that div u = div v, (9)
and

u L ∞ ∩ Ḃd/p,p 2 v Ḃd/p,p q .
In the case where d = 2 our method does not provide solutions with the spectrum in ∆. Nevertheless, one can obtain solutions with the spectrum in a different type of sets. For each δ ∈ (0, π/4) let C δ be the symmetric cone

C δ := {(ξ 1 , ξ 2 ) ∈ R 2 | |ξ 1 | ≤ (tan δ) |ξ 2 |}.
With this notation we have: Theorem 8. Consider the numbers δ ∈ (0, π/8), ε ∈ (0, 1] and some parameters p ∈ [2, ∞) and q ∈ (1, 2). Then, for any vector field

v ∈ S (R 2 ) ∩ Ḃ2/p,p q (R 2 ), with spec(v) ⊆ C δ , there exists a vector field u ∈ L ∞ (R 2 ) ∩ Ḃ2/p,p 2 (R 2 ), with spec(u) ⊆ C (1+ε)δ , such that div u = div v, and u L ∞ ∩ Ḃ2/p,p 2 v Ḃ2/p,p q .
When compared with Theorem 3 one can observe that Theorem 7 (or Theorem 8) has two major drawbacks. First, we are not alowed to take p < 2 or q ≥ 2 as parameters for the space Ḃd/p,p q on the source side. Secondly, for the space in which we obtain the solution we lose some control of the "third parameter". In other words, we would prefer to obtain L ∞ ∩ Ḃd/p,p q for the solution space instead of L ∞ ∩ Ḃd/p,p 2 which is a slighlty larger space. (See however, Lemma 35 for a "perfect" version of our results in the case p = q = 2.) On the other hand it is unlikely that one can deduce Theorem 7 directly from Theorem 3. Indeed, given a vector field v ∈ Ḃd/p,p q , with spec(v) ⊆ ∆, by Theorem 3 one can find some vector field u ∈ L ∞ ∩ Ḃd/p,p q such that div u = div v, however, not necessarly with spec(u) ⊆ ∆. It is not obvious that one can obtain a solution u with spec(u) ⊆ ∆ by direct methods: suppose P ∆ is the Fourier projection on ∆, i.e., P ∆ = I -P + , where P + is the Riesz projection and I is the identity operator. We have P ∆ v = v and we can write div P ∆ u = div v.

However, since P D is not bounded on L ∞ , we may not have P ∆ u ∈ L ∞ , i.e., P ∆ u is not in general a candidate for a solution. The same observation applies to or Theorem 8. To our knowledge, except for the method we give in this paper, there is no other method in the literature able to prove results like Theorem 7 or Theorem 8.

In fact, when d ≥ 3, we will prove a more general result than Theorem 7. Our methods alow us to work with more general Fourier multipliers than the usual derivatives. In order to formulate our result we first need some preparations.

Let σ ∈ C 2 (R d , R) be a function. We consider the following properties (that may or not be satisfied by σ):

(P1) The function σ satisfies the estimate

|∇ α σ(ξ)| |ξ| 1-|α| , on R d , for any multiindex α ∈ N d with |α| ≤ 2;
(P2) The function σ is odd in the variable ξ 1 and even in any other variable, i.e.,

σ( 1 ξ 1 , 2 ξ 2 , ..., d ξ d ) = 1 σ(ξ 1 , ξ 2 , ..., ξ d ), on R d , for any signs 1 , ..., d ∈ {-1, 1}.
Introduce the new functions σ 1 , ..., σ d defined by

σ j (ξ 1 , ξ 2 , ..., ξ d ) := σ(ξ j , ξ 2 , ...ξ j-1 , ξ 1 , ξ j+1 ..., ξ d ),
on R d , for any index j ∈ {1, 2, ..., d}.

Consider some half-spaces D 1 , ..., D d ⊂ R d and a family of functions G 1 , ..., G d : R d → R d-1 . We say that the function G j is adapted to the half-space D j if there exists a rotation R on R d (depending on j) and a function σ : R d → R (depending on j) satisfying (P1), (P2) such that D j = R(U ), where U := R d-1 × (0, ∞), and

G j = (σ 1 • R, ..., σ d-1 • R).
We say that the family of functions G 1 , ..., G d : R d → R d-1 is adapted to the family of halfspaces D 1 , ..., D d ⊂ R d , if for each j ∈ {1, ..., d} the function G j is adapted to D j and Let us denote by S c, the space of all Schwartz function f whose Fourier transform f is compactly supported and vanishing in a neighborhood of 0. Suppose E and F are some Banach function spaces on R d such that S c, is dense in E and

d j=1 |G j (ξ)| 1 D j (ξ) ∼ |ξ| 1 D (ξ), (10)
m(∇)f F f E ,
for any f ∈ S c, . Then, by linearity and density m(∇) can be uniquely extended to a bounded operator m(∇) : E → F (see also Remark 13). We will often say that m is the symbol of the Fourier multiplier m(∇).

To a vector valued function

G : R d → R d-1 , with G = (G 1 , ..., G d-1
), where G 1 , ..., G d-1 : R d → R are scalar functions of polynomial growth, we associate the vector-valued Fourier multiplier

G(∇) := (G 1 (∇), ..., G d-1 (∇)).
In other words, if f ∈ S c, , by G(∇)f we mean

G(∇)f := (G 1 (∇)f, ..., G d-1 (∇)f ).
Suppose u 1 , ..., u d-1 ∈ S c, and let u be the (d -1)-vector field2 u := (u 1 , ..., u d-1 ). By G(∇)

• u we mean G(∇) • u := G 1 (∇)u 1 + ... + G d-1 (∇)u d-1 .
Now we can formulate our generalisation of Theorem 7.

Theorem 9. Let d ≥ 3 be an integer and consider some parameters p ∈ [2, ∞) and q ∈ (1, 2). Suppose that the family of functions G 1 , ..., G d : R d → R d-1 is adapted to the family of half-spaces

D 1 , ..., D d ⊂ R d .
Then, for any system of (d -1)-vector fields (v j ) j=1,..,d with v j ∈ S (R d ) ∩ Ḃd/p,p q (R d ) and spec(v j ) ⊆ D j , there exists a system of (d -1)-vector fields (u j ) j=1,..,d , with

u j ∈ L ∞ (R d ) ∩ Ḃd/p,p 2 (R d ) and spec(u j ) ⊆ D j , such that d j=1 G j (∇) • u j = d j=1 G j (∇) • v j , (11)
and

d j=1 u j L ∞ ∩ Ḃd/p,p 2 d j=1 v j Ḃd/p,p q .
In this paper equations such as [START_REF] Burkholder | A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions[END_REF] or [START_REF] Curcȃ | On the interpolation of the spaces W l,1 (R d ) and W r,∞ (R d )[END_REF] will be called divergence-like equations.

In subsection 5.1 we will see that Theorem 9 easily implies Theorem 7.

About the proofs

Our proofs of Theorem 9, Theorem 7 and Theorem 8 are based on two ingredients:

1. The W-method of interpolation. This is the key method that we are using throughout the paper. Let (A 0 , A 1 ) and (B 0 , B 1 ) be Banach couples and T : A 0 + A 1 → B 0 + B 1 be a linear operator such that T (B j ) → T (A j ), for any j = 0, 1. Suppose we want to see under which conditions on the spaces involved and the operator T we have

T F 1 θ (B 0 , B 1 ) → T F 2 θ (A 0 , A 1 ) , (12)
for some θ-interpolation functors F 1 θ , F 2 θ . One can say that, in some sense, we "interpolate" linear equations or that we preserve some form of surjectivity of the operator T . In order to give reasonable sufficient conditions for [START_REF] Diestel | Vector measures[END_REF] to hold for some convenient interpolation functors we introduce a variant of the complex interpolation method which will be called the W-method 3 . In our case F 1 θ will be given by the usual complex method of interpolation and F 2 θ will be given by our W-method. Roughly speaking this method consists in the following. Suppose (A 0 , A 1 ) is a compatible couple of Banach spaces. In order to define the interpolation space of the couple (A 0 , A 1 ) via the W-method we use the three lines lemma on the strip as in the standard complex method of interpolation. However, instead of quantifying the endpoint regularity of the analytic functions involved via the norms L ∞ (R, A j ) (j = 0, 1) we use slightly more complicated quantities that depend on some prescribed pair of Banach spaces (X 0 , X 1 ). In this way, for each θ ∈ (0, 1), we obtain an interpolation space that will be denoted by (A 0 , X 0 | A 1 , X 1 ) θ . The efficiency of the W-method relies (between other facts) on properly choosing the spaces X 0 , X 1 .

When the spaces X j have the U M D property, under some additional embedding assumptions concerning the Banach spaces involved, the W-method of interpolation preserves the "surjectivity" of operators (i.e., (12) holds). The main requirements for applying the W-method are twofold:

(i) On one hand one needs to verify some embedding conditions for the domains and the co-domains of the operator. We give some simple necessary conditions that are easy to formulate, however not sharp. We also mention that, in the absence of any such conditions, it is not possible to preserve surjectivity (see the examples in the second part of subsection 3.3).

(ii) On the other hand explicitly computing the space (A 0 , X 0 | A 1 , X 1 ) θ seems to be difficult in practice. However, there are particular situations in which we can embed (A 0 , X 0 | A 1 , X 1 ) θ in some convenient space. More precisely, when A j are of the form A j = A ∩ X j , for some Banach space A, we have

(A 0 , X 0 | A 1 , X 1 ) θ = (A ∩ X 0 , X 0 | A ∩ X 1 , X 1 ) θ → A ∩ (X 0 , X 1 ) θ .
For instance, when we have

X 0 = Ḃd/r,r 2 , X 1 = Ḃd/2,2 2
and A = L ∞ , the above embedding becomes

(L ∞ ∩ Ḃd/r,r 2 , Ḃd/p,p 2 | L ∞ ∩ Ḃd/2,2 2 , Ḃd/2,2 2 ) θ → L ∞ ∩ ( Ḃd/r,r 2 , Ḃd/2,2 2
) θ .

By using only the result of Mazya (Theorem 3 in the case p = q = 2) and the W-method together with the embedding Ḃd/p,p

1 (R d ) → L ∞ (R d )
we easily obtain the following:

Theorem 10. Let d ≥ 2 be an integer and consider some parameters p ∈ [2, ∞) and q ∈ (1, 2). Then, for any vector field

v ∈ S (R d )∩ Ḃd/p,p q (R d ) there exists a vector field u ∈ L ∞ (R d )∩ Ḃd/p,p 2 (R d ) such that div u = div v, and
u L ∞ ∩ Ḃd/p,p 2 v Ḃd/p,p q .
One can even obtain an analogue of Theorem 10 for a class of Lorentz-Sobolev spaces (for definitions see subsection 2.2). Namely, by using only the result of Mazya and the W-method, together with some standard facts in the theory of Lorentz spaces, we easily obtain the following: Theorem 11. Let d ≥ 2 be an integer and consider some parameters p ∈ [2, ∞) and q ∈ (1, 2). Then, for any vector field

v ∈ S (R d ) ∩ Ẇ d/p L p,3/2 (R d ) there exists a vector field u ∈ L ∞ (R d ) ∩ Ẇ d/p L p,2 (R d ) such that div u = div v, and u L ∞ ∩ Ẇ d/p L p,2 v Ẇ d/p L p,q .
The conditions p ≥ 2 and q < 2 in Theorem 10, Theorem 11, as well as in Theorem 7 and Theorem 9, are induced by some technical limitations of the W-method (see subsection 5.2).

The Bourgain-Brézis technique.

In [5, Section 4, p. 403] Bourgain and Brézis proved the torus analogue of Theorem 3 in the case where p = q = d = 2. They conluded the existence of solutions for the divergence equation by duality. Namely, they proved that (see [START_REF] Bourgain | On the equation div Y = f and application to control of phases[END_REF]

, Lemma 2, p. 403]) u L 2 (T 2 ) ∇u L 1 (T 2 )+W -1,2 (T 2 ) , (13)
for any u ∈ L 1 (T 2 ), with u(0) = 0. In order to obtain this, they used the following key estimate (see [5, (4.20), p. 405]):

n∈Z 2 \{0} n 1 n 2 |n| 4 sin n 1 θ 1 sin n 2 θ 2 ≤ C,
uniformly in θ 1 , θ 2 ∈ T, for some numerical constant C > 0. By convexity this allows us to write

n∈Z 2 \{0} n 1 n 2 |n| 4 F 1 (n) F 2 (n) ≤ C F 1 L 1 (T 2 ) F 2 L 1 (T 2 ) , for any F 1 , F 2 ∈ L 1 (T 2
). Thanks to this bilinear estimate, after decomposing ∇u in the space L 1 (T 2 ) + W -1,2 (T 2 ), we can deal with the space L 1 (T 2 ) in [START_REF] Grafakos | Classical Fourier Analysis[END_REF].

We use the technique introduced by Bourgain and Brézis in [5, Section 4, p. 403] and we prove a version of Theorem 9 (and Theorem 8) in the case where the source space is Ẇ d/2,2 . As we will see, thanks to this technique we are able to work with more general Fourier multipliers than derivatives. Also, it is this technique that allows us to gain some control on the Fourier spectrum of solutions. The results obtained by this method are "perfect" in the sense that that the source space is Ẇ d/2,2 and the solution space is L ∞ ∩ Ẇ d/2,2 ; there is no loss of regularity in the third parameter. The drawback of this technique is the fact that it does not apply to the case where p = 2.

As in the case of Theorem 10, we can easily obtain Theorem 9 using the W-method. This time however, insted of using Mazya's result we use the more general results that we obtain via the Bourgain-Brézis technique. Using the properties of Lorentz spaces we can give a Lorentz-Sobolev version of Theorem 9. In fact, our methods will provide more general results. On one hand the function spaces we work with can be more general than those in the statements of our final results (see for instance Theorem 37). One the other hand, the conditions imposed on Fourier multipliers and the Fourier spectrum of the solutions can be more general. Also, by using the technique of Mazya, one can easily obtain a version of Theorem 3 in the case p = q = 2 that concerns general Hodge systems. Combining this result with the W-method one can obtain an analogue of Theorem 10 for Hodge systems. We will not consider however, such issues here. In this paper, we limit ourselves to some model situations that are easier to describe.

Notation. Throughout the paper we use mainly standard notation. For instance, we often use the symbols and ∼. For two nonnegative variable quantities a and b we write a b if there exists a constant C > 0 such that a ≤ Cb. If a b and b a, then we write a ∼ b. For simplicity we denote by spec(f ) the Fourier spectrum of a tempered distribution f ; in other words, spec(f ) = supp f . Everywhere in this paper S (R d ) is the space of tempered distributions. When X is a function space on R d and u = (u 1 , ..., u d ) is a vector filed on R d where each u j belongs to X, we write u ∈ X instead of u ∈ X d . A similar convention will be made for the (d -1)-vector fields. Other notation will be introduced when needed.

Function spaces

In this section we quickly recall the definition and some properties of some standard function spaces.

Sobolev and Besov spaces

Let S be the space of all Schwartz functions f on R d such that f vanishes in a neighborhood of 0. When 1 < p < ∞ and α ∈ R the homogeneous space Ẇ α,p (R d ) is obtained by completion of S under the norm

f Ẇ α,p := |∇| α f L p .
We can see that we can also define the above homogeneous spaces Ẇ α,p by completion of the the normed function spaces Ẇ α,p c (R d ). Here, Ẇ α,p c (R d ) is the space of all the compactly supported functions whose Ẇ α,p -norm is finite. The spaces Ẇ α,p as defined here are complete.

We continue by briefly recalling the definition of the Besov spaces (we do not define here the Triebel-Lizorkin spaces; see [START_REF] Triebel | Theory of function spaces II[END_REF] for details). Consider a radial function Φ ∈ C ∞ c (R d ) such that supp Φ ⊂ B(0, 2) and Φ ≡ 1 on B(0, 1). For k ∈ Z we define the operators P k , acting on the space of tempered distributions on R d , by the relation

P k f (ξ) := Φ ξ 2 k -Φ ξ 2 k-1 f (ξ) , (14)
for any Schwartz function f on R d . The operators P k will be called Littlewood-Paley "projections" adapted to R d . For any Schwartz function f we have that

f = k∈Z P k f ,
in the sense of tempered distributions. The homogeneous Besov space Ḃα,p q (R d ) (with 1 ≤ p, q ≤ ∞ and α a real number) is obtained by completion of S under the norm

f Ḃα,p q := j∈Z 2 αkq P k f q L p 1/q . We have Ḃα,2 2 (R d ) = Ẇ α,2 (R d
) with equivalent norms. The main advantage of our definition of the homogeneous Besov spaces is the fact that, whenever α 0 -d/p 0 = α 1 -d/p 1 and α 1 > α 0 we have the embedding Ḃα 1 ,p 1

q 1 (R d ) → Ḃα 0 ,p 0 q 0 (R d ), (15)
for any q 0 , q

1 ∈ [1, ∞) with q 0 ≤ q 1 .
Note that we have the following dilation properties:

f (λ•) Ḃα,p q ∼ λ α-d/p f Ḃα,p q , (16)
for any f ∈ Ḃα,p q (R d ) respectively and any λ > 0. In particular, when α = d/p the spaces Ḃα,p q have the same scaling property as L ∞ . In what follows the spaces of the form Ḃd/p,p q will be called critical. It is worth recalling here, that, by a direct application of the Bernstein inequalities we get the embedding Ḃd/p,p

1 (R d ) → L ∞ (R d ). When q > 1 the critical spaces Ḃd/p,p q do not embed in L ∞ .
Remark 12. Note that the spaces Ḃd/p,p q (with q > 1) as defined here contain elements that are not tempered distributions. However, when α < d/p the elements of the space Ḃα,p q are all tempered distributions (see for instance [1, Remark 2.26, p. 68] or [START_REF] Bourdaud | Réalisations des espaces de Besov homogènes[END_REF]).

Remark 13. Since the operator div : S ∩ Ḃd/p,p q → Ḃd/p-1,p q is linear and bounded (here, S ∩ Ḃd/p,p q is endowed with the norm induced by Ḃd/p,p q ), by density of S ∩ Ḃd/p,p q in Ḃd/p,p q it extends uniquely to an operator div : Ḃd/p,p q → Ḃd/p-1,p q . Similar facts hold for other spaces and other operators. In this way we can remove from the hypotheses of Theorem 7, Theorem 8 and Theorem 9 the fact that the source v belongs to S . For instance [START_REF] Burkholder | A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions[END_REF] , then v ∈ Ḃd/p-2,p q is a tempered distribution and we can define the spectrum of v as spec(v) := supp v. This observation will be applied for other function spaces as well.

It is easy to see that any space of the form Ḃα,p q with p, q ∈ (1, ∞) is embedded in l α q (L p ) and hence it has the U M D property (see for instance [START_REF] Pisier | Martingales in Banach Spaces[END_REF]).

Lorentz-Sobolev spaces

Consider4 some parameters p ∈ (1, ∞), q ∈ [1, ∞] and α ≥ 0. The homogeneous Lorentz-Sobolev spaces Ẇ α L p,q (R d ) is the completion of the normed space of Schwartz functions f on R d under the norm

f Ẇ α L p,q := |∇| α f L p,q ,
where L p,q is the usual Lorentz space of parameters p and q.

Remark 15. One can easily adapt the Remarks 12, 13 and 14 to the case of the Lorentz-Sobolev spaces.

Many of the embedding properties of the Besov and Triebel-Lizorkin spaces hold for the Lorentz-Sobolev spaces (see for instance [START_REF] Seeger | Embeddings for spaces of Lorentz-Sobolev type[END_REF] for detailes). We mention below some properties of Lorentz-Sobolev spaces that will be needed in the proof of Theorem 11. All of them are direct consequences of well-known facts from the theory of Lorentz spaces.

Lemma 16. For any r ∈ [2, ∞), we have that

Ẇ d/2,2 (R d ) → Ẇ d/r L r,2 (R d ).
Proof.

It
f L r,2 |∇| d(1/2-1/r) f L 2 ,
for any Schwartz function f on R d . This can be rewritten as

f Ẇ d/r L r,2 = |∇| d/r f L r,2 |∇| d/2 f L 2 = f Ẇ d/2,2 , obtaining that Ẇ d/2,2 → Ẇ d/r L r,2 .
Lemma 17. For any r ∈ (1, ∞), we have that

Ẇ d/r L r,1 (R d ) → L ∞ (R d ).
Proof. For any Schwartz function f on R d we have that

I d/r * f L ∞ f L r,1 , (17)
where

I d/r (x) = |x| d/p-d = |x| -d/p
, for any x ∈ R d . Indeed, using [13, Theorem 1.4.17 (v), p. 52], we have

R d f (y) |x -y| d/r dy ≤ I d/r (L r,1 ) * f L r,1 = I d/r L r ,∞ f L r,1
, and we can easily see that

I d/r L r ,∞ = sup λ>0 λ x ∈ R d | |x| < (1/λ) r /d 1/r ∼ 1.
Hence, (17) holds. We can reformulate [START_REF] Mironescu | On some inequalities of Bourgain, Brézis, Maz'ya, and Shaposhnikova related to L 1 vector fields[END_REF] as

f L ∞ |∇| d/r f L r,1 = f Ẇ d/r L r,1 , This shows that Ẇ d/r L r,1 → L ∞ .
Lemma 18. Suppose p 0 , p 1 , q 0 , q 1 ∈ [1, ∞) and α 0 , α 1 ≥ 0. Then, for any θ ∈ (0, 1) we have

( Ẇ α 0 L p 0 ,q 0 (R d ), Ẇ α 1 L p 1 ,q 1 (R d )) θ = Ẇ α L p,q (R d ), (18)
where α = (1 -θ)α 0 + θα 1 , 1/p = (1 -θ)/p 0 + θ/p 1 and 1/q = (1 -θ)/q 0 + θ/q 1 .

Proof. This can be proved by Stein's method of interpolation (see for instance [13, Theorem 1.3.7, p. 37]) as follows. Note that the function ξ → |ξ| it defined on R d \ {0} satisfies

∇ k |ξ| it ≤ C(1 + |t|) d+2 |ξ| -k , a.e. in ξ ∈ R d \ {0},
|∇| it L a,b →L a,b a,b C(1 + |t|) d+2 . ( 19
)
Let us consider the analytic family of operators (T z ) z∈S with

T z := |∇| (1-z)α 0 +zα 1 ,
for all z ∈ S. Thanks to [START_REF] Muskhelishvili | Singular Integral Equations[END_REF], the analytic family (T z ) z∈S satisfies the hypothesis of [13, Theorem 1.3.7, p. 37]. Hence, we get T θ ( Ẇ α 0 L p 0 ,q 0 , Ẇ α 1 L p 1 ,q 1 ) θ → (L p 0 ,q 0 , L p 1 ,q 1 ) θ = L p,q , and, in a similar way (applying Stein's method for the family (T -z ) z∈S ),

T -θ (L p,q ) = T -θ (L p 0 ,q 0 , L p 1 ,q 1 ) θ → ( Ẇ α 0 L p 0 ,q 0 , Ẇ α 1 L p 1 ,q 1 ) θ . Hence, T θ ( Ẇ α 0 L p 0 ,q 0 , Ẇ α 1 L p 1 ,q 1 ) θ = L p,q ,
and ( 18) is proven.

The spaces Ẇ α L p,q have scaling properties that are similar to those of the Besov spaces (see ( 16)). In particular, the spaces Ẇ d/p L p,q have the same scaling as L ∞ . As we have seen in Lemma 17 we have Ẇ d/p L p,1 → L ∞ . However, when q > 1 the critical spaces Ẇ d/p L p,q do not embed in L ∞ .

Let us see that the spaces Ẇ α L p,q have the U M D property when p, q ∈ (1, ∞). For this is sufficient to see that L p,q has the U M D property. Consider some p 0 , p 1 ∈ (1, ∞) such that p 0 < p < p 1 . Since, L p 0 and L p 1 are U M D spaces, by Burkholder's theorem (see [START_REF] Burkholder | A geometric condition that implies the existence of certain singular integrals of Banach-space-valued functions[END_REF]) the Hilbert transform is bounded on L 2 (T,L p 0 ) and L 2 (T,L p 1 ) respectively. Hence, the Hilbert transform is bounded on the space

(L 2 (T,L p 0 ), L 2 (T,L p 1 )) η,q = L 2 (T, (L p 0 , L p 1 ) η,q ) = L 2 (T,L p,q ),
where η ∈ (0, 1) is such that 1/p = (1 -η) /p 0 + η/p 1 (see for instance [2, Theorem 5.6.2, p. 123]). By Bourgain's theorem ( [START_REF] Bourgain | Some remarks on Banach spaces in which martingale difference sequences are unconditional[END_REF]), we get that L p,q has the U M D property.

Some quotient spaces

f Y /D := inf f -∈Y D c f + f - Y .
In this paper we will work with quotient spaces of the form Ẇ α,2 /D and (L 1 + Ẇ α,2 )/D. One can easily see that for Y = Ẇ α,2 or Y = L 1 + Ẇ α,2 we have the following norming property sup

f Y /D ≤1 f, g = g Y * ,
for any g ∈ Y * D , where the supremum is taken over all Schwartz functions f with f Y /D ≤ 1. In the case where Y is the Sobolev space Ẇ α,2 it is easy to compute the seminorm induced by Y /D. Namely, let us see that for any u ∈ S and any measurable set D ⊆ R d we have

u Ẇ α,2 /D = R d |ξ| 2α 1 D (ξ) | u(ξ)| 2 dξ 1/2 . ( 20
)
Indeed, we have

u 2 Ẇ α,2 /D = inf v∈ Ẇ α,2 D c R d |ξ| 2α | u(ξ) + v(ξ)| 2 dξ = D |ξ| 2α | u(ξ)| 2 dξ + inf v∈ Ẇ α,2 D c D c |ξ| 2α | u(ξ) + v(ξ)| 2 dξ = D |ξ| 2α | u(ξ)| 2 dξ.
We recall that, by P D we denote the Fourier projection on the set D, i.e., we have

P D f (ξ) = 1 D (ξ) f (ξ),
for any ξ ∈ R d and any Schwartz function f . Note that, in the case where D = (0, ∞) d the operator P (0,∞) d is the Riesz projection. In this case we will write P + in the place of P (0,∞) d .

3 The W-method of complex interpolation

In this section we introduce a variant of the complex interpolation and we prove several of its properties. We call this new method of interpolation the W-method and, as stated in the introduction (see subsection 1.4), this will be used in the proof of Theorem 9, Theorem 7, Theorem 8, Theorem 11. We mainly study here only the properties of the W-method that are used in the proof of our main results. In subsection 3.1 we show that the W-method is indeed an interpolation method. However, we ignore some issues specific to the interpolation methods in general such as computing dual interpolation spaces or reiteration theorems. These problems do not concern us here.

An important aspect is the relation of the W-method with the classical complex method. We do not know in general how to compute efficiently the interpolation spaces obtained via the Wmethod. However, as we will see in subsection 3.2 the space obtained via the W-method is, in many "convenient" cases, the same as the space obtained via the classical complex method.

The main feature of the W-method is that one can use it to "interpolate" linear equations. It is one of the main ingredients that enter in the proof of our main results and it is the final goal of this section.

Construction of the interpolation space

We describe here the W-method and prove some basic properties. The proofs we give are straightforward adaptations of those that correspond to the classical complex interpolation as found in [2, Chapter 4]. Following the general presentation in [2, Chapter 4] let us introduce now the W-method.

For the beginning, fix two Banach spaces X 0 and X 1 and suppose (A 0 , A 1 ) is a Banach couple. Let F 2 = F 2 (A 0 , X 0 | A 1 , X 1 ) be the linear space of all bounded continuous functions f with values in A 0 + A 1 , defined on the strip

S := {z ∈ C | 0 ≤ z ≤ 1} , that are analytic in the open strip S 0 := {z ∈ C | 0 < z < 1} ,
and moreover, such that f (j + it) ∈ A j for any j = 0, 1 and any t ∈ R, and

f F 2 := max j=0,1 sup Λ j ≤1 R Λ j f (j + it) 2 X j dt 1/2 < ∞, (21)
where, for each j = 0, 1, Λ j : A j → X j are linear bounded operators. One can easily verify that

• F 2 defines a norm on F 2 . Fix 0 < θ < 1. Consider the linear space C θ (A 0 , X 0 | A 1 , X 1 ) defined by C θ (A 0 , X 0 | A 1 , X 1 ) := a ∈ A 0 + A 1 | a = f (θ), for some f ∈ F 2 (A 0 , X 0 | A 1 , X 1 ) .
and define, for each a ∈ C θ (A 0 , X 0 | A 1 , X 1 ), the quantity

a θ := inf f F 2 | a = f (θ), f ∈ F 2 (A 0 , X 0 | A 1 , X 1 ) . Lemma 19. The mapping a → a θ is a norm on C θ (A 0 , X 0 | A 1 , X 1 ).
In order to prove Lemma 19 we rely on the following basic fact (and at least implicitely well-known): Lemma 20. Fix some 1 ≤ p < ∞ and let Z be a Banach space. Suppose F : S → Z is a bounded continuous function which is analytic in S 0 such that the functions t → F (j + it) belong to the space L p (R,Z).

Then, for any z ∈ S 0 , we have

F (z) = - 1 2πi R F (it) it -z dt + 1 2πi R F (1 + it) 1 + it -z dt. (22)
In particular, for any θ ∈ (0, 1),

F (θ) Z θ,p max j=0,1 R F (j + it) p Z dt 1/p . ( 23
)
Proof of Lemma 20. Fix some z ∈ S 0 . Consider some arbitrary R > 0 and the curve γ R given by the boundary of the rectangle

[0, 1] × [-R, R], oriented anti-clockwise. For R sufficeintly large we have z ∈ [0, 1] × [-R, R]
. By Cauchy's formula we get

F (z) = 1 2πi γ R F (ζ) ζ -z dζ,
and we can rewrite this as

F (z) = - 1 2πi R -R F (it) it -z dt + 1 2πi R -R F (1 + it) 1 + it -z dt + 1 2πi 1 0 F (x + iR) x + iR -z dx - 1 2πi 1 0 F (x -iR) x -iR -z dx. (24)
Note that, the functions t → F (j + it)/ (j + it -θ) belong to the space L 1 (R, Z). Indeed, by Hölder's inequality (since we always have p > 1) we can write

R F (j + it) j + it -z Z dt ≤ R 1 |j + it -z| p dt 1/p R F (j + it) p Z dt 1/p θ,z R F (j + it) p Z dt 1/p < ∞, (25)
with the natural modification in the case where p = ∞.

Also, 1 2πi

1 0 F (x ± iR) x ± iR -z dx Z ≤ 1 |x ± iR -z| F L ∞ (S,Z) → 0, (26)
when R → ∞.

Using ( 24), [START_REF] Ziemer | Weakly Differentiable Functions[END_REF], letting R → ∞ and using the dominated convergence theorem, we get the representation formula [START_REF] Peetre | Sur la transformation de Fourier des fonctions à valeurs vectorielles[END_REF]. Using ( 22) and ( 25), for z = θ, we obtain [START_REF] Seeger | Embeddings for spaces of Lorentz-Sobolev type[END_REF].

Proof of Lemma 19. Clearly, • θ is a seminorm on C θ (A 0 , X 0 | A 1 , X 1 ). It remains to see that, if a θ = 0, for some a ∈ C θ (A 0 , X 0 | A 1 , X 1
), then a = 0. We prove this by showing that

a A 0 +A 1 θ a θ , (27)
for all a ∈ C θ (A 0 , X 0 | A 1 , X 1 ). For this purpose fix a ∈ C θ (A 0 , X 0 | A 1 , X 1 ) and consider a functional λ ∈ (A 0 + A 1 ) * , with λ = 1, such that a A 0 +A 1 ≤ 2λ (a). Consider also a function

f ∈ F 2 (A 0 , X 0 | A 1 , X 1 ), such that f (θ) = a and f F 2 ≤ 2 a θ .
Let us define, for each j = 0, 1, the linear operators Λ j : A j → X j by Λ j (a j ) = λ(a j )e j , for any a j ∈ A j , where e j ∈ X j are some fixed vectors with e j X j = 1. Clearly, for any j = 0, 1,

Λ j (a j ) X j = |λ(a j )| ≤ a j A 0 +A 1 ≤ a j A j ,
for any a j ∈ A j , and we get Λ j ≤ 1.

Using this observation and introducing the function

F : S → C defined by F (z) := λ (f (z)), one can write, max j=0,1 R |F (j + it)| 2 dt 1/2 = max j=0,1 R |λ (f (j + it))| 2 dt 1/2 = max j=0,1 R Λ j (f (j + it)) 2 X j dt 1/2 ≤ f F 2 ≤ 2 a θ < ∞. ( 28
)
This shows, in particular, that the functions t → F (j + it) belong to the space L 2 (R, C). We also see immediately that F is bounded, continuous on S and analytic in S 0 . Hence, by applying Lemma 20 for Z = C and p = 2 (more precisely (23)), using (28), we get

a A 0 +A 1 ≤ 2λ (a) = 2F (θ) max j=0,1 R |F (j + it)| 2 dt 1/2 a θ ,
which proves (27). Now, thanks to Lemma 19, we can define the interpolation space (A 0 , X 0 | A 1 , X 1 ) θ as being the completion of the normed space (C θ (A 0 , X 0 | A 1 , X 1 ) , • θ ).

One can easily see that (A 0 , X 0 | A 1 , X 1 ) θ is an intermediate space:

A 0 ∩ A 1 → (A 0 , X 0 | A 1 , X 1 ) θ → A 0 + A 1 . ( 29
)
The second embedding in (29), follows directly from the inequality (27). In order to see the first embedding, pick a ∈ A 0 ∩ A 1 and consider the function f (z) := exp (z 2 -θ 2 ) a. One can easily check that f (θ) = a, f ∈ F 2 and

a θ ≤ max j=0,1 sup Λ j ≤1 R Λ j f (j + it) 2 X j dt 1/2 ≤ max j=0,1 R f (j + it) 2 A j dt 1/2 ∼ θ a A 0 ∩A 1 .

This gives us that

A 0 ∩ A 1 → C θ (A 0 , X 0 | A 1 , X 1 ) → (A 0 , X 0 | A 1 , X 1 ) θ .
Let us see now that the W-method provides an exact interpolation functor:

Proposition 21. Consider some Banach spaces X 0 , X 1 . Let (A 0 , A 1 ) , (B 0 , B 1 ) be two Banach couples and T : A 0 + A 1 → B 0 + B 1 be a linear operator such that T : A j → B j is bounded for any j = 0, 1, of norm T j→j .

Then, the operator

T : (A 0 , X 0 | A 1 , X 1 ) θ → (B 0 , X 0 | B 1 , X 1 ) θ ,
is bounded and of norm T θ→θ satisfying

T θ→θ ≤ T 1-θ 0→0 T θ 1→1 .
Proof. Without loss of generality we suppose that T j→j > 0, for any j = 0, 1. For the brevity of notation we denote by

• θ , • F 2 , • θ and • F 2 the norms on the spaces (A 0 , X 0 | A 1 , X 1 ) θ , F 2 (A 0 , X 0 | A 1 , X 1 ), (B 0 , X 0 | B 1 , X 1 ) θ and F 2 (B 0 , X 0 | B 1 , X 1 ) respectively. Pick some a ∈ C θ (A 0 , X 0 | A 1 , X 1 ) and fix some ε > 0. Consider a function f ∈ F 2 (A 0 , X 0 | A 1 , X 1 ) such that f (θ) = a and f F 2 ≤ (1 + ε) a θ . The function F (z) := T z-1 0→0 T -z 1→1 T f (z) belongs to F 2 (B 0 , X 0 | B 1 , X 1 )
. Indeed, F is bounded and continuous on S with values in B 0 +B 1 , analytic in S 0 and, for any linear operators Λ j : B j → X j of norm at most 1, we have

R Λ j F (j + it) 2 X j dt 1/2 ≤ R T -1 j→j Λ j • T f (j + it) 2 X j dt 1/2 ≤ sup Λ j ≤1 R Λ j f (j + it) 2 X j dt 1/2 , ( 30
)
for any j = 0, 1, where the supremum is taken over all linear bounded operators Λ j : A j → X j with Λ j ≤ 1. Here, we have used the fact that T -1 j→j Λ j • T : A j → X j is a linear operator of norm at most 1, for any j = 0, 1. Now, by (30), we get

T θ-1 0→0 T -θ 1→1 T f (a) θ = F (θ) θ ≤ F F 2 ≤ f F 2 ≤ (1 + ε) a θ , and letting ε → 0 one obtains, T a θ ≤ T 1-θ 0→0 T θ 1→1 a θ , for any a ∈ C θ (A 0 , X 0 | A 1 , X 1 ). Since, by definition, C θ (A 0 , X 0 | A 1 , X 1 ) is dense in (A 0 , X 0 | A 1 , X 1 ) θ ,
we get the conclusion.

A particular case

In general, computing the interpolation space (A 0 , X 0 | A 1 , X 1 ) θ seems to be a nontrivial task. However, there are some particular cases where an explicit computation is easy.

Let us restrict to the case, where A 0 = X 0 and A 1 = X 1 and let us denote, for simplicity, the space (X 0 , X 0 | X 1 , X 1 ) θ by (X 0 |X 1 ) θ . Also, instead of F 2 (X 0 , X 0 | X 1 , X 1 ) we write F 2 (X 0 |X 1 ) and instead of C θ (X 0 , X 0 | X 1 , X 1 ) we write C θ (X 0 |X 1 ). In this case, formula [START_REF] Runst | Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations[END_REF] becomes

f F 2 = max j=0,1 R f (j + it) 2 X j dt 1/2 . ( 31
)
Indeed, for any j = 0, 1, we have sup

Λ j ≤1 R Λ j f (j + it) 2 X j dt 1/2 ≤ R f (j + it) 2 X j dt 1/2
, and sup

Λ j ≤1 R Λ j f (j + it) 2 X j dt 1/2 ≥ R id X j f (j + it) 2 X j dt 1/2 = R f (j + it) 2 X j dt 1/2
, where id X j : X j → X j is the identity mapping on X j .

It turns out that, the space (X 0 |X 1 ) θ coincides with the space (X 0 , X 1 ) θ obtained via the classical complex interpolation method. The proof of this fact is easy, however we give it below for the sake of completeness. Proposition 22. Suppose (X 0 , X 1 ) is a compatible couple of Banach spaces. Then, for any θ ∈ (0, 1), (X 0 |X 1 ) θ = (X 0 , X 1 ) θ , with equivalence of norms.

Remark 23. Implicitly, the embedding (X 0 |X 1 ) θ → (X 0 , X Proof. Consider some a ∈ (X 0 , X 1 ) θ ∩ C θ (X 0 |X 1 ) and let some f ∈ F 2 (X 0 |X 1 ) be such that

f (θ) = a and f F 2 ≤ 2 a (X 0 |X 1 ) θ .
By [2, Lemma 4.3.2 (ii), p. 93] (or [10, Section 9.4, (ii)]) we have

a (X 0 ,X 1 ) θ ≤ 1 1 -θ R f (iτ ) X 0 P 0 (θ, τ )dτ 1-θ 1 θ R f (1 + iτ ) X 1 P 1 (θ, τ )dτ θ , (32)
where P j (j = 0, 1) are the real Poisson kernels defined by P j (s + it, τ ) := e -π(τ -t) sin πs sin 2 πs + (cos πs -e ijπ-π(τ -t) ) 2 ,

for s ∈ (0, 1), t, τ ∈ R. Note that P j (θ, •) ∈ L 2 (R, R) and by the Cauchy-Schwarz inequality, R f (iτ )

X 0 P 0 (θ, τ )dτ ≤ R f (iτ ) 2 X 0 dτ 1/2 R P 2 0 (θ, τ )dτ 1/2 f F 2 a (X 0 |X 1 ) θ .
In a similar way we get

R f (iτ ) X 0 P 0 (θ, τ )dτ a (X 0 |X 1 ) θ ,
and combining with (32) one obtains

a (X 0 ,X 1 ) θ a (X 0 |X 1 ) θ . (33)
By taking the closure we get (X 0 |X 1 ) θ → (X 0 , X 1 ) θ .

Converselly, if a ∈ (X 0 , X 1 ) θ , then there exists g ∈ F (X 0 , X 1 ) (see [START_REF] Bergh | Interpolation spaces. An introduction[END_REF]Chapter 4] for the standard notation F (X 0 , X 1 )) such that g (θ) = a and max j=0,1

sup t∈R g(j + it) X j ≤ 2 a (X 0 ,X 1 ) θ . ( 34
)
Introduce the function g : S → X 0 + X 1 defined by g(z) := exp(z 2 -θ 2 )g(z), for z ∈ S. We observe that, for any j = 0, 1,

R g(j + it) 2 X j dt 1/2 R e -2t 2 g(j + it) 2 X j dt 1/2 ≤ R e -2t 2 dt 1/2 sup t∈R g(j + it) X j ∼ sup t∈R g(j + it) X j . (35)
Hence, g ∈ F 2 (X 0 |X 1 ), a = g(θ) ∈ C θ (X 0 |X 1 ) and by (34), (35),

a (X 0 |X 1 ) θ a (X 0 ,X 1 ) θ , (36)
We have now (X 0 , X 1 ) θ → (X 0 |X 1 ) θ and Proposition 22 is proven.

An immediate consequence of Proposition 22 is the following useful embedding result:

Corollary 24. Suppose (X 0 , X 1 ) is a compatible couple of Banach spaces. Then, for any Banach space A, we have the embedding

(A ∩ X 0 , X 0 | A ∩ X 1 , X 1 ) θ → A ∩ (X 0 , X 1 ) θ .
Proof. Consider the canonical inclusion ι : A ∩ X 0 + A ∩ X 1 → X 0 + X 1 as a linear bounded operator ι : A ∩ X j → X j and apply Proposition 21. We get

(A ∩ X 0 , X 0 | A ∩ X 1 , X 1 ) θ → (X 0 , X 0 | X 1 , X 1 ) θ = (X 0 |X 1 ) θ .
Since by Proposition 22 we have (X 0 |X 1 ) θ = (X 0 , X 1 ) θ , we now obtain the embedding

(A ∩ X 0 , X 0 | A ∩ X 1 , X 1 ) θ → (X 0 , X 1 ) θ . ( 37
)
Also, using the fact that

(A ∩ X 0 , X 0 | A ∩ X 1 , X 1 ) θ is an intermediate space (see (29)), we have (A ∩ X 0 , X 0 | A ∩ X 1 , X 1 ) θ → A ∩ X 0 + A ∩ X 1 → A,
which together with (37) proves Corollary 24.

Solutions of linear equations

In this subsection we highlight the main strength of the W-method. Namely, we show here how the W-method can be used in order to "interpolate " underdetermined equations. Here we make an essential use of the fact that the Hilbert transform is bounded on spaces of the form L 2 (R, Z), where Z is an U M D space. The U M D property plays here a key role.

Before stating the results in this subsection let us make some (common) notational conventions. The space C l b (R, Z), where l ∈ N, is the space of all the functions f : R →Z for which the k-th derivative f (k) is a continuous and bounded Z-valued function on R for all k ∈ N with k ≤ l. We endow C l b (R, Z) with the norm

f C l b (R,Z) := l k=0 f (k) L ∞ (R,Z) .
Given a measurable function ω : R →(0, ∞) we denote by L 2 (ω, Z) the space of the strongly (Bochner-Lebesgue) measurable functions f : R →Z for which the norm

f L 2 (ω,Z) := R f (t) 2 Z ω(t)dt 1/2 , is finite. When ω(t) = exp (t 2 ) or ω(t) = exp (-t 2 ) the space L 2 (ω, Z) is denoted by L 2 (exp (t 2 ) , Z)
or by L 2 (exp (-t 2 ) , Z) respectively. However, when ω ≡ 1 we prefer to write L 2 (R, Z) instead of L 2 (1, Z).

Boundary values of functions on the strip

Let us recall now some (at least implicitely) well-known facts related to the Hilbert transforms of vector-valued functions. Let Z be a Banach space and consider a function

f ∈ C 4 b (R, Z) ∩ L 2 (exp (t 2 ) , Z).
The Hilbert transform of f is defined by

Hf (t) := 1 π lim ε→0 ε<|t-s|<1/ε f (s) t -s ds = 1 π lim ε→0 ε<|s|<1/ε f (t -s) s dy,
for t ∈ R. As one can immediately check, for such f the above limit exists, for every x ∈ R (the convergence being in the norm of Z). Also, we get that

Hf ∈ C b (R, Z). (38)
Indeed, for every t ∈ R,

Hf (t) Z ≤ 1 π lim ε→0 ε<|t-s|<1 f (s) t -s ds Z + 1 π lim ε→0 1<|t-s|<1/ε f (s) t -s ds Z = 1 π lim ε→0 ε<|t-s|<1 f (s) -f (t) t -s ds Z + 1 π lim ε→0 1<|t-s|<1/ε f (s) t -s ds Z ≤ 2 π f Lip(R,Z) + 1 π f L 2 (exp(t 2 ),Z) f C 3 b (R,Z) + f L 2 (exp(t 2 )
,Z) and hence, Hf (t) is uniformly bounded in Z. The continuity of Hf can be proved in a similar way, by estimating the expression Hf (t 1 ) -Hf (t 2 ), when t 1 , t 2 ∈ R are close to each other.

In what follows we need a vector-valued version of the Plemelj formula. The proof we give is completely similar to the one in the scalar valued case, however, we include it here for completeness (see for instance [START_REF] Muskhelishvili | Singular Integral Equations[END_REF]). Lemma 25. Suppose Z is a Banach space and consider a function f ∈ C 3 b (R, Z)∩L 2 (exp (t 2 ) , Z). Then, when ε 0 we have

1 2πi R f (s) s -(t ± iε) ds → ±f (t) + iHf (t) 2 ,
in the norm of Z, uniformly in t ∈ R.

Proof. We only consider the case of the sign "+", the other one being similar. We first show that

1 2πi R f (s) s -iε dy - f (0) 2 - i 2π |s|>ε f (s) -s ds Z f C 1 b (R,Z) ε 1/2 , ( 39
)
for any ε ∈ (0, 1), where the implicit constant does not depend on f or ε. Changing the variables (s = ετ ), this is equivalent to

R f (ετ ) 1 τ -i -1 |τ |>1 (τ ) 1 τ dτ -πif (0) Z f C 1 b (R,Z) ε 1/2 . ( 40
) Notice that R 1 τ -i -1 |τ |>1 (τ ) 1 τ dτ < ∞ and R 1 τ -i -1 |τ |>1 (τ ) 1 τ dτ = πi,
and hence, it remains to show that

R (f (ετ ) -f (0)) 1 τ -i -1 |τ |>1 (τ ) 1 τ dτ Z f C 1 b (R,Z) ε 1/2 .
One can see this by a direct computation. Indeed, the quantity

R (f (ετ ) -f (0)) 1 τ -i -1 |τ |>1 (τ ) 1 τ dτ Z is bounded by R f (ετ ) -f (0) Z 1 τ -i -1 |τ |>1 (τ ) 1 τ dτ [-R,R] f (ετ ) -f (0) Z 1 1 + |τ | dτ + [-R,R] c f (ετ ) -f (0) Z 1 τ 2 dτ , (41)
for any R > 1. Since,

f (ετ ) -f (0) Z ≤ ε f C 1 b (R,Z) |τ |, we get [-R,R] f (ετ ) -f (0) Z 1 1 + |τ | dτ f C 1 b (R,Z) εR. Also, since f (ετ ) -f (0) Z ≤ 2 f C 1 b (R,Z) , we have [-R,R] c f (ετ ) -f (0) Z 1 τ 2 dτ f C 1 b (R,Z) /R.
From (41) we get now,

R f (ετ ) -f (0) Z 1 τ -i -1 |τ |>1 (τ ) 1 τ dτ f C 1 b (R,Z) (εR + 1/R),
where the implicit constant does not depend on ε or R. Setting R = ε -1/2 we obtain (40) and hence (39). Now consider some function f ∈ C 1 b (R, Z) and for each t ∈ R, define f t : R → Z by f t (x) := f (x + t) for all x ∈ R. We can see that

f t ∈ C 1 b (R, Z) and f t C 1 b (R,Z) = f C 1 b (R,Z)
, for any t ∈ R. By a simple change of variables we can also observe that

1 2πi R f t (s) s -iε ds - f t (0) 2 - i 2π |s|>ε f t (s) -s ds Z equals 1 2πi R f (s) s -(t + iε) ds - f (t) 2 - i 2π |t-s|>ε f (s) t -s ds Z .
Hence, by applying (39) to f t and letting ε → 0 we obtain the conclusion.

Let us introduce the some operators that quantify the boundary behaviour of analytic functions on the strip. For each j = 0, 1 let

H j : L 2 exp t 2 , Z ∩ C 3 b (R, Z) → C b (R, Z) ,
be defined by

H j f (t) := -if (t) -(-1) j Hf (t) 2 ,
and

R j : L 2 (exp (t 2 ) , Z) ∩ C 3 b (R, Z) → C b (R, Z), be defined by R j f (t) := ρ j * f (t),
where ρ j : R → C are the bounded functions

ρ j (t) := 1 2πi 1 1 -(-1) j it , for all t ∈ R. It is easy to see that, for f ∈ L 2 (exp (t 2 ) , Z), the quantity R j f is indeed well-defined and R j f ∈ C b (R, Z).
An easy consequence of (38) and Lemma 25 is the following fact:

Lemma 26. Suppose (B 0 , B 1 ) is a compatible couple of Banach spaces. Consider some functions

u j ∈ C 3 b (R, B j ) ∩ L 2 (exp (t 2
) , B j ), j = 0, 1, and define u :

S → B 0 + B 1 by u(z) := - 1 2πi R u 0 (t) it -z dt + 1 2πi R u 1 (t) 1 + it -z dt,
for all z ∈ S 0 , and u (j + it) := H j u j (t) + R j u 1-j (t),

for all t ∈ R. Then, u ∈ C b (S, B 0 + B 1 ).
Proof. Clearly, by (38) (for

Z = B 0 + B 1 ) we have H j u j ∈ C b (R, B 0 + B 1 )
. Also, by Lemma 25 (for Z = B 0 + B 1 ) we have

(-1) j+1 2πi R u j (t) j + it -(j + iτ + (-1) j ε) dt → H j u j (τ ), in B 0 +B 1 uniformly in τ ∈ R, when ε 0. On the other hand, we have R j u 1-j ∈ C b (R, B 0 + B 1
) and, as one can easily see,

(-1) j+1 2πi R u 1-j (t) j + it -(1 -j + iτ + (-1) j ε) dt → R j u 1-j (τ ), in B 0 + B 1 uniformly in τ ∈ R, when ε 0.
Hence, u is approaching its boundary values uniformly. Since u is analytic on S 0 we obtain the conclusion.

Interpolation of equations

Let us illustrate by some examples the fact that, in general, the surjectivity of operators is not preserved by interpolation: Example 1. Consider the operator T s : L 2 (T) × L 2 (T) → L 2 (T), defined by the formula T s (f, g) := f + g, for any (f, g) ∈ L 2 (T). One can easily see that T s : L 2 (T) × L 4 (T) → L 2 (T) and T s : L 4 (T) × L 2 (T) → L 2 (T) are surjective operators however, the operator

T s : (L 2 (T) × L 4 (T), L 4 (T) × L 2 (T)) 1/2 → (L 2 (T), L 2 (T)) 1/2 , (42)
is not surjective. Indeed, if T s in (42) is surjective, then

T s : L 3 (T) × L 3 (T) → L 2 (T)
is surjective, and we get the false embedding L 2 (T) → L 3 (T).

Example 2. Let us consider now another example which is more closely related to the equations we treat in this paper. For any p ≥ 1 let W -1,p (T 2 ) be the spaces of those distributions f that are divergences of L p -vector fields on T 2 and with f (0) = 0 (in general, if Z is a function space on T 2 , we denote by Z the space of those f ∈ Z with f (0) = 0). The norm on

W -1,p (T 2 ) is given by f W -1,p (T 2 ) = inf f 1 L p (T 2 ) + f 2 L p (T 2 ) | f = ∂ 1 f 1 + ∂ 2 f 2 .
We have that div :

L 1 (T 2 ) → W -1,1 (T 2 ) and div : L 3 (T 2 ) → W -1,3 (T 2
) are surjective operators. However, the operator div :

L 2 (T 2 ) → (W -1,1 (T 2 ), W -1,3 (T 2 )) 1/2 , (43)
cannot be surjective. Indeed, since div :

L 2 (T 2 ) → W -1,2 (T 2 ),
the surjectivity of the operator in (43) would imply that

(W -1,1 (T 2 ), W -1,3 (T 2 )) 1/2 → W -1,2 (T 2 ),
which, by duality is equivalent to

W 1,2 (T 2 ) → (W 1,∞ (T 2 ), W 1,3/2 (T 2 )) 1/2 . (44)
Note that (44) is false (see cite [START_REF] Curcȃ | On the interpolation of the spaces W l,1 (R d ) and W r,∞ (R d )[END_REF]Section 4]).

In what follows we will work in a slightly different setting. Given two pairs of Banach spaces A j → B j , j = 0, 1 and an operator T defined on B 0 + B 1 such thatT : A j → T ((B j ) is surjective, we study the surjectivity of T : (A 0 , A 1 ) θ → T ((B 0 , B 1 ) θ ). Lemma 28 below gives some sufficient additional conditions under which surjectivity is preserved by the complex interpolation. To state and prove Lemma 28 we need the technical Lemma 27.

We introduce first some notation needed in the statement of Lemma 27. Let ϕ ∈ C ∞ c ([-1, 1] , R) be a function of integral 1 and ε > 0. Define the function ϕ ε by ϕ ε (t) := ε -1 ϕ(ε -1 t), for any t ∈ R. For any ε > 0, and any (other) function g : R → Z taking values in some Banach space Z, we define the function g ε := g * ϕ ε . With this notation we state the following: Lemma 27. Let A, X 0 , X 1 , B 0 , B 1 , E, F be Banach spaces such that A, B 0 , B 1 → E and consider a bounded linear operator T : E → F . Denote A 0 := A ∩ X 0 and A 1 := A ∩ X 1 . Suppose moreover that the following conditions are satisfied:

(i) B 1 → X 1 → X 0 and B 0 → A 0 ;
(ii) A and X 1 have a separable preduals;

(iii) T : A j → F and T : B j → F are bounded for each j = 0, 1 and T (B 1 ) → T (A 1 ).

Then, for each j = 0, 1 we have the following:

For any v j ∈ L 2 (exp(t 2 ), B j ) and any ε > 0 there exists a function

u ε j ∈ L 2 (exp(t 2 ), A j ) ∩ C 3 b (R,A j ), such that T u ε j (t) = T v j,ε (t), (45)
for any in t ∈ R, and satisfying the estimates:

u ε j L 2 (A j ) v j L 2 (B j ) , (46)
and

R 1-j u ε j L 2 (exp(-t 2 ),A 1-j ) v j L 2 (exp(t 2 ),B j ) + δ j0 R 1-j v j L 2 (exp(-t 2 ),B 1-j ) . ( 47
)
where all the implicit constants do not depend on v j and ε.

(Here, v j,ε (t) = v j * ϕ ε (t) and δ j0 is the Kronecker symbol, i.e., we have δ j0 = 1 if j = 0 and δ j0 = 0 if j = 0.) Roughly speaking the conditions (45)-( 47) are describing the fact that the equation T u = T v can be solved efficiently on the boundary of the strip S. The role of Lemma 27 is to transform the easy to state conditions (i)-(iii) into the more technical conditions (45)-(47).

In order to prove Lemma 27 we need some simple facts that are easy consequences of classical inequalities.

Fact 1. Suppose Z is a Banach space and consider some function g ∈ L 2 loc (R, Z). Then, we have:

(i) g ε L 2 (exp(-t 2 ),Z) g L 2 (exp(-t 2 ),Z) , uniformly in ε > 0; (ii) g ε L 2 (exp(t 2 ),Z) g L 2 (exp(t 2 ),Z) , uniformly in ε > 0.
Proof of Fact 1. We prove only item (i), item (ii) being similar. By Minkowski's inequality we have

g ε L 2 (exp(-t 2 ),Z) = R e -t 2 B(0,1) g(t -εs)ϕ(s)ds Z dt 1/2 ≤ B R (0,1) R e -t 2 g(t -εs) 2 Z ϕ(s)dt 1/2 ds = B R (0,1) R e -(t+εs) 2 g(t) 2 Z dt 1/2 ϕ(s)ds R e -(t+εs) 2 g(t) 2 Z dt 1/2 g ε L 2 (exp(-t 2 ),Z) ,
where we have used the fact that e -(t+εs) 2 ∼ e -t 2 , when s ∈ B R (0, 1).

Fact 2. Suppose Z is a Banach space and consider some function g ∈ L 2 (exp(-t 2 ), Z). Then, we have R j g L 2 (exp(-t 2 ),Z) g L 2 (exp(t 2 ),Z) , for any j = 0, 1.

Proof of Fact 2. Fix j ∈ {0, 1}. Note first that if g ∈ L 2 (exp(-t 2 ), Z), then g ∈ L 1 (R, Z).

Using the boundedness of the function ρ j on R, one writes

R j g L 2 (exp(-t 2 ),Z) = R e -t 2 R ρ j (t -s)g(s)ds 2 Z dt 1/2 R e -t 2 R g(s) Z ds 2 dt 1/2 R g(s) Z ds.
By the Cauchy-Schwarz inequality we get

R g(s) Z ds ≤ g L 2 (exp(t 2 ),Z) ,
and Fact 2 is proved.

Let us introduce some more notation. Let Z be a Banach space and fix some N ∈ N * . For any function g ∈ L 1 loc (R, Z) we denote by E N g the conditional expectation of g with the respect to the σ-algebra generated by the intervals I k N := [k/N, (k + 1) /N ), were k ∈ Z. In other words, if (g) I is the mean of g on one of these intervals I, i.e., (g)

I := 1 |I| I g(t)dt,
we define the corresponding conditional expectation of g by

E N g := k∈Z (g) I k N 1 I k N .
See [START_REF] Diestel | Vector measures[END_REF]Chapter 5] for some fundamental properties of the conditional expectation operator E N . Now we can pass to the proof of Lemma 27.

Proof of Lemma 27. For each j = 0, 1 consider some functions v j ∈ L 2 (exp (t 2 ) , B j ). In the case where j = 0 one can simply set

u ε 0 := v 0,ε . Clearly, u ε 0 ∈ L 2 (exp(t 2 ), A 0 ) ∩ C 3 b (R,A 0 ).
It is also clear that the conditions (45), (46) are satisfied thanks to the fact that B 0 → A 0 . Let us verify (47). We have

R 1 u ε 0 L 2 (exp(-t 2 ),A 1 ) ∼ R 1 v 0,ε L 2 (exp(-t 2 ),A) + R 1 v 0,ε L 2 (exp(-t 2 ),X 1 ) , (48)
and it remains to bound each term in the right hand side of (48). Since B 0 → A ∩ X 0 , we have in particular that B 0 → A and hence,

R 1 v 0,ε L 2 (exp(-t 2 ),A) R 1 v 0,ε L 2 (exp(-t 2 ),B 0 ) v 0,ε L 2 (exp(t 2 ),B 0 ) v 0 L 2 (exp(t 2 ),B 0 ) , (49)
where for the second " " we have used Fact 2 and for the third " " we have used Fact 1 (ii).

Since

B 1 → X 1 , R 1 v 0,ε L 2 (exp(-t 2 ),X 1 ) R 1 v 0,ε L 2 (exp(-t 2 ),B 1 ) = (R 1 v 0 ) ε L 2 (exp(-t 2 ),B 1 ) R 1 v 0 L 2 (exp(-t 2 ),B 1 ) , (50)
where for "=" we have used the relation ρ * ϕ ε = ϕ ε * ρ and for the last " " we have used Fact 1 (i). From (48), ( 49) and (50) we obtain (47) in the case j = 0. Now, we deal with the case j = 1. By using the open mapping theorem one gets that, if b ∈ B 1 , then there exists a ∈ A 1 such that T a = T b and a A 1 ≤ C b B 1 for some constant C > 0. As a consequence, for each k ∈ Z we can find some elements

a k N ∈ A 1 with T a k N = T (v 1 ) I k N and such that a k N A 1 ≤ C (v 1 ) I k N B 1 .
Hence, defining u N : R → A 1 by

u 1,N := k∈Z a k N 1 I k N ,
we have

T u 1,N (t) = T E N v 1 (t), (51)
for any t ∈ R, and

u 1,N (t) A 1 E N v 1 (t) B 1 , (52)
uniformly in t ∈ R.

Define now the function u ε 1,N := u 1,N * ϕ ε = (u 1,N ) ε , the convolution being in the t variable. Thanks to (51) we have

T u ε 1,N (t) = T (E N v 1, ) ε (t), (53)
for any t ∈ R.

Let us observe that, when N → ∞,

(E N v 1 ) ε (t) -v 1,ε (t) B 1 → 0, (54)
uniformly in t ∈ R.

Indeed, by Jensen's inequality and [12, Corollary 2, p. 126] we have

(E N v 1 ) ε (t) -v 1,ε (t) B 1 ≤ B R (t,1/ε) (E N v 1 ) ε (s) -v 1,ε (s) B 1 ϕ ε (t -s)ds ≤ B R (t,1/ε) (E N v 1 ) ε (s) -v 1,ε (s) 2 B 1 ϕ ε (t -s)ds 1/2 ε E N v 1 -v 1 L 2 (R,B 1 ) → 0.
Also one easily observe that the sequence of functions (u ε 1,N ) N ≥1 is equi-continuous and uniformly bounded. Indeed, using the Cauchy-Schwaz inequality,

u ε 1,N (t 1 ) -u ε 1,N (t 2 ) A 1 ≤ R u 1,N (s) A 1 |ϕ ε (t 1 -s) -ϕ ε (t 2 -s)| ds ≤ u 1,N L 2 (R,A 1 ) R |ϕ ε (t 1 -s) -ϕ ε (t 2 -s)| 2 ds 2 , ( 55
)
4 Spectral analysis

In this section we study the solutions of divergence-like equations via L 2 -based Fourier analysis methods. This is done by a slight modification of the ideas of Bourgain and Brézis used in the proof of [START_REF] Bourgain | On the equation div Y = f and application to control of phases[END_REF]Lemma 2]. While in this case the techniques we use are essentially those of Bourgain and Brézis, we give more general existence results that take into account the shape of the Fourier spectrum of solutions. The final results of this section will represent the "1-endpoint" when we apply the W-method (see Lemma 35 and Lemma 36 below).

Symbols with bounded Fourier transform

Let 1 ≤ ≤ d be some integers and let m : R d → C be a function. We say that m is an -BB symbol 5 if the following conditions are satisfied:

(i) there exists a constant C > 0 such that, in the case < d,

R d- |∂ α 1 1 ...∂ α l m(ξ , ξ )| dξ ≤ C |ξ | +|α| , (71)
for all α = (α 1 , ..., α ) ∈ {0, 1} and all ξ ∈ (0, ∞) , and, in the case = d,

|∂ α 1 1 ...∂ α d d m(ξ)| ≤ C |ξ| +|α| , (72)
for all α = (α 1 , ..., α d ) ∈ {0, 1} d and all ξ ∈ (0, ∞) d ;

(ii) m is an odd function in each of the components ξ 1 , ξ 2 ,....,ξ , i.e., m (ξ 1 , ..., ξ j-1 , -ξ j , ξ j+1 ...., ξ d ) = -m (ξ 1 , ..., ξ j-1 , ξ j , ξ j+1 ...., ξ d ) ,

for all 1 ≤ j ≤ , and all ξ 1 , ξ 2 , ...., ξ d ∈ R.

For any integer ν we denote by I ν the interval [2 ν-1 , 2 ν ]. For every k = (k 1 , ..., k ) ∈ Z we consider the positive dyadic box I k := I k 1 × ... × I k and we associate to it the -symmetric set:

s (I k ) := α 1 ,...,α ∈{0,1} ((-1) α 1 I k 1 ) × ... × ((-1) α I k ) ⊂ R .
The next technical Lemma is the basis for all of our results in this section. Its proof consists in slightly adapting some arguments of Bourgain and Br´ezis (see [ Proof. We first prove Lemma 30 in the case where m is a d-BB symbol. In this case we have to prove that

k∈Z d s d (I k ) m(ξ)e i ξ,x dξ C, (74)
Now, let us notice that whenever a : R d → C, b 1 , ..., b d : R → C are sufficiently smooth functions and J j = [q j , r j ], (q j < r j ), j = 1, ..., d are d intervals, we have

J 1 ×...×J d a (ξ) d j=1 b j (ξ j ) dξ = α∈{0,1} d J 1 ×...×J d (-∇) α a (ξ) d j=1 [b j (ξ j )] α j J j dξ, (76)
where, for each 1 ≤ j ≤ d, the quantity [b j (ξ j )]

α j J j is defined 6 as follows [b j (ξ j )] 0 J j := r j q j b j (t) dt δ r j (ξ j ) and [b j (ξ j )] 1 J j := ξ j q j b j (t) dt,
where δ r j is the Dirac measure on R concentrated in r j . The formula (76) easily follows by induction on d and integration by parts.

Fix some integers k 1 , ..., k d and consider the intervals I k j = 2 k j -1 , 2 k j , j = 1, ..., d. By applying (76) to the functions a = m and b j (ξ j ) = sin (ξ j x j ), we obtain

I k 1 ×...×I k d m(ξ) d j=1 sin (ξ j x j ) dξ = α∈{0,1} d I k 1 ×...×I k d (-∇) α m (ξ) d j=1
[sin (ξ j x j )]

α j I k j dξ. ( 77
)
By a direct computation,

J sin (t • x j ) dt min 4 k j |x j | , 1 |x j | ,
for any subinterval J ⊂ I k j , and consequently, [sin (ξ j x j )]

α j I k j min 4 k j |x j | , 1 |x j | , ( 78
)
for any j = 1, ..., d, any α j ∈ {0, 1} and any ξ j ∈ I k j .

Let us fix some integer 0 ≤ l ≤ d and consider the case of α = (1, ..., 1, 0, ..., 0) ∈ {0, 1} d (l values equal to 1). By (78) we get that the quantity is bounded by

I k 1 ×...×I k d (-∇) α m (ξ) d j=1 [sin (ξ j x j )]
I k 1 ×...×I k l ∂ 1 ...∂ l m ξ 1 , ..., ξ l , 2 k l+1 , ..., 2 k d dξ 1 ...dξ l d j=1 min 4 k j |x j | , 1 |x j | . ( 79
)
Fix some index µ ∈ {1, ..., d}. Using the fact that m is a d-BB symbol one can write (see (72))

I k 1 ×...×I k l ∂ 1 ...∂ l m ξ 1 , ..., ξ l , 2 k l+1 , ..., 2 k d dξ 1 ...dξ l ≤ C I k 1 ×...×I k l 1 2 kµ(d+l) dξ 1 ...dξ l ∼ C 2 k 1 +...+k l 2 kµ(d+l) .
Combining this with (79) we get that

I k 1 ×...×I k d (-∇) α m (ξ) d j=1
[sin (ξ j x j )]

α j I k j dξ C 2 k 1 +...+k l 2 kµ(d+l) d j=1 min 4 k j |x j | , 1 |x j | ,
and hence, for

Γ µ := k = (k 1 , ..., k d ) ∈ Z d | k µ = max 1≤j≤d k j , k∈Γµ I k 1 ×...×I k d (-∇) α m (ξ) d j=1 [sin (ξ j x j )] α j I k j dξ is bounded by C k∈Γµ 2 k 1 +...+k l 2 kµ(d+l) d j=1 min 4 k j |x j | , 1 |x j | = C k∈Γµ 2 k 1 +...+k l 2 kµ(d+l) 2 k 1 +...+k d d j=1 min 2 k j |x j | , 1 2 k j |x j | . (80)
For any k ∈ Γ µ we have k µ = max 1≤j≤d k j . Hence, we can write

2 k 1 +...+k l 2 kµ(d+l) 2 k 1 +...+k d = l j=1 2 k j -kµ d j=1 2 k j -kµ ≤ 1,
and therefore, the right hand side of (80) is at most

C k∈Γµ d j=1 min 2 k j |x j | , 1 2 k j |x j | ≤ C k 1 ,k 2 ,...,k d ∈Z d j=1 min 2 k j |x j | , 1 2 k j |x j | = C d j=1 k j ∈Z min 2 k j |x j | , 1 2 k j |x j | C.
In other words, we have seen that for any µ ∈ {1, ..., d} and any multiindex α ∈ {0, 1} d of the form α = (1, ...1, 0, ..., 0),

k∈Γµ I k 1 ×...×I k d (-∇) α m (ξ) d j=1
[sin (ξ j x j )]

α j I k j dξ C. (81)
Clearly, (81) remains true for any α ∈ {0, 1} d . Now, observing that Z d is covered by the union of the sets Γ 1 , ...Γ d , one can write

k∈Z d I k 1 ×...×I k d (-∇) α m (ξ) d j=1 [sin (ξ j x j )] α j I k j dξ ≤ d µ=1 k∈Γµ ... C.
This, together with (75) and (77) proves Lemma 30 in the case = d (i.e., (74)). The case where 1 ≤ < d, can be obtained from the case = d as follows.

Suppose that 1 ≤ < d and m is an -BB symbol. Let M be a measurable subset of R d-of finite measure. For each x ∈ R d-define the function m x : R → C by m x (ξ ) := M m(ξ , ξ )e i ξ ,x dξ , for all ξ ∈ R . Thanks to the fact that m satisfies the condition (71), m x is well-defined and it satisfies uniformly in θ the condition (72) as a symbol on R . Indeed,

|∂ α 1 1 ...∂ α m x (ξ )| = M ∂ α 1 1 ...∂ α m(ξ , ξ )e i ξ ,x dξ ≤ M |∂ α 1 1 ...∂ α m(ξ , ξ )| dξ ≤ R d- |∂ α 1 1 ...∂ α m(ξ , ξ )| dξ ≤ C |ξ | +|α| , (82)
for all α = (α 1 , ..., α ) ∈ {0, 1} and all ξ ∈ (0, ∞) . (Note that the final estimate in (82) does not depend on the set M .) Also, m θ is odd in each variable and hence, we have (73). Now, using (72) for m θ , we have Proof. For each n ∈ N, we consider the functions K n defined by

k ∈Z s (I k )×M m(ξ)e i ξ,x dξ = k ∈N s (I k ) M m(ξ , ξ )e i ξ ,x dξ e i ξ ,x dξ = k ∈N s (I k ) m x (ξ ) e i ξ ,x dξ C, uniformly in x ∈ R d ,
K n (ξ) = m(ξ)1 J d n (ξ) for all ξ ∈ R d , where J n := [-2 n , -2 -n ] ∪ [2 -n , 2 n ] and J d n = J n × ... × J n (d times).
It is easy to see that K n are well defined continuous functions. One can also see that K n are uniformly bounded. Indeed, applying Lemma 30 for M = J d- n (we suppose that < d; the case = d is similar) we can write (using the triangle inequality),

|K n (x)| ∼ J n ×J d- n m(ξ)e i ξ,x dξ = k ∈Z s (I k )⊆J n s (I k )×J d- n m(ξ)e i ξ,x dξ ≤ k ∈Z s (I k )×J d- n m(ξ)e i ξ,x dξ C,
uniformly in x ∈ R d , and in n ∈ N. (Here, for "=", we have used the fact that the sets s (I k ) with s (I k ) ⊆ J n are pairwise almost disjoint and they cover J n .) Hence, K n L ∞ C uniformly in n ∈ N. Using the sequential Banach-Alaoglu theorem, we can find some K ∈ L ∞ , with K L ∞ C and such that K n → K in the w * -topology on L ∞ , up to a subsequence.

Consider now some ψ ∈ S . Clearly, for any n,

K n , ψ = m1 J d n , ψ . (83)
Since ψ ∈ L 1 we have K n , ψ → K, ψ up to a subsequence when n → ∞. Also, we have

m1 J d n , ψ → m, ψ ,
and by (83) we get K, ψ = m, ψ .

The uniqueness of K immediately follows from (84). Lemma 31 is proved.

Divergence-like equations in

Ẇ d/2,2
We can now prove our existence result for divergence-like equations in a particular type of critical spectral spaces. We start by an analogue of [5, Lemma 2]. The proof we give below rests on some elaborations of the main ideas used by Bourgain and Brézis in the proof of [5, Lemma 2]. Lemma 31 from the previous subsection will play here an important role.

In what follows let us denote by ∇ σ 2 u the first two components of the "σ-gradient" of u, namely,

∇ σ 2 u := (∂ σ 1 u, ∂ σ 2 u),
where ∂ σ j is the Fourier multiplier ∂ σ j := σ j (∇).

Lemma 33. Let d ≥ 3 be an integer. Consider the set U :

= R d-1 × (0, ∞). If σ : R d → R satisfies (P1),(P2) then, |∇| -1 |∇ σ 2 | 2 u Y * /U ∇ σ 2 u (L 1 +Y * )/U + ∇ σ 2 u 1/2 Y * /U ∇ σ 2 u 1/2 (L 1 +Y * )/U , (85)
for any u ∈ S , where Y := Ẇ d/2,2 .

(For the meaning of the notation Y * /U and (L 1 + Y * )/U see subsection 2.3.)

Proof. Clearly, Y * = Ẇ -d/2,2 . First we show that

R d σ 2 1 (ξ)σ 2 2 (ξ) |ξ| d+2 1 U (ξ) | u(ξ)| 2 dξ ∇ σ 2 u 2 (L 1 +Y * )/U + ∇ σ 2 u Y * /U ∇ σ 2 u (L 1 +Y * )/U , (86)
for any function u ∈ S .

Consider some functions

F 1 , F 2 , h 1 , h 2 , F - 1 , F - 2 ∈ S with spec(F - 1 ), spec(F - 2 ) ⊆ U c such that ∇ σ 2 u = (F 1 , F 2 ) + (h 1 , h 2 ) + (F - 1 , F - 2 ), (87)
and

F 1 L 1 + F 2 L 1 + h 1 Y * + h 2 Y * ≤ 2 ∇ σ 2 u (L 1 +Y * )/U . (88)
We have

R d σ 2 1 (ξ)σ 2 2 (ξ) |ξ| d+2 1 U (ξ) | u(ξ)| 2 dξ = c R d σ 1 (ξ)σ 2 (ξ) |ξ| d+2 1 U (ξ) ∂ σ 1 u(ξ) ∂ σ 2 u(ξ)dξ,
Using this, (87) and the fact that spec(F - j ) ⊆ U c (and hence 1 U (ξ) F - j (ξ) = 0, for j = 1, 2 and for all ξ ∈ R d ), we can write

R d σ 2 1 (ξ)σ 2 2 (ξ) |ξ| d+2 1 U (ξ) | u(ξ)| 2 dξ = I + II, where
I := c R d σ 1 (ξ)σ 2 (ξ) |ξ| d+2 1 U (ξ) F 1 (ξ) F 2 (ξ)dξ,
and II is the sum of a finite number of terms of the form

c R d σ 1 (ξ)σ 2 (ξ) |ξ| d+2 1 U (ξ)g 1 (ξ)g 2 (ξ)dξ, (89)
where each g k : R d → C is one of the functions

h 1 , h 2 , h 1 , h 2 , F 1 , F 2 , F 1 , F 2
and at least one g k is h j or h j for some j ∈ {1, 2}.

One can verify immediately that the symbol m defined by

m(ξ) := σ 1 (ξ)σ 2 (ξ) |ξ| d+2 1 U (ξ), ξ ∈ R d \ {0} ,
satisfies the conditions in Lemma 31. Hence, by a applying Lemma 31 and (88),

|I| ∼ | K * F 1 , F 2 | ≤ K * F 1 L ∞ F 2 L 1 ≤ K L ∞ F 1 L 1 F 2 L 1 ∇ σ 2 u 2 (L 1 +Y * )/U . (90)
(Here, we have used the notation from Lemma 31: K = m.)

In order to estimate II we estimate each of its terms of the form (89). By the Cauchy-Schwarz inequality, we get

R d σ 1 (ξ)σ 2 (ξ) |ξ| d+2 1 U (ξ)g 1 (ξ)g 2 (ξ)dξ ≤ 2 k=1 R d 1 U (ξ) |g k (ξ)| 2 |ξ| d dξ 1/2 , ( 91
)
where we have used the inequality σ 1 (ξ)σ 2 (ξ) |ξ| 2 1, which follows directly from (P1).

Note that, if |g k | = | h j |, for some j ∈ {1, 2}, then, by (88)

R d 1 U (ξ) |g k (ξ)| 2 |ξ| d dξ 1/2 ≤ h j Y * ≤ 2 ∇ σ 2 u (L 1 +Y * )/U . (92)
If |g k | = | F j |, for some j ∈ {1, 2}, then, since F j 1 U = ( ∂ σ j u -h j )1 U , the triangle inequality together with (88) gives One can immediately check that the function

σ 1 σ 2 := (σ 1 • R) 2 -(σ 2 • R) 2
is odd in each of the variables ξ 1 , ξ 2 . Using this, we easily observe that the symbol m defined by m (ξ) := σ 1 (ξ)σ 2 (ξ)

|ξ| d+2 1 U (ξ), ξ ∈ R d \ {0} ,
satisfies the conditions in Lemma 31. Hence, as in (86), we obtain that

R d σ 1 (ξ) 2 σ 2 (ξ) 2 |ξ| d+2 1 U (ξ) | v(ξ)| 2 dξ ∇ σ 2 v 2 (L 1 +Y * )/U + ∇ σ 2 v Y * /U ∇ σ 2 v (L 1 +Y * )/U , (95)
for any function v ∈ S , where ∇ σ 2 v := (∂ σ 1 v, ∂ σ 2 v), and ∂ σ 1 is the Fourier multiplier of symbol σ j , for any j = 0, 1. Since the spaces Y * /U and (L 1 + Y * )/U are invariant under the rotation R, by applying (95) to the function v = u • R t we obtain (by changing the variables) that

R d (σ 1 (ξ) -σ 2 (ξ)) 2 (σ 1 (ξ) + σ 2 (ξ)) 2 |ξ| d+2 1 U (ξ) | u(ξ)| 2 dξ is bounded by ∇ σ 2 u 2 (L 1 +Y * )/U + ∇ σ 2 u Y * /U ∇ σ 2 u (L 1 +Y * )/U , (96)
for any function u ∈ S . By adding up, we get from ( 86) and (96) that With the same methods one can prove an analogue of Theorem 8 when d = 2 and the source space is Ẇ 1,2 (R 2 ): Lemma 36. Consider the numbers δ ∈ (0, π/8) and ε ∈ (0, 2]. Then, for any vector field v ∈ Ẇ 1,2 (R 2 ), with spec(v) ⊆ C δ , there exist a vector field u ∈ L ∞ (R 2 ) ∩ Ẇ 1,2 (R 2 ), with spec(u j ) ⊆ C (1+ε)δ , such that div u = div v,

R d σ(ξ) |ξ| d+2 1 U (ξ) | u(ξ)| 2 dξ ∇ σ 2 u 2 (L 1 +Y * )/U + ∇ σ 2 u Y * /U ∇ σ 2 u (L 1 +Y * )/U , (97)
and u L ∞ ∩ Ẇ 1,2 v Ẇ 1,2 .

14 3. 1 24 4analysis 31 4 . 1 5 Solutions in interpolation spaces 42 5 . 1

 141244151 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 A naive interpolation strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 The main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 About the proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Function spaces 2.1 Sobolev and Besov spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Lorentz-Sobolev spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Some quotient spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 The W-method of complex interpolation Construction of the interpolation space . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 A particular case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.3 Solutions of linear equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.3.1 Boundary values of functions on the strip . . . . . . . . . . . . . . . . . . . 21 3.3.2 Interpolation of equations . . . . . . . . . . . . . . . . . . . . . . . . . . . Spectral Symbols with bounded Fourier transform . . . . . . . . . . . . . . . . . . . . . . . 31 4.2 Divergence-like equations in Ẇ d/2,2 . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Proof of the main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 5.2 Remark concerning the "third" parameter . . . . . . . . . . . . . . . . . . . . . . 46 Suppose d ≥ 2 is an integer and consider some compactly supported function f ∈ L d R d . Standard Calderón-Zygmund theory shows that there exists a vector field u

1 ( 1 ( 1 v

 111 R d ) there exists a vector field u ∈ L ∞ (R d ) ∩ Ḃd/r,r R d ) such that div u = div v,and u L ∞ ∩ Ḃd/r,r Ḃd/r,r 1 .

  on R d , where D := ∪ d j=1 D j . Let us recall now some standard notation concerning the Fourier multipliers. To a scalar valued function m ∈ L 1 loc (R d \{0}, R) we associate the Fourier multiplier m(∇) defined by the relation m(∇)f (ξ) := m(ξ) f (ξ), on R d , for any Schwartz function f whose Fourier transform f is compactly supported and vanishing in a neighborhood of 0. In most of the cases one can extend the meaning of m(∇) as follows.

Let

  Y be a Banach function space on R d and let D ⊆ R d be a measurable set. Relative to the set D we define the closed subspace Y D of Y by Y D := {f ∈ Y | spec(f ) ⊆ D} , the norm being the one induced by Y . For simplicity we will denote the quotient space Y /Y D c by Y /D. In the case where f ∈ Y is a Schwartz function, we define its Y /D-seminorm by

5 ,

 5 Lemma 3, p. 404]): Lemma 30. Let d ≥ 1 be an integer and let m : R d → C be an -BB symbol for some 1 ≤ ≤ d and some constant C. Then, k ∈Z s (I k )×M m(ξ)e i ξ,x dξ C, for any measurable subset M ⊆ R d-of finite measure, uniformly in x ∈ R d and in M . (By convention, if = d, then s (I k ) × M is replaced by s d (I k ).)

e

  uniformly in x = (x 1 , ..., x d ) ∈ R d .Since m is odd in the variables ξ 1 , ..., ξ d , for any k ∈ Z d one can write,s d (I k ) m(ξ)e i ξ,x dξ = s d (I k ) iξ j x j dξ = (2i) j x j ) dξ.

α j I k j dξ 6

 6 Strictly speaking we should write [b j (•)] αj Jj (ξ j ) instead of [b j (ξ j )] αj Jj . However, for simplicity we prefer here to use the last notation.

  which proves Lemma 30. By applying Lemma 30 we can deduce the following useful fact: Lemma 31. Let d ≥ 1 and 1 ≤ ≤ d be some integers, and let m : R d → C be an -BB symbol satisfying condition (71) or (71) (when = d) for some constant C. Then, there exists some kernel K ∈ L ∞ R d such that K(ξ) = m(ξ) on R d and K L ∞ C, the implicit constant not depending on C. Remark 32. The meaning of Lemma 31 is that there exists a unique function K ∈ L ∞ , such that K L ∞ C and K, ψ = m, ψ , for any function ψ ∈ S .

R d 1 U 2 |ξ| d dξ 1 / 2 = R d 1 U 2 dξ 1 / 2 ≤

 12121212 (ξ) |g k (ξ)| (ξ) |ξ| -d ∂ σ j u(ξ) -h j (ξ) ∂ σ j u Y * /U + h j Y * ≤ ∇ σ 2 u Y * /U + 2 ∇ σ 2 u (L 1 +Y * )/U ≤ 3 ∇ σ 2 u Y * /U .(93)Since |g k | = | h j | (for some j) for at least one k, we get from (91), (92) and (93) thatR d σ 1 (ξ)σ 2 (ξ) |ξ| d+2 1 U (ξ)g 1 (ξ)g 2 (ξ)dξ ∇ σ 2 u Y * /U ∇ σ 2 u (L 1 +Y * )/U . Hence, |II| ∇ σ 2 u Y * /U ∇ σ 2 u (L 1 +Y * )/U .(94)By (93) and (94) we get (86).Consider the rotationR (ξ) = (ξ 1 -ξ 2 , ξ 1 + ξ 2 , ξ 3 , ..., ξ d ),for any ξ ∈ R d . Consider now the functionsσ 1 := σ 1 • R -σ 2 • R, and σ 2 := σ 1 • R + σ 2 • R.

1 Uσ 2 u 2 (L 1 1 U 2 Y 2 (L 1 2 Y 2 Y 2 (L 1 |∇| - 1 |∇| - 1 2 L 2 ξ∼ |ξ| - 1 d 2 L 2 ξ∼ |ξ| -1 |ξ| 2 1 D 2 L 2 ξ= |ξ| 1 D 2 L 2 ξ∼ |ξ| 1 D|∇| - 1

 12112212221112212212212211 where σ(ξ) := σ 2 1 (ξ)σ 2 2 (ξ) + (σ 1 (ξ) -σ 2 (ξ))2 (σ 1 (ξ) + σ 2 (ξ))2 , for any ξ ∈ R d . Since for any real numbers a, b we havea 2 b 2 + (a -b) 2 (a + b) 2 ∼ a 4 + b 4 ,we obtain σ(ξ) ∼ σ 4 1 (ξ) + σ 4 2 (ξ), for all ξ ∈ R d , and now (97) gives us (ξ) | u(ξ)| 2 dξ ∇ +Y * )/U + ∇ σ 2 u Y * /U ∇ σ 2 u (L 1 +Y * )/U .(98)Note that by[START_REF] Pisier | Martingales in Banach Spaces[END_REF] (with D = U ), we have|∇| -1 |∇ σ 2 (ξ) | u(ξ)| 2 dξ,and together with (98) this concludes the proof of Lemma 33.By composition with rotations and by adding up inequalities of the form (85), Lemma 33 easily implies the following: Lemma 34. Let d ≥ 3 be an integer. Suppose that the family of functions G 1 , ..., G d : R d → R d-1 is adapted to the family of half-spaces D 1 , ..., D d ⊂ R d . For any j ∈ {1, ..., d} we have|∇| -1 |G j (∇)| 2 u Y * /U G j (∇)u (L 1 +Y * )/D j + G j (∇)u 1/* /D j G j (∇)u 1/+Y * )/D j ,for any u ∈ S , whereY := Ẇ d/2,2 .(For the properties of the functions G 1 , ..., G d and their relation with the half-spaces D 1 , ..., D d see the subsection 1.3 in the introduction of this paper.)We can now state and prove the existence results of this subsection:Lemma 35. Let d ≥ 3 be an integer. Suppose that the family of functions G 1 , ..., G d : R d → R d-1is adapted to the family of half-spaces D 1 , ..., D d ⊂ R d . Then, for any system of (d -1)-vector fields (v j ) j=1,..,d with v j ∈ Ẇ d/2,2 (R d ) and spec(v j ) ⊆ D j , there exists a system of (d -1)-vector fields (u j ) j=1,..,d , withu j ∈ L ∞ (R d ) ∩ Ẇ d/2,2 (R d ) and spec(u j ) ⊆ D j , such that d j=1 G j (∇) • u j = d j=1 G j (∇) • v j , and d j=1 u j L ∞ ∩ Ẇ d/2,2 d j=1 v j Ẇ d/2,2 .Proof. As before, let Y be the space Ẇ d/2,2 . According to Lemma 34, for any ϕ ∈ S , we have|∇| -1 |G j (∇) |G j (∇) ϕ Y * /D j G j (∇) ϕ (L 1 +Y * )/D j + G j (∇) ϕ 1/* /D j G j (∇) ϕ 1/2 (L 1 +Y * )/D j = G j (∇) ϕ (L 1 +Y * )/D j +(ε 1/2 G j (∇) ϕ 1/* /D j )(ε -1/2 G j (∇) ϕ 1/+Y * )/D j ) ≤ G j (∇) ϕ (L 1 +Y * )/D j +ε G j (∇) ϕ Y * /D j + ε -1 G j (∇) ϕ (L 1 +Y * )/D j ,for any ε ∈ (0, 1) and any j ∈ {1, ..., d}. By adding up these inequalities we getd j=1 |G j (∇) |G j (∇) ϕ Y * /D j ε d j=1 G j (∇) ϕ Y * /D j +ε 1-N d j=1 G j (∇) ϕ (L 1 +Y * )/D j .(99)Since the family G 1 ,...,G d is adapted to D 1 ,...,D d , (see[START_REF] Calderón | Intermediate spaces and interpolation, the complex method[END_REF]) we easily getd j=1 |G j (ξ)| β 1 D j (ξ) ∼ β |ξ| β 1 D (ξ),(100)on R d , for any β > 0. Using now (100), with β = 2, we can writed j=1 |G j (∇) |G j (∇) ϕ Y * /D j ∼ d j=1 |ξ| -1 |G j (ξ) | 2 1 D j (ξ) | ϕ (ξ) | |ξ| -d/j=1 |G j (ξ) | 2 1 D j (ξ) | ϕ (ξ) | |ξ| -d/(ξ) | ϕ (ξ) | |ξ| -d/(ξ) | ϕ (ξ) | |ξ| -d/2way (by (10)) we can writed j=1 G j (∇) ϕ Y * /D j ∼ d j=1 |G j (ξ) |1 D j (ξ) ϕ (ξ) | |ξ| -d/2 L 2 ξ ∼ d j=1 |G j (ξ) |1 D j (ξ) ϕ (ξ) | |ξ| -d/(ξ) | ϕ (ξ) | |ξ| -d/2 |G j (∇) |G j (∇) ϕ Y * /D j ∼ d j=1 G j (∇) ϕ Y * /D j ,and together with (99) yields d j=1 G j (∇) ϕ Y * /D j ε d j=1 G j (∇) ϕ Y * /D j + ε -1 d j=1 G j (∇) ϕ (L 1 +Y * )/D j . Choosing ε sufficiently small one can write d j=1 G j (∇) ϕ Y * /D j d j=1 G j (∇) ϕ (L 1 +Y * )/D j .By duality (using the closed range theorem) we get Lemma 35.

  for any t ∈ R and any nonnegative integer k ≤ d + 2, where C is a positive constant depending only on d. It follows that (see[START_REF] Muscalu | Classical and Multilinear Harmonic Analysis[END_REF] Theorem 8.2 , p. 197]) for any a ∈ (1, ∞), the norm of the operator |∇| it : L a → L a satisfies

	|∇| it	L a →L a

a C(1 + |t|) d+2 .

This implies, via the real method of interpolation (see for instance [2, Theorem 5.2.1 (2), p. 109 ]), that for any a ∈ (1, ∞) and any b ∈ [1, ∞], we have

  1 ) θ was already proved and used by Peetre in a different context (see [22, Lemme 1.1]). Both proofs, the one that we give below and Peetre's, are easy consequences of the ideas of Calderón from [10, Section 9.4].

In this paper the distributions with d -1 components will be called (d -1)-vector fields.

Here, "W" stands for "weak".

The results of this subsection will be used only in the proof of Theorem 11.

Here, BB stands for "Bourgain-Brézis".

Here we use the fact that if the dual X * of a Banach space X is separable, then, X is separable (see for instance [14, Theorem 4.6-8, 245]).

for any t 1 , t 2 ∈ R and it remains to notice that, by (52), we have

Since (u ε 1,N ) N ≥1 is equi-continuous and uniformly bounded sequence, and A, X 1 have a separable preduals, there exists some u ε 1 ∈ L 2 (R, A 1 ) such that u ε 1,N (t) → u ε 1 (t) in the w * -topology on A and in the w * -topology on X 1 , up to a subsequence, for all t ∈ R. By an argument similar to the one used in (55), it is easy to see that one can choose u ε 1 ∈ C 3 b (R, A 1 ). Thanks to (53) and (54) one can write T u ε 1 (t) = T v 1,ε (t), for all t ∈ R, which proves (45). In order to verify (46) one uses the Young inequality and (52):

lim inf

Observe now that, as in (56), we get

where for the second " " we have used Fact 1 (ii). In particular, we have that

Let us verify now that u ε 1 also satisfies (47). We can write

where for the first " " we have used Fact 2 and for the second " " we have used the embedding X 1 → X 0 (that implies A 1 → A 0 ). Combining (57) with (58) we get

and (47) is proved in the case j = 1. Lemma 27 is proved.

We are now able to state and prove the main result of subsection 3.3:

Lemma 28. Fix some number θ ∈ (0, 1). Let the Banch spaces A, X 0 , X 1 , A 0 , A 1 , B 0 , B 1 , E, F and the operator T be as in Lemma 27. Moreover, we assume that X 0 , X 1 and B 1 are U M D spaces and that (X 0 , X 1 ) θ has a separable predual. Then, for any b ∈ (B 0 , B 1 ) θ there exists some a ∈ A ∩ (X 0 , X 1 ) θ such that T a = T b,

Since we can replace (if necessary) v by exp(z 2 -θ 2 )v, we can assume without loss of generality that v j ∈ L 2 (exp (t 2 ) , B j ), where v j (t) := v(j + it), for all t ∈ R. Define, for each ε ∈ (0, 1), the function v ε on S by v ε := v * ϕ ε , as in the statement of Lemma 27.

Note that, thanks to Lemma 26,

for all t ∈ R. From this identity, since B 1 is an U M D space, we can write

where for the second " " we have used Young's inequality and for the last " " we have used (60). In particular, we get

By Lemma 27 there exist some functions 45), ( 46), (47). Define ũε :

for all z ∈ S 0 , and ũε

We show that, for any j = 0, 1, we have the estimate:

for any bounded linear operator Λ j : A j → X j with Λ j ≤ 1, the implicit constant not depending on Λ j .

Using (63) we write:

Since the spaces X j have the U M D property, we get

where for the third "≤" we have used (46) and for the last " " we have used (60). It remains to estimate the second term in the right hand side of (65):

where for the first " " we have used (47) and for the last " " we have used (60) and (61). By (65), (66), (67) we have proved (64). Hence, we have obtained

This implies that for a ε := ũε (θ) we have

Note that, by Proposition 22, (B 0 |B 1 ) θ = (B 0 , B 1 ) θ , and by Corollary 24, (A 0 , X 0 | A 1 , X 1 ) θ → A ∩ (X 0 , X 1 ) θ . From this and (68) we get

We observe that for b ε := v ε (θ) we have

Indeed, by applying Lemma 20, (62), the continuity of T : E → F and (45), one gets

We let ε → 0. Since v j,ε → v j in L 2 (R, B j ), for each j = 0, 1 we get that b ε → b in (B 0 |B 1 ) θ = (B 0 , B 1 ) θ . Also, thanks to (69), since A and(X 0 , X 1 ) θ have separable preduals, by the sequential Banach-Aloglu theorem, there exists some a ∈ A ∩ (X 0 , X 1 ) θ such that a 1/n → a (n ∈ N * ) in the w * -topology on A and in the w * -topology on (X 0 , X 1 ) θ , up to a subsequence. Also, by (69) we get

It follows that T b 1/n → T b and T a 1/n → T a in the w * -topology of F , up to a subsequence. Consequently, by (70) we have T a = T b.

θ we can use the above compactness argument in order to obtain a solution for any b ∈ (B 0 |B 1 ) θ . Lemma 28 is proved.

Remark 29. One can easily adapt Lemma 27 and Lemma 28 to the more general case of the equations T u = Lv, where T, L : E → F are possibly different operators. For this we have to change the conditions (i) and (iii) in Lemma 27 by (i') B 1 → X 1 → X 0 and there exists an operator L T : E → F such that L T : B j → A j is bounded for each j = 0, 1 and T • L T = L on B 0 ;

(iii') T : A j → F and L : B j → F are bounded for each j = 0, 1 and L(B 1 ) → T (A 1 ).

The modifications needed for the corresponding proofs are minor. However, for the sake of simplicity, we preferred to present the proofs only in the case T = L.

(For the meaning of C δ see subsection 1.3.) Sketch of the proof. First we establish the result for ε = 2. Let us denote by D(C δ ) the set defined by

the union being taken after all the dyadic boxes

) that are included in C δ and are maximal (with respect to the inclusion relation) with this property. One can find a finite number of rotations R 1 , ..., R n : R 2 → R 2 such that

for some sufficiently large r > 0. As in the proof of (86) we get

for any u ∈ S , where Y = Ẇ d/2,2 . As in the proof of Lemma 31, by Lemma 30 (applied in the case = d = 2) one can see that there exists K ∈ L ∞ such that

with the same meaning as in Lemma 31. We use then the same method as in (90). The rest of the argument remains essentially the same as the one used in the proof of (86).

Note that, by (104) we get (by composition with rotations) that

for all j ∈ {1, ..., n}, where R l j (ξ) is the l-th coordinate of the vector R j (ξ). Using the (103

for all j ∈ {1, ..., n}, and one can write

for all j ∈ {1, ..., n}. One can easily check that

for all ξ ∈ R 2 . Hence, by adding up the inequalities (105) we get

which can be rewritten as

By duality (as in the proof of Lemma 35) we obtain that for any vector field

Thanks to (103) this immediately implies Lemma 36 in the case ε = 2. To obtain the result for any ε ∈ (0, 1) simply cover the symmetric cone C δ with a small union of rotated copies of the symmetric cones C δ/n for some large integer n > 0. It suffices now to apply the result corresponding to the case ε = 2 to each rotated copy of C δ/n and then add the obtained solutions.

Solutions in interpolation spaces

Proof of the main results

We now discuss some immediate applications of the W-method to the divergence-like equation. First we formulate a general result: Theorem 37. Let X, X, Y , Ỹ , F be Banach function spaces on R d satisfying the embeddings X → X, Y → Ỹ → X and consider a bounded linear operator T : X → F . Suppose moreover that the following conditions are satisfied:

(ii) T is bounded from X to F and from Y to F , and

Fix some θ ∈ (0, 1). Then, for any vector field v ∈ (L ∞ ∩ X, Y ) θ , there exists a vector field

Proof. We apply Lemma 28 for the Banach spaces X 0 = X,

Y and the operator T . One can easily observe that in this setting the conditions of Lemma 28 (part of them are explicitly stated in Lemma 27) are satisfied. Indeed, in order to verify the condition (i) in Lemma 27 it suffices to see that X, Y → X and hence,

The space Ỹ is reflexive (since it has the U M D property) and hence, it has a separable predual 7 . Also, A = (L 1 ) * has a separable predual. Thus, the condition (ii) in Lemma 27 is verified. Condition (iii) in Lemma 27 is ensured by condition (ii) in Theorem 37.

Notice that X ∩ Ỹ is a separable space that is dense in ( X, Ỹ ) θ (see [2, Theorem 4.2.2 (a), p. 91]). It follows that ( X, Ỹ ) θ is a reflexive and separable space and consequently it has a separable predual. We also have by (i) that the spaces X, Y , Ỹ have the U M D property. We can apply now Lemma 28 and we get Theorem 37.

Let us see now that Theorem 37 above implies Theorem 10, Theorem 11, Theorem 9 and Theorem 8. In what follows we will ignore the space F since it is easy for the operators T we use to find a space F sufficiently large such that T : X → F (one can simply set F of the form F = B -a,b ∞ for some a, b ∈ (1, ∞), with a sufficiently large).

Proof of Theorem 10. Let us consider some parameter r ∈ [2, ∞) such that 1/p = (1 -θ)/r + θ/2. We apply Theorem 37 for the Banach spaces X = Ḃd/r,r . Hence, X → X and Y → Ỹ → X. Also, by Mazya's theorem (Theorem 3 in the case p = q = 2) we have T (Y ) → T (L ∞ ∩ Ỹ ), for the operator T = div. Now the hypotheses of Theorem 37 are satisfyed.

Observe that, since X = Ḃd/r,r

and we can write

2

Since we also have ( Ḃd/r,r 2 , Ḃd/2,2

2

it remains to apply Theorem 37 and Theorem 10 is proved.

Proof of Theorem 11. As in the proof of Theorem 10 let us consider r ∈ [2, ∞) such that 1/p = (1 -θ)/r + θ/2. We apply Theorem 37 for the Banach spaces

and Y = Ỹ = Ẇ d/2,2 . It remains to verify that the hypotheses of Theorem 37 are satisfyed. Indeed, by the monotonicity properties of the Lorentz spaces we also have Ẇ d/r L r,1 → Ẇ d/r L r,2 , i.e., X → X. By Lemma 16 we get Ẇ d/2,2 → X, i.e., Y = Ỹ → X. Also, by Mazya's theorem (Theorem 3 in the case p = q = 2) we have T (Y ) → T (L ∞ ∩ Ỹ ) and now the hypotheses of Theorem 37 are satisfyed.

By Lemma 17 we have Ẇ d/2,2 → X and X = Ẇ d/r L r,1 → L ∞ and hence L ∞ ∩ X = Ẇ d/r L r,1 . From this and Lemma 18 we get

Lemma 18 also gives

and now one can easily conclude the proof of Theorem 11 by a direct application of Theorem 37.

Proof of Theorem 9. The proof is very similar to the one of Theorem 10. Suppose p, r, θ are as in the proof of Theorem 10. We put

Now, the operator T is the operator formaly defined for the systems of (d -1)-vector fields by the formula

where v 1 , ..., v d ∈ S are (d -1)-vector fields. In order to verify the item (ii) in Theorem 37 we use Lemma 35 instead of Mazya's theorem. It remains to apply Theorem 37 and to observe that, by the retraction method, (( Ḃd/r,r

for any j ∈ {1, ..., d}.

Proof of Theorem 8. Again, suppose p, r, θ are as in the proof of Theorem 10. We put

) C (1+ε)δ . The operator T is the usual divergence operator T = div. It remains to apply Theorem 37 and to observe that, by the retraction method, (( Ḃd/r,r

Both of these equalities rest on the fact that the Fourier projections P C δ on the sets C δ are a sum of two rotated and dilated Riesz projections. Hence, P C δ is bounded on each of the spaces Ḃd/r,r Let us see now that Theorem 9 implies Theorem 7. For this we need only some elementary geometry. Suppose d ≥ 3 and consider the unit vectors ν j := (1, ..., 1, 2, 1, ..., 1)/ √ d + 1 in R d (with value 2 on the j-th position), j ∈ {1, ..., d}. For each j ∈ {1, ..., d} define the half-spaces

and let D be the set D := D 1 ∪ ... ∪ D d . By p D j (ξ) we denote the orthogonal projection of the point ξ on the support hyperplane Π j of D j :

for any j ∈ {1, ..., d}.

Consider the function σ : R d → R defined by σ(ξ) = ξ 1 . We can immediately see that this σ satisfies the conditions (P1), (P2). We have now σ j (ξ) = ξ j , for all j ∈ {1, ..., d}. Consider G 0 := (σ 1 , ..., σ d-1 ) and let G j be obtained by composing G 0 with a rotation that transforms

for all j ∈ {1, ..., d}. In order to see that the family of functions G 1 , ..., G d is adapted to the family D 1 , ..., D d of half-spaces it remains to prove the following equivalence that corresponds to (10): Lemma 38. With the above notation we have

Proof. Observe that |ν i -ν j | < 1 and | ν i , ν j | < 1 for any i, j ∈ {1, ..., d}, i = j. It follows from this that we can find some sufficiently small number α ∈ (0, 1) such that

for any i, j ∈ {1, ..., d}, and

Let c ∈ (0, 1) such that √ 1 -c equals the left hand side of (110). In order to prove Lemma 38 it suffices to see that, for any ξ ∈ R d ,

with

then the left hand side of (111) is at least

and we are done. Else, we have | ξ, ν 1 | > √ 1 -α |ξ| and decomposing ξ as ξ = βν 1 + w, for some β ∈ R and w ∈ R d with w⊥ν 1 , we can rewrite this inequality as

or, equivalently,

Now, note that, using (110)

As above we get Remark 39. It would be interesting if one could replace the set ∆ = R d \(-∞, 0) d in Theorem 7 with the set (-∞, 0) d . This will give a stronger version of Theorem 7. It is not known whether this stronger version is true or not. The methods used in this paper seem to not apply in the case of the set (-∞, 0) d .

Remark concerning the "third" parameter

Let us consider here the problem related to the nonoptimality of the third parameter. For the sake of simplicity we are concerned here only with the divergence equation. Similar observations can be made for the case of the divergence-like equations.

Recall that, in Theorem 10 in contrast to Theorem 3, we lose some control of the parameter q of the Besov spaces involved: we start with a source term in Ḃd/p,p q and we end up with a solution in Ḃd/p,p 2 which, despite the fact that it has the "right" differential regularity (the exponents p and s = d/p are the right ones), it is a space strictly larger than Ḃd/p,p q . This is due to the fact that in order to easily compute the source space we have chosen X such that X → L ∞ . Consequently, we have to take X strictly larger than X. Indeed, choosing X = X the hypotheses of Theorem 37 imply that Ẇ d/2,2 → X → L ∞ , however, Ẇ d/2,2 is not embedded in L ∞ . By the method we used to prove Theorem 10 it is unlikely to improve the solution space to L ∞ ∩ Ḃd/p,p q . A similar remark can be made for Theorem 11.

When we use Theorem 37, in order to not lose any regularity, we would like to have that X

Since Ẇ d/2,2 → X we cannot impose the condition X → L ∞ . Apart from this situation, there are other natural candidates for the space X that one may expect to satisfy (113). However, this condition (113) is too restrictive. For instance we cannot pick X = Ḃd/r,r r for some r ∈ (2, ∞). Indeed, in this case, we have the following negative result: Proposition 40. Let r ∈ (2, ∞) and θ ∈ (0, 1) be some fixed parameters. Then,

(The corresponding norms on the two interpolation spaces are not equivalent.)

Proof. Suppose by contradiction that we have

where 1/p = (1 -θ) /r + θ/2. On the other hand, since p > 2, there exists some η ∈ (0, 1) such that ( Ḃd/p,p p , Ḃd,1

This, together with (114) and T. Wolff's interpolation theorem (see [START_REF] Wolff | A note on interpolation spaces[END_REF]Theorem 2]) implies that, there exists some θ 1 ∈ (0, 1) such that

Consider some function ψ ∈ C ∞ c (B(0, 1)) such that ψ ≡ 1 on B(0, 1/2) and define the operator T ψ by T ψ f = f -f * ψ, for any Schwartz function f . We extend T ψ by continuity to the spaces L ∞ ∩ Ḃd/r,r r , Ḃd,1 ) must be embedded in L ∞ . In other words,

for any Schwartz function f . Young's inequality and the fact that ψ is Schwartz gives us that

which together with (115) yields

In other words we have obtained the embedding W d/2,2 → L ∞ , which is false.

Open problem. Suppose D = R d-1 × (0, ∞) and X is a function space on R d such that ) θ vector field)?

If the answer to this question is yes, then, by using Theorem 37 we would be able to provide a version of Theorem 7 with no loss of regularity in the third parameter: (And a similar statement with Ḃd/p,p q in place of Ḟ d/p,p q .)

One can formulate similar conjectures corresponding to the statements of Theorem 9 and Theorem 8.