Katarzyna Pietruska-Pa

Andrzej Stos
email: stos@math.univ-bpclermont.fr

Poincaré Inequality and Haj lasz-Sobolev spaces on nested fractals

Keywords: Nested fractals, Poincaré inequality 2010 MS Classification: Primary 46E35, Secondary 31E05, 28A80

Given a nondegenerate harmonic structure, we prove a Poincaré-type inequality for functions in the domain of the Dirichlet form on nested fractals. We then study the Haj lasz-Sobolev spaces on nested fractals. In particular, we describe how the "weak"-type gradient on nested fractals relates to the upper gradient defined in the context of general metric spaces.

Introduction

The interest in analysis on fractals arose from mathematical physics, and dates back to the 80's of the past century. The first object to be meticulously defined was the Kigami Laplacian on the Sierpiński gasket [START_REF] Kigami | A harmonic calculus on the Sierpi"nski spaces[END_REF], and, somehow in parallel, the Brownian motion on the gasket [START_REF] Barlow | Brownian motion on the Sierpinski gasket[END_REF]. Since then, we have seen an outburst of papers focusing both on analytic and probabilistic aspects of stochastic processes with fractal state-space. The analytic approach, concerned mostly with Dirichlet forms, their domains and generators, proved particularly useful while constructing processes on fractals. On the other hand, derivatives on fractals have been defined [START_REF] Kigami | Harmonic calculus on on p.c.f. self-similar sets[END_REF][START_REF] Kusuoka | Dirichlet forms on fractals and products of random matrices[END_REF][START_REF] Strichartz | Taylor approximations on Sierpiński gasket type fractals[END_REF][START_REF] Teplyaev | Gradients on fractals[END_REF] and their properties studied. For an account of results from that time, as well as an extended list of references, we refer to [START_REF] Kigami | Analysis on fractals[END_REF] (analytic) and [START_REF] Barlow | Diffusion on fractals[END_REF] (probabilistic).

In present paper, departing from the definition of the gradient on nested fractals from [START_REF] Kusuoka | Dirichlet forms on fractals and products of random matrices[END_REF][START_REF] Teplyaev | Gradients on fractals[END_REF], we prove certain Poincaré-type inequalities on nested fractals, for functions belonging to the domain of the Brownian Dirichlet form (which can be seen as a fractal counterpart of the Sobolev space W 1,2 (R d )). We will then be concerned with Poicaré-Sobolev spaces and spaces of Korevaar-Schoen type, and our analysis will be much in spirit of [START_REF] Koskela | Mac Manus Quasiconformal mappings and Sobolev spaces[END_REF] and [START_REF] Koskela | Dirichlet forms, Poincaré inequalities and the Sobolev spaces of Korevaar and Schoen[END_REF].

In the last paper mentioned, the authors consider general metric measure spaces equipped with a Dirichlet structure (E, D(E)), much alike the nested fractals we consider. However, in order to proceed, they make a standing assumption that the intrinsic metric related to the Dirichlet structure,

d E (x, y) = sup{φ(x) -φ(y) : φ ∈ Γ; dη R (φ, φ) ≤ dµ},
where Γ is a µ-separating core of E, induces the topology equivalent to the initial one. This assumption fails for fractals: the metric d E is degenerate there, see [START_REF] Barlow | Stability of parabolic Harnack inequalities on metric measure spaces[END_REF], p.6. So, in order to extend the results from [START_REF] Koskela | Dirichlet forms, Poincaré inequalities and the Sobolev spaces of Korevaar and Schoen[END_REF], one should either modify the definition of d E or choose a different approach.

A discussion of gradients with connetcion to the Poincaré inequality and relation between various function spaces can be found in the recent paper [START_REF] Gogatishvili | Characterisation of Besov and Trieble-Lizorkin Spaces on Metric Measure Spaces[END_REF]. While three types of gradients are considered, the one used in P.I. is the so called upper gradient, a notion that depends on rectifiable curves. In the context of nested fractals there may be no such curves at all. Again, for a meaningful theory a different notion of gradient should be considered.

We propose a hands-on approach based on discrete approximations of nested fractals and Kusuoka gradients. By a limiting procedure, the gradient can be reasonably defined for functions belonging to the domain of the Dirichlet form, although it is usually hard to decide whether the limit exists at a given point for functions other than m-harmonic. This gradient can be used in Poincaré-type inequalities and in defining variants of Sobolev spaces on fractals.

We start with a local version of Poincaré inequality (P.I., for short), which then yields a global P.I. on nested fractals. We obtain inequalities of the form where µ is the d-dimensional Hausdorff measure on the fractal, ν is the Kusuoka energy measure on the fractal (see Section 2.2.3 for a precise definition), d w is the walk dimension of the fractal we are considering, d its Hausdorff dimension, and ∇f, Z ∇f replaces the square of the norm of the gradient. The measure ν is typically singular with respect to the Hausdorff measure, but does not charge points.

Observe that in the Euclidean case we have d w = 2, and so the scale function in P.I. will be linear as it should. Poincaré inequalities involving the Dirichlet energy measure in a general setting have been investigated in the paper [START_REF] Barlow | Stability of parabolic Harnack inequalities on metric measure spaces[END_REF], but that paper did not relate to the definition of gradients on fractal sets. A choice, or even the existence, of a gradient is not obvious on fractals. We propose to use a weak-type gradient with the energy measure (cf. Section 2 2.2).

As an application, in the second part of our paper, we compare several possible definitions of Sobolev-type functions on fractals. On metric spaces, several definitions of Sobolev-type spaces have been considered (see e.g. [START_REF] Franchi | Definitions of Sobolev classes on metric spaces[END_REF], [START_REF] Hajlasz | Sobolev spaces on metric-measure spaces. Heat kernels and analysis on manifolds, graphs, and metric spaces[END_REF], [START_REF] Korevaar | Sobolev spaces and harmonic spaces for metric space targets[END_REF]), and nested fractals are of particular interest in this context. In present paper, we introduce Poincaré-inequality based Sobolev spaces on fractals and examine their relation with Korevaar-Schoen spaces and Hajlasz-Sobolev spaces. While in a typical situation on metric spaces the scaling factor in a Poincaré inequality is r, the radius of a given ball, it turns out that on nested fractals it doesn't yield an interesting inequality. To deal with relevant Sobolev spaces, one should take into account the specific geometry of the fractal and use a scaling factor r dw /2 , as in (1.1). For some preliminary relations between Hajlasz-Sobolev and Korevaar-Schoen Sobolev spaces on fractals we refer to a paper by Hu [START_REF] Hu | A note on Haj lasz-Sobolev spaces on fractals[END_REF].

Preliminaries

We use C or c to denote a positive constant depending possibly on the fractal set, whose exact value is not important for our purposes and which may change from line to line. We will write f ≍ g (on a set D) if there exists a constant C > 0 such that for every x ∈ D one has C -1 g(x) ≤ f (x) ≤ Cg(x). For an m-integrable function f and a set A of finite measure we adopt the notation

f A = A f dm = 1 m(A)
f dm.

Nested fractals

The framework of nested fractals is that of Lindström [START_REF] Lindstrøm | Brownian motion on nested fractals[END_REF]. Suppose that φ 1 , ..., φ M , M ≥ 2, are similitudes of R N with a common scaling factor L > 1. When A ⊂ R n , then we write Φ(A) for M i=1 φ i (A), and Φ m for Φ composed m times. There exists a unique nonempty compact set (see [START_REF] Falconer | The Geometry of Fractal Sets[END_REF], [START_REF] Lindstrøm | Brownian motion on nested fractals[END_REF])

K ⊂ R N such that K = M i=1 φ i (K) = Φ(K). (2.1)
It is called the self-similar fractal generated by the family of similitudes φ 1 , ...φ M .

Since the set K has a finite nonzero diameter, for simplicity we can and will assume that diam K = 1. Each of the mappings φ i has a unique fixed point v i . Such a point is called an essential fixed point if there exists another fixed point v j such that for some transformations φ k , φ l one has φ k (v i ) = φ l (v j ). The set of all essential fixed points will be denoted by V (0) = {v 1 , ..., v r }. For m = 1, 2, ... we set V (m) = Φ m (V (0) ) and V (∞) = m≥0 V (m) . For nondegeneracy, we assume that r = #V (0) ≥ 2.

The system {φ 1 , ..., φ M } is said to satisfy the open set condition if there exists an open, nonempty set U such that Φ(U ) ⊂ U and for all i = j one has φ i (U )∩φ j (U ) = ∅. If the open set condition is satisfied, then the Hausdorff dimension of the selfsimilar fractal K is equal to d = d(K) = log M log L . By µ we denote the d-dimensional Hausdorff measure on K normalized so that µ(K) = 1.

For m ≥ 1, by a word of length m we mean a sequence w = (w 1 , ..., w m ) ⊂ {1, ..., M } m . Collection of all words of length m is denoted by W m ; W * = m≥1 W m consists of all words of finite length, W is the collection of all infinite words. When w ∈ W * is a finite word, then |w| denotes its length. If w ∈ W is an infinite word, then [w] m denotes its restriction to first m coordinates, i.e. for w = (w 1 , w 2 , ...), [w] m = (w 1 , ..., w m ). When w = (w 1 , . . . , w m ) is given, then we will use the notation φ w = φ w1 • • • • • φ wm , and for a set A, A w = φ w (A). (1) An m-simplex is any set of the form φ w (K) with w ∈ W m (m-simplices are just scaled down copies of K). The collection of all m-simplices will be denoted by T m . The 0-simplex is just K.

(2) For an m-simplex S = φ w (K), w ∈ W m , let V (S) = φ w (V (0) ) be the set of its vertices. An m-cell is any of the sets φ w (V (0) ). Two points x, y ∈ V (m) are called m-neighbors, denoted x m ∼ y, if they belong to a common m-cell.

(3) If ∆ ∈ T m , m ≥ 1, we denote by ∆ * the union of ∆ and all the adjacent m-simplices, and by ∆ * * -the union of ∆ * and all m-simplices adjacent fo ∆ * .

(4) For any x ∈ K \ V (∞) and m ≥ 1, set ∆ m (x) to be the unique m-simplex that contains x.

(

) For any x, y ∈ K \ V ∞ , define ind(x, y) = min{m ≥ 1 : ∆ m (x) ∩ ∆ m (y) = ∅}. When ind(x, y) = n, we set S(x, y) = ∆ n-1 (x) ∪ ∆ n-1 (y). 5
(6) When an m-simplex ∆ = K w = φ w (K), w ∈ W m is given and w ∈ W n is another finite word, then by ∆ w we denote the (m + n)-simplex φ w w (K).

From now on we will assume that for every S, T ∈ T m , m ≥ 1, with S = T , one has S ∩ T = V (S) ∩ V (T ) (nesting). Define the graph structure E (1) on V (1) as follows: we say that (x, y) ∈ E (1) , if x and y are 1-neighbors. Then we require the graph (V (1) , E (1) ) to be connected. For x, y ∈ V (0) , let R x,y be the reflection in the hyperplane bisecting the segment [x, y]. Then we stipulate that ∀ i∈{1,...,M} ∀ x,y∈V (0) , x =y ∃ j∈{1,...,M} R x,y (φ i (V (0) )) = φ j (V (0) ) (natural reflections map 1-cells onto 1-cells).

The self-similar fractal K is called a nested fractal, if it satisfies the above open set condition, nesting, invariance under local isometries, and the connectivity assumption.

Part of our results will require the following Property (P) of the fractal: Property (P). There exist α > 0 such that for all n = 1, 2, ... and x, y -nonvertex points such that y ∈ ∆ * n (x) \ ∆ * n+1 (x) one has

ρ(x, y) ≥ α L n . (2.2)
Remark 1. Property (P) holds true for nested fractals such that the similitudes (φ i ) i=1,...,M have the same unitary part. This class of fractals contains the wellknows examples such as the Sierpiński gaskets, snowflakes, the Vicsek set etc. Proof of this statement is given in the Appendix. Clearly, if ind(x, y) = n, then ∆ n-1 (x)∩∆ n-1 (y) = ∅. These sets either coincide or are adjacent (ie. they meet at exactly one point). Moreover, under Property (P), the index ind (x, y) is closely related to the Euclidean distance of x, y. Lemma 2.2. 1. For any fixed x ∈ K \ V (∞) and n ≥ 2, one has

{y : ind(x, y) = n} = ∆ * n-1 (x) \ ∆ * n (x). (2.3)
2. Assume additionally that the fractal K satisfies the property (P).

If ind (x, y) = n then ρ(x, y) ≍ L -n , (2.4)
ρ(x, y) being the Euclidean distance.

Proof. Fix x ∈ K \ V (∞) and n ≥ 2. Observe that y ∈ ∆ * n (x) if and only if ∆ n (x) ∩ ∆ n (y) = ∅, which is equivalent to ind (x, y) ≥ n + 1. Since {∆ n (x)} n is a decreasing sequence of sets, (2.3) follows. Relation (2.4) follows from (2.3) and Property (P).

Gradients of nested fractals

To proceed, we need to define the gradient. The material in this section is classic and follows mainly [START_REF] Kigami | Analysis on fractals[END_REF] and [START_REF] Teplyaev | Gradients on fractals[END_REF]. For other results concerning gradients on fractals we refer to [START_REF] Kusuoka | Dirichlet forms on fractals and products of random matrices[END_REF][START_REF] Kigami | Harmonic calculus on on p.c.f. self-similar sets[END_REF][START_REF] Strichartz | Taylor approximations on Sierpiński gasket type fractals[END_REF].

Nondegenerate harmonic structure on K

Suppose that K is the nested fractal associated with the system {φ 1 , ..., φ M }. Let A = [a x,y ] x,y∈V (0) be a conductivity matrix on V (0) , i.e. a symmetric real matrix with nonnegative off-diagonal entries and such that for any x ∈ V (0) , y∈V (0) a x,y = 0.

For f : V (0) → R, set E (0) A (f, f ) = 1 2 x,y∈V (0) a x,y (f (x) -f (y)) 2 .
Then we define two operations:

(1) Reproduction. For f ∈ C(V (1) ) we let

E (1) A (f, f ) = M i=1 E (0) A (f • φ i , f • φ i ).
The mapping

E (0) A → E (1)
A is called the reproduction map and is denoted by R.

(2) Decimation. Given a symmetric form E on C(V (1) ), define its restriction to C(V (0) ), E V (0) , as follows. Take f :

V (0) → R, then set E| V (0) (f, f ) = inf{E(g, g) : g : V (1) → R and g| V (0) = f }.
This mapping is called the decimation map and will be denoted by De.

Let G be the symmetry group of V (0) , i.e. the group of transformations generated by symmetries R x,y , x, y ∈ V (0) . Then we have ( [START_REF] Lindstrøm | Brownian motion on nested fractals[END_REF], [START_REF] Sabot | Existence et unicité de la diffusion sur un ensemble fractal[END_REF]):

Theorem 2.3. Suppose K is a nested fractal. Then there exists a unique number ρ = ρ(K) > 1 and a unique, up to a multiplicative constant, irreducible conductivity matrix A on V (0) , invariant under the action of G, and such that

(De • R)(E (0) A ) = 1 ρ E (0)
A .

(2.5)

A is called the symmetric nondegenerate harmonic structure on K. By analogy with the electrical circuit theory, ρ is called the resistance scaling factor of K. The number

d w = d w (K) def = log(Mρ) log L > 1 is called the walk dimension of K. For further use, note that ρ = L dw-d .

The canonical Dirichlet form on

K Suppose A is the nondegenerate harmonic structure on K. Define E (0) = E (0) A , then let E (m) (f, f ) = ρ m |w|=m E (0) (f • φ w , f • φ w ), f ∈ C(V (m) ).
The sequence E (m) is nondecreasing, i.e. for every f :

V (∞) → R, one has E (m) (f, f ) ≤ E (m+1) (f, f ), m = 0, 1, 2, . . . Set D = {f : V (∞) → R : sup m E (m) (f, f ) < ∞} and for f ∈ D E(f, f ) = lim m→∞ E (m) (f, f ). (2.6) Further, D = D(E) = {{ ∈ C(K) : {| V (∞) ∈ D}, E(f, f ) = E(f | V (∞) , f | V (∞) ) for f ∈ D.
Then (E, D) is a regular local Dirichlet form on L 2 (K, µ), which agrees with the group of local symmetries of K. This Dirichlet form is also called 'the Brownian Dirichlet form on K', and will be essential in defining the gradient. It satisfies the following scaling relation:

for any f ∈ D, E(f, f ) = ρ m w∈Wm E(f • φ w , f • φ w ).
(2.7)

Harmonic functions on K and energy measure

Definition 2.4. Suppose f :

V (0) → R is given. Then h ∈ D(E) is called harmonic on K with boundary values f, if E(h, h) minimizes the expression E(g, g) among all g ∈ D(E) such that g| V (0) = f.
The unique harmonic function that agrees with f on V (0) will be denoted by Hf.

Denote by H the space of all harmonic functions on K. It is an r-dimensional linear space, which can be equipped with the norm

h 2 H = E(h, h) + ( x∈V (0) h(x)) 2 .
Further, H denotes the orthogonal complement in H of the (one-dimensional) subspace of constant functions, and let P : H → H be the orthogonal projection onto H. The norm on H is given by h 2 = E(h, h) (note that • is a seminorm on H, vanishing on constant functions), and the corresponding scalar product on H will be denoted by •, • .

Next, for i = 1, ..., M, we define the map

M i : H → H by M i h = h • φ i , and
M i : H → H by M i = P • M i . From the scaling relation (2.7) we deduce that for h ∈ H and m ≥ 0, h 2 = ρ m |w|=m M w h 2 , (2.8)
where by M w we have denoted

M wm • • • • • M w1 h = P (h • φ w ).
For f ∈ D, let's define the energy measure associated with f as the measure whose value on any given m-simplex K w = K w1...wm is equal to

ν f (K w ) = ρ m E(f • φ w , f • φ w ).
(2.9)

When h ∈ H is a harmonic function and w ∈ W m , then ν h (K w ) = ρ m M w h 2 . Let h 1 , ..., h r-1 be an orthonormal basis in H. Then the expression

ν def = r-1 i=1 ν hi (2.10)
does not depend of the choice of the orthonormal basis and its value on an m-simplex

K w is equal to ν(K w ) = ρ m Tr M * w M w .
The measure given by (2.10) is called the Kusuoka measure, or the energy measure on K. This measure has no atoms, and typically is singular with respect to the measure µ.

Gradients

When x ∈ K is a nonlattice point, then x has a unique address: it is an (infinite

) sequence w = w 1 w 2 . . . such that x = ∞ m=1 K [w]m (recall that we have denoted [w] m = (w 1 ...w m )). For such a nonlattice point, let Z m (x) =      M * [w]m M [w]m Tr M * [w]m M [w]m if Rank M [w]m > 0; 0 otherwise. (2.11)
It can be shown that Z m (•) is a bounded, matrix-valued martingale with respect to ν, and as such it is convergent ν-a.s. to an integrable function Z(•).

For a nonlattice point x with address w, set

∇ m f (x) = M -1 [w]m ( P H)(f • φ [w]m ), m = 1, 2, ...,
then the gradient of f at point x is the element of H given by

∇f (x) = lim m→∞ ∇ m f (x),
provided the limit exists. For the discussion of the 'pointwise gradients' and their properties we refer to [START_REF] Teplyaev | Gradients on fractals[END_REF], [START_REF] Pelander | Procucts of random matrices and derivatives on p.c.f. fractals[END_REF] and [START_REF] Hino | Energy measures and indices of Dirichlet forms, with applications to derivatives on some fractals[END_REF]. But even if the pointwise limits of ∇ m are not known to exist, we do know (see [START_REF] Kusuoka | Dirichlet forms on fractals and products of random matrices[END_REF], Lemmas 3.5 and 5.1, and also the discussion in [START_REF] Teplyaev | Gradients on fractals[END_REF], p. 137) that when f ∈ D, then there exists a measurable mapping

Y (•, f ) such that E(f, f ) = K Y (•, f ), Z(•)Y (•, f ) dν(•). (2.12)
With an abuse of notation, we will write ∇f for the object Y (•, f ), which is defined ν-a.e. When we will use the pointwise value, it will be clearly indicated.

Definition 2.5.

1. A continuous function f : K → R is called m-harmonic if f • φ w is harmonic for any w = (w 1 ...w m ) ∈ W m .
2. There exists a unique m-harmonic function with given values at points from V (m) . For a continuous function f on K, by H m f we denote the unique mharmonic function that agrees with f on V (m) .

Remark 2. When f is m-harmonic, then for any nonlattice point x ∈ K with address

w ∈ W ∞ one has ∇ m f (x) = ∇ m+n f (x)
for any n ≥ 0, and so ∇f (x) exists at nonlattice points (which are of full ν-measure); note also that ∇ m ff (and thus also ∇ff ) is constant inside each K w with |w| = m.

Poincaré inequality on nested fractals

Poincaré inequalities on nested fractals that one can find in the literature (see e.g. [START_REF] Barlow | Stability of parabolic Harnack inequalities on metric measure spaces[END_REF] and its references) are usually written in the form

B |f -f B | 2 dµ ≤ cΨ(R) B dΓ(f, f ), (3.1)
where B is a ball of radius R, Ψ : R + → R + is a scale function (most commonly, Ψ(R) = R σ ), and Γ(f, f ) is the energy measure associated with the Brownian Dirichlet form on fractals. Poincaré inequalities P (q, p), on a metric measure space (X, ρ, µ), with a doubling measure µ and another Radon measure ν, are similar in spirit, but involve usually two functions. One says that a pair of measurable functions (f, g) satisfies the (q, p)-Poincaré inequality, when

B |f -f B | q dµ 1/q ≤ CR σB |g| p dν 1/p , (3.2)
where σ ≥ 1 is a given number, and σB denotes the ball concentric with B, but with radius σ times the radius of B. For an account of Poincaré inequalities in metric spaces, we refer mainly to [START_REF] Hajlasz | Sobolev met Poincaré[END_REF], and also to [START_REF] Hajlasz | Sobolev spaces on metric-measure spaces. Heat kernels and analysis on manifolds, graphs, and metric spaces[END_REF]. Poincaré inequalities on nested fractals we will be concerned with will be variants of two-weight inequalities.. The measure µ appearing on the left-hand side will be the Hausdorff measure on K, while the measure ν on the right-hand side will be the Kusuoka energy measure. Recall that the measure ν in most cases is not absolutely continuous with respect to µ. The difference from the classical case is that the integral on the right-hand side will not be a barred integral with respect to the measure ν, but it will be divided by the measure µ of the underlying set.

We start with a fractal version of Poincaré inequality -where balls are replaced with simplices. This version does not require property (P) of the underlying fractal. The precise statement reads as follows.

Theorem 3.1. Let f ∈ D(E), and let ∆ be any m-simplex, m ≥ 0. Then we have

∆ |f (x) -f ∆ |dµ(x) ≤ C (diam ∆) dw/2 1 µ(∆ * ) ∆ * ∇f, Z ∇f dν 1/2 ≤ CL -mdw/2 L -md ∆ * ∇f, Z∇f dν 1/2 , (3.3)
where ∆ * denotes the union of ∆ and all m-simplices adjacent to ∆.

The proof will be given later on. Now, we start with a local version of Poincaré inequality for adjacent lattice points. Proposition 3.2. Suppose f ∈ D(E), and let x m ∼ y. Let K w be the m-simplex that contains both points x, y, with address w ∈ W m . Then

|f (x) -f (y)| 2 ≤ C(diam K w ) dw-d Kw ∇f, Z ∇f dν.
(3.4)

Proof. Set c(x, y) = a -1
x ′ y ′ where x ′ , y ′ ∈ V (0) are such that x = φ w (x ′ ) and y = φ w (y ′ ) (the matrix A = [a x,y ] was introduced in Section 2.2.1). Then we have:

|f (x) -f (y)| 2 ≤ c(x, y)) u,v∈V (0) a uv |f • φ w (u) -f • φ w (v)| 2 = c(x, y))E (0) (f • φ w , f • φ w ) ≤ E(f • φ w , f • φ w ) = c(x, y) K ∇(f • φ w ), Z ∇(f • φ w ) dν ≤ c 1 K ∇(f • φ w ), Z ∇(f • φ w ) dν,
where c 1 = sup{a x,y : x, y ∈ V (0) }.

Since diam K w = L -m , the scaling relation from Lemma 3.3 below gives the desired statement. Lemma 3.3. Let f ∈ D(E), and let K w be an m-simplex.

Then K ∇(f • φ w ), Z ∇(f • φ w ) dν = L -m(dw-d) Kw ∇f, Z ∇f dν (3.5)
Remark 3. The right hand side of (3.5) is well-defined since ∇f, Z ∇f exists ν-a.e. and K ∇f, Z ∇f dν < ∞, see Theorem 4 of [START_REF] Teplyaev | Gradients on fractals[END_REF]. Remark 4. While ν(K w ) depends in general on w, the scaling factor on the right hand side of (3.5) depends only on m = |w|. Thus, the lemma is not tantamount to a simple change of variables but reflects an interplay between ∇f and Z.

Proof.

Step 1. Assume that f is m-harmonic. Then ∇f (y) exists at all nonlattice points y and ∇f (y) = ∇ m f (y). Observe that ∇ m f (•) is constant (ν-a.e.) inside each m-simplex K w and that it differs there from M -1 w f (•) by a constant only. It follows that

Kw ∇f, Z ∇f dν = Kw ∇ m f, Z ∇ m f dν = lim n→∞ Kw ∇ m f, Z n ∇ m f dν.
To justify the last statement, observe that the random variables

X n = ∇ m f, Z n ∇ m f converge to X = ∇ m f, Z∇ m f in L 1 (K, dν). This is so because X n ≥ 0, X n → X in measure ν and K X n dν = K ∇ m f, Z n ∇ m f dν = E(H n f, H n f ) → E(f, f ) = K X dν.
The convergence in L 1 (K, dν) follows then from Scheffé's theorem.

For short, let us write F = ∇ m f ∈ H. Let n > m be fixed, and let i = (i m+1 , . . . , i n ) ∈ W n-m so that wi ∈ W n . Z n is constant on n-simplices and, once n > m, we have

Kw F, Z n F dν = |i|=n-m Kwi F, Z n F dν = |i|=n-m M wi F 2 Tr( M * wi M wi ) • L n(dw-d) Tr( M * wi M wi ) = L n(dw-d) |i|=n-m M wi F 2 = L n(dw-d) |i|=n-m E(F • φ wi , F • φ wi ).
From the scaling property of E,

|i|=n-m E(F • φ wi , F • φ wi ) = |i|=n-m E((F • φ w ) • φ i , (F • φ w ) • φ i ) = L -(n-m)(dw-d) E(F • φ w , F • φ w ).
We know that F • φ w and f • φ w differ by a constant only, so that

E(F • φ w , F • φ w ) = E(f • φ w , f • φ w ).
Piecing everything together, we obtain

Kw F, Z n F dν = L m(dw-d) E(f • φ w , f • φ w ) = L m(dw-d) K ∇(f • φ w ), Z ∇(f • φ w ) dν.
The right-hand side does not depend on n, thus we can pass with n to infinity, obtaining (3.5).

Step 2. Let now f be n-harmonic, with n > m.

Then K w = |i|=n-m K wi and Kw ∇f, Z ∇f dν = |i|=n-m Kwi ∇f, Z ∇f dν. (3.6)
To each of the integrals on the right-hand side of (3.6) we apply Step 1, obtaining

(3.6) = L n(dw-d) |i|=n-m K ∇(f • φ wi ), Z∇(f • φ wi ) dν = L m(dw-d) |i|=n-m L (n-m)(dw-d) E((f • φ w ) • φ i , (f • φ w ) • φ i ),
which is, from the scaling property, equal to

L m(dw-d) E(f • φ w .f • φ w ) = L m(dw-d) K ∇(f • φ w ), Z ∇(f • φ w ) dν.
Step 3. Let now f be any function from D(E). Then

E(f, f ) = lim n→∞ E(H n f, H n f ) and E(f • φ w , f • φ w ) = lim n→∞ E(H n (f • φ w ), H n (f • φ w )). (3.7)
From Step 2 we have: for n ≥ m,

K ∇ n (H n f • φ w ), Z ∇ n (H n f • φ w ) dν = L -m(dw-d) Kw ∇(H n f ), Z ∇(H n f ) dν, (3.8
) and the assertion follows from the limiting procedure: the left-hand side of (3.8) is equal to

E(H n f • φ w , H n f • φ w ) n→∞ -→ E(f • φ w , f • φ w ). As to the right-hand side, since ∇(H n f ) = ∇ n f, and ∇ n f converges to ∇f in the seminorm K •, Z• dν 1/2 ,
we also have the convergence in the restricted seminorm Kw •, Z• dν

1/2
, which gives the desired convergence.

From Proposition 3.2 we derive the local Poincaré inequality for nonlattice points. Theorem 3.4. Supopose that K satisfies property (P). Let f ∈ D(E) and x, y

∈ K \ V (∞) . Then |f (x) -f (y)| 2 ≤ Cρ(x, y) dw 1 µ(S(x, y)) S(x,y) ∇f, Z ∇f dν.
where S(x, y) was introduced in Definition 2.1 (6).

Proof.

Step 1. Suppose z ∈ V (m) is a vertex of ∆ ∈ T m and let y ∈ Int ∆. Then one finds a chain z = z 0 , z 1 , ..., z k → y such that for all k = 1, 2, ... the points z k-1 and z k are (m + k)-neighbors. Denote by ∆(z k-1 , z k ) the (m + k)-simplex they belong to. From Proposition 3.2 we have, since ∆(z k+1 , z k ) ⊂ ∆,

|f (z k-1 ) -f (z k )| 2 ≤ C (diam ∆(z k-1 , z k )) dw-d ∆(z k-1 ,z k ) ∇f, Z ∇f dν ≤ C (diam ∆(z k-1 , z k )) dw-d ∆ ∇f, Z ∇f dν.
Since f is continuous, summing over k we obtain

|f (z) -f (y)| ≤ ∞ k=1 |f (z k-1 ) -f (z k )| ≤ ∞ k=1 (diam ∆(z k-1 , z k )) dw -d 2 ∆ ∇f, Z∇f dν 1/2 ≤ ∞ k=1 L -m+k 2 (dw-d) ∆ ∇f, Z ∇f dν 1/2 = CL -m(dw -d) 2 ∆ ∇f, Z ∇f dν 1/2
and consequently

|f (z) -f (y)| 2 ≤ CL m(d-dw) ∆ ∇f, Z ∇f dν. (3.9)
Step 2. Suppose x, y belong to a common m-simplex ∆. Then choose a vertex

v ∈ V (∆), write |f (x) -f (y)| 2 ≤ 2(|f (x) -f (v)| 2 + |f (v) -f (y)| 2
), and apply Step 1 in order to get (3.9) for x and y.

Step 3. The result of Step 2 extends immediately to the case when x, y belong to two adjacent m-simplices: m) . One applies Step 1 to the pair (x, z) and then to (y, z), getting

when x ∈ ∆ 1 ∈ T m , y ∈ ∆ 2 ∈ T m and ∆ 1 , ∆ 2 are adjacent, then ∆ 1 and ∆ 2 share a vertex z ∈ V (
|f (x) -f (y)| 2 ≤ CL m(d-dw) ∆1∪∆2 ∇f, Z ∇f dν. (3.10)
Step 4. Now take any x, y ∈ K \ V (∞) . Let ind (x, y) = m. Then S(x, y) = ∆ m-1 (x)∪∆ m-1 (y) is composed either of a common (m-1)-simplex or two adjacent (m-1)-simplices. In the first case, apply Step 2, in the latter case -Step 3. In either case, µ(S(x, y)) ≍ L -(m-1)d and ρ(x, y) ≍ L -m , so the theorem is proven.

Proof of Theorem 3.1. Choose ∆ ∈ T m . By Jensen's inequality we have

∆ |f (x) -f ∆ |dµ(x) ≤ ∆ |f (x) -f ∆ | 2 dµ(x) 1/2
, and further:

∆ |f (x) -f ∆ | 2 dµ(x) = ∆ |f (x) - ∆ f (y)dµ(y)| 2 dµ(x) = ∆ | ∆ (f (x) -f (y))dµ(y)| 2 dµ(x) ≤ ∆ ∆ |f (x) -f (y)| 2 dµ(y)dµ(x) = 1 µ(∆) 2 ∆ ∆ |f (x) -f (y)| 2 dµ(y)dµ(x).
Points x and y under the integral belong to a common m-simplex ∆, and so ind(x, y) > m (without loss of generality we can and do assume that x, y are nonvertex points). Using Lemma 2.2, we split the inner integral as follows.

∆ |f (x) -f (y)| 2 dµ(y) = ∞ n=m+1 {y∈∆: ind (x,y)=n} |f (x) -f (y)| 2 dµ(y) = ∞ n=m+1 (∆ * n-1 (x)\∆ * n (x))∩∆ |f (x) -f (y)| 2 dµ(y). (3.11)
When ind(x, y) = n, then ρ(x, y) ≍ L -n and moreover there exist two adjacent (n -1)-simplices, say S and T, such that x ∈ S, y ∈ T (S = T is permitted). Let v ∈ V (n-1) be a common vertex of S and T. Then, according to (3.10) (which is true without property (P) as well)

|f (x) -f (y)| 2 ≤ CL -n(dw-d) S∪T ∇f, Z ∇f dν ≤ CL -n(dw-d) ∆ * n-1 (x)
∇f, Z ∇f dν.

As µ(∆ ∩ (∆ * n-1 (x) \ ∆ * n (x))) ≤ µ(∆ * n-1 (x)) ≍ L -nd , each of the integrals in (3.11) is bounded by CL -ndw ∆ * n-1 (x)
∇f, Z ∇f dν.

Consequently,

∆ ∆ |f (x) -f (y)| 2 dµ(y)dµ(x) ≤ C ∞ n=m+1 L -ndw ∆ ∆ * n-1 (x)
∇f, Z ∇f dνdµ(x).

(3.12)

Let w ∈ W m be such that ∆ = φ w (K) and for i

∈ W n-1-m set ∆ i = φ wi (K) ⊂ ∆. Observe that on each ∆ i the mapping x → ∆ * n-1 (x) is constant and equal to ∆ * i . It follows ∆ ∆ * n-1 (x) ∇f, Z ∇f dνdµ(x) = i∈Wn-1-m ∆i ∆ * i ∇f, Z ∇f dνdµ(x) = i∈Wn-1-m ∆ * i ∇f, Z ∇f dνµ(∆ i ) ≤ C i∈Wn-1-m L -nd ∆ * i ∇f, Z ∇f dν. (3.13)
Sets ∆ * i are not pairwise disjoint, but each of them is consists of at most M + 1 simplices from T n-1 . Therefore, if in (3.13) we decompose each of the integrals over ∆ * i into a number of integrals over corresponding (n -1)-simplices, then each of these (n -1)-simplices will appear at most M + 1 times in the sum. Furthermore, since for any i

∈ W n-1-m one has ∆ * i ⊂ ∆ * , and i∈Wn-1-m ∆ i = ∆, it follows that i∈Wn-1-m ∆ * i ∇f, Z ∇f dν ≤ C ∆ * ∇f, Z ∇f dν. (3.14)
Collecting (3.12), (3.13), (3.14) we obtain

∆ ∆ |f (x) -f (y)| 2 dµ(y)dµ(x) ≤ C ∞ n=m+1 L -ndw L -nd ∆ * ∇f, Z ∇f dν = CL -mdw L -md ∆ * ∇f, Z ∇f dν.
To complete the proof, observe again that L -md = cµ(∆).

Below we derive a Poincaré inequality that uses balls instead of simplices. This statement requires property (P) and will be used throughout for the results of next section.

Theorem 3.5. Suppose that K satisfies (P). Suppose f ∈ D(E). Let x 0 ∈ K \ V (∞) be a nonvertex point and let r > 0 be given. Denote B = B(x 0 , r) = {y ∈ K : ρ(x 0 , y) ≤ r}. Then there exist C > 0 and A ≥ 1 (independent of x 0 and r) such that

B |f -f B |dµ ≤ Cr dw 2 1 r d B(x0,Ar) ∇f, Z ∇f dν 1/2 . ( 3.15)
Proof. Only minor changes need to be introduced in the proof of Theorem 3.1. From property (P) there exists α ∈ (0, 1) such that for every nonlattice x ∈ K, and any m ≥ 1

B(x, α L m ) ⊆ ∆ * m (x) ⊆ B(x, 2 L m ). (3.16)
Let n 0 be the unique integer such that L -(n0+1) < r α ≤ L -n0 , so that

B(x, r) ⊆ B(x, αL -n0 ) ⊆ ∆ * n0 (x).
As before, we get

B |f -f B | 2 dµ ≤ 1 µ(B) 2 B B |f (x) -f (y)| 2 dµ(x)dµ(y). Since B ⊂ ∆ * n0 (x 0 ) and ∆ * n0 (x 0 ) = S 1 ∪ . . . ∪ S K
is the sum of a finite number of neighboring n 0 -simplices, we estimate the inner integral as

∆ * 0 |f (y) -f (x)| 2 dµ(x)dµ(y) = i Si |f (x) -f (y)| 2 dµ(y). (3.17)
Now we work with the integral over each S i separately. Observe that when x, y are as in the integral in (3.17), then ∆ n0 (x) ∩ ∆ n0 (y) = ∅, so that ind (x, y) ≥ n 0 + 1. Therefore, for any i = 1 . . . K, we have

Si |f (x) -f (y)| 2 dµ(y) = ∞ n=n0+1 Si∩{y:ind(x,y)=n} |f (x) -f (y)| 2 dµ(y) = ∞ n=n0 Si∩(∆ * n (x)\∆ * n+1 (x)) |f (x) -f (y)| 2 dµ(y) ≤ c ∞ n=n0 L -ndw Si∩∆ * n (x)
∇f, Z ∇f dν.

From now on we proceed identically as in the proof of (3.3), ending up with

B B |f -f B | 2 dµdµ ≤ B i Si |f (x) -f (y)| 2 dµ(y)dµ(x) ≤ L -n0dw L -n0d+f i S * i ∇f, Z ∇f dν ≤ cL -n0d L -n0d B(x0, 2L α r) ∇f, Z ∇f dν ≤ cr dw 1 µ(B(x 0 , 2L α r) B(x0, 2L α r) ∇f, Z ∇f dν,
where we have used the inclusions

S * i ⊆ ∆ * n0 ⊆ B(x 0 , 2L -n0 ) ⊆ B(x 0 , 2L α r). Set A = 2L
α . The proof is complete.

Sobolev spaces on fractals

On metric spaces, several definitions of Sobolev-type spaces are possible (see e.g. [START_REF] Franchi | Definitions of Sobolev classes on metric spaces[END_REF], [START_REF] Hajlasz | Sobolev spaces on metric-measure spaces. Heat kernels and analysis on manifolds, graphs, and metric spaces[END_REF], [START_REF] Korevaar | Sobolev spaces and harmonic spaces for metric space targets[END_REF]). We recall some of them below. Their mutual relations and connections with the Poincaré inequality form now a well established theory ( [START_REF] Hajlasz | Sobolev met Poincaré[END_REF], [START_REF] Hajlasz | Sobolev spaces on metric-measure spaces. Heat kernels and analysis on manifolds, graphs, and metric spaces[END_REF]). Below, we briefly recall the relevant definitions. Suppose (X, ρ, µ) is a metric measure space, where µ is a doubling Radon measure on a metric space (X, ρ). Any nested fractal K fits into this definition, with µ not only doubling but even Ahlfors regular. In the following definitions of Sobolevtype spaces we suppose p ≥ 1.

1. The Hajlasz-Sobolev spaces M 1,p (X) consists of those functions f ∈ L p (X), for which there exists a function g ∈ L p (X), g ≥ 0 such that

|f (x) -f (y)| ≤ Cρ(x, y)(g(x) + g(y)) (4.1)
for µ-almost all x, y ∈ X.

2. The space P 1,p (X) consists of those functions f ∈ L 1 loc (X), for which there exist σ ≥ 1 and g ∈ L p (X) such that for every ball B = B(x, r)

B |f -f B |dµ ≤ r B(x,σr) g p dµ 1/p . (4.2)
3. The Korevaar-Schoen Sobolev space, KS 1,p (X) consists of those functions f ∈ L p (X) for which lim sup

ǫ→0 X B(x,ǫ) |f (x) -f (y)| p ǫ p dµ(x)dµ(y) < ∞.
One considers also the Newtonian spaces N 1,p (X). The upper gradient those spaces are based on involves integrals over rectifiable curves. On nested fractals, the family of rectifiable curves might be empty or not rich enough to yield a non-degenerate object.

In general, the inclusions M 1,p (X) ⊂ P 1,p (X) ⊂ KS 1,p (X) hold true, but not always they can be reversed. In some cases however -for example in R d -all three definitions yield the same function spaces. We refer to [START_REF] Koskela | Mac Manus Quasiconformal mappings and Sobolev spaces[END_REF] and [START_REF] Hajlasz | Sobolev spaces on metric-measure spaces. Heat kernels and analysis on manifolds, graphs, and metric spaces[END_REF] for more details.

We are now going to adapt definitions of the spaces M 1,p , P 1,p and KS 1,p to the fractal setting. As we have already mentioned in the Introduction, the scale r is not a natural scale here, and it will be replaced by r dw 2 . Let us mention that in many cases (the Euclidean spaces, some manifolds) the walk dimension d w , read off from the heat kernel estimates on the underlying space, is equal to 2, so that the scale r dw 2 is just r. Definition 4.1. Let K be the nested fractal defined in Section 2.1; let p ≥ 1 and σ > 0 be given. Recall that µ denotes the normalized d-dimensional Hausdorff measure on K and ν -the Kusuoka measure. We say that a function f ∈ L p (K, µ) belongs to:

-the space M 1,p σ (K, µ), when there exists a nonnegative function g ∈ L p (K, µ) such that for µ-a.e. x, y ∈ K, |f (x)f (y)| ≤ ρ(x, y) σ (g(x) + g(y));

(4.3)

-the space P 1,p σ (K), when there exists a nonnegative function g ∈ L p (K, ν) such that for any x ∈ K and 0 < r < diam K,

B(x,r) |f -f B(x,r) |dµ ≤ r σ 1 µ(B(x, Ar) B(x,Ar) g p dν 1/p , (4.4)
with some A ≥ 1; the inequality (4.4) will be called the (1, p, σ)-Poincaré inequality;

-the space KS 1,p σ (K), when lim sup

ǫ→0 K B(x,ǫ) |f (x) -f (y)| p ǫ pσ dµ(x)dµ(y) < ∞,
-the Besov-Lipschitz space Lip(σ, p, ∞), σ > 0 (see [START_REF] Jonsson | Brownian motion on fractals and function spaces[END_REF]), if

f Lip = sup m≥0 a (p) m (f ) < ∞,
where

a (p) m (f ) = L mσ L md ρ(x,y)≤ c 0 L m |f (x) -f (y)| p dµ(x)dµ(y) 1/p
, with some c 0 > 0. Note that different values of this constant yield the same function space with equivalent norms.

It is immediate to see that the spaces Lip(σ, p, ∞)(K) and KS 1,p σ (K) coincide and that their norms are equivalent.

We now turn to relations between the Poincaré-Sobolev and Korevaar-Schoen Sobolev spaces on fractals. The inclusion P 1,p σ (K) ⊂ KS 1,p σ (K) is true under usual constraints on parameters (p ≥ 1, σ > d/p), and it can be reversed for p = 2, σ = dw 2 .

Proposition 4.2. Suppose that the fractal K satisfies property (P). Let p ≥ 1 and σ > 0 be given.

(1) If σ > d/p, then P 1,p σ (K) ⊂ KS 1,p σ (K). (2) When σ = dw 2 , then P 1,2 σ (K) = KS 1,2 σ (K).

Proof. Once (1) is proven, then the inclusion '⊂' in [START_REF] Barlow | Brownian motion on the Sierpinski gasket[END_REF] would follow from the relation d w > d (true for any nested fractal). As to the opposite inclusion, Theorem 3.5 gives that the (1, 2, dw 2 )-Poincaré inequality holds true for any f ∈ D(E). As D(E) = Lip( dw 2 , 2, ∞) = KS 1,2 dw/2 (K) (Theorem 5 of [START_REF] Pietruska-Pa Luba | On function spaces related to fractional diffusions on dsets[END_REF]), the inclusion '⊃' in (2) follows.

Therefore we need to prove (1). Our proof is a modification of the proof of Theorem 4.1 of [START_REF] Koskela | Mac Manus Quasiconformal mappings and Sobolev spaces[END_REF]. See also [START_REF] Hajlasz | Sobolev met Poincaré[END_REF], Theorem 5.3 and its proof.

Assume that f ∈ P 1,p σ (K) and that the pair (f, g) satisfies the (1, p, σ)-Poincaré inequality. Introduce a fractal version of Riesz potentials:

J p (g, n, x) = ∞ m=0 L -(m+n)σ 1 µ(∆ * n+m (x)) ∆ * n+m (x)
g p (z)dν(z)

1/p .
The potentials J p (g, n, x), are well-defined for all nonlattice points of K (this is a set of full measure µ).

We will show that there exists a constant k 0 ≥ 0 such that for µ-a.a. x, y ∈ K with ind(x, y) ≥ k 0 one has:

|f (x) -f (y)| ≤ C (J p (g, ind (x, y) -k 0 , x) + J p (g, ind (x, y) -k 0 , y)) (4.5)
Since by assumption f ∈ L p (K, µ) ⊂ L 1 (K, µ), µ-almost every point of K is a µ-Lebesgue point for f (cf. [START_REF] Toledano | A note on the Lebesgue differentiation in spaces of homogeneous type[END_REF]):

f (x) = lim r→0 B(x,r) f (y)dµ(y) = lim r→0 f B(x,r) .
Let x, y be two nonlattice Lebesgue points for f and let n 0 = ind(x, y). We use a classical chaining argument. Denote r m = α AL m , where A ≥ 1 is the constant from the Poincaré inequality (4.4), and α ∈ (0, 1) comes from (3.16). Using the Jensen's inequality, the doubling property for µ, the Poincaré inequality (4.4) and (3.16), we obtain the following chain of inequalities:

|f (x) -f B(x,rn 0 ) | ≤ ∞ m=0 |f B(x0,rn 0 +m) -f B(x,rn 0 +m+1) | ≤ ∞ m=0 B(x,rn 0 +m+1) |f (z) -f B(x,rn 0 +m) |dµ(z) ≤ ∞ m=0 B(x,rn 0 +m) |f (z) -f B(x,rn 0 +m) |dµ(z) ≤ C ∞ m=0 r σ n0+m 1 µ(B(x, Ar n0+m )) B(x,Arn 0 +m) g(z) p dν(z) 1/p ≤ C ∞ m=0 L -(m+n0)σ 1 µ(∆ * n0+m (x)) ∆ * n 0 +m (x)
g(z) p dν(z) From Lemma 2.2, there exists a universal constant C 1 > 0 such that when ind(x, y) = n 0 , then ρ(x, y) ≤ C 1 L -n0 = C1A α r n0 . For short, denote R = (1 + C1A α )r n0 . Let k 0 be the smallest number such that for any z ∈ K, B(z, AR) ⊂ ∆ * n0-k0 (z), cf. (3.16). Using the Poincaré inequality (4.4) and the Ahlfors-regularity of µ we get:

|f B(x,rn 0 ) -f B(y,rn 0 ) | (4.8) ≤ |f B(x,rn 0 ) -f B(x,R) | + |f B(y,rn 0 ) -f B(x,R) | ≤ B(x,rn 0 ) |f (z) -f B(x,R) | dµ(z) + B(y,rn 0 ) |f (z) -f B(x,R) | dµ(z) ≤ µ(B(x, R)) µ(B(x, r n0 )) + µ(B(x, R)) µ(B(y, r n0 )) B(x,R) |f (z) -f B(x,R) | dµ(z) ≤ CR σ 1 µ(B(x, AR) B(x,AR) g(z) p dν(z) 1/p ≤ CJ p (g, ind(x, y) -k 0 , x). (4.9)
The estimate (4.5) follows when we sum up (4.6), (4.7), (4.8). The proposition will be proven once we show that

sup m≥k0 a (p) m (f ) p < ∞, where
a (p) m (f ) p = L m(σp+d) ρ(x,y)≤ α L m |f (x) -f (y)| p dµ(x)dµ(y).
We have:

a (p) m (f ) p ≤ K ∆ * m (x) |f (x) -f (y)| p dµ(y)dµ(x) ≤ K ∞ k=m+1 ∆ * k-1 (x)\∆ * k (x) |f (x) -f (y)| p dµ(y) dµ(x) (4.10) Since y ∈ ∆ * k-1 (x) \ ∆ * k (x) is tantamount to ind(x, y) = k + 1
, we can use the previously obtained estimate (4.5) and get

∆ * k-1 (x)\∆ * k (x) |f (x) -f (y)| p dµ(y) ≤ C ∆ * k-1 (x)\∆ * k (x)(x) J p p (g, k -k 0 , x)dµ(y) + ∆ * k (x)\∆ * k+1 (x) J p p (g, k -k 0 , y)dµ(y) = C(I k (x) + II k (x)). ( 4

.11)

To estimate these two parts we need a lemma, which is similar to Lemma 4.3 (ii), (iii) of [START_REF] Koskela | Mac Manus Quasiconformal mappings and Sobolev spaces[END_REF]:

Lemma 4.3. Let N ≥ 1, p ≥ 1, σ > 0 be given and let the functions f ∈ L p (K, µ), g ∈ L p (K, ν) satisfy the (1, p, σ)-Poincaré inequality. Then for µ-almost all x ∈ K ∆ * N (x) J p p (g, N, y)dµ(y) ≤ CL -N σp ∆ * * N (x) g p dν (4.
J p (g, N, y) = ∞ m=0 L -(m+N )σ 1 µ(∆ * N +m (y)) ∆ * N +m (y) g p (z)dν(z) 1/p ≤ C ∞ m=0 L -(σ-d p )(N +m) ∆ * * N (x) g p (z)dν(z) 1/p = CL -(σ-d p )N ∆ * * N (x) g p (z)dν(z) 1/p . Since µ(∆ * N (x)) ≤ CL -N d , ( 4
.12) follows. To see (4.13), observe that, using (4.12):

K J p p (g, N, y)dµ(y) = ∆∈TN ∆ J p p (g, N, y)dµ(y) ≤ C ∆∈TN L -N pσ ∆ * * g p dν.
A covering argument as the one used to conclude the proof of Theorem 3.1 gives (4.13).

Conclusion of the proof of Proposition 4.2. Since

I k (x) ≤ µ(∆ * k-1 (x))J p p (g, k -k 0 , x) ≤ CL -kd J p p (g, k -k 0 , x),
one has, using (4.13)

K I k (x)dµ(x) ≤ K ∆ * k (x) J p p (g, k -k 0 , x)dµ(y)dµ(x) ≤ C K J p p (g, k -k 0 , x)µ(∆ * k (x))dµ(x) ≤ CL -k(d+σp) K g p dν. (4.14)
To estimate the other part, we use (4.12):

K II k (x)dµ(x) ≤ K ∆ * k (x) J p p (g, k -k 0 , y)dµ(y)dµ(x) ≤ C K ∆ * k-k 0 (x)
J p p (g, kk 0 , y)dµ(y)dµ(x) We now turn our attention to the relation of Poincaré-Sobolev spaces P 1,p σ (K) to Haj lasz-Sobolev spaces M 1,p σ (K). It has been proven by Hu that M 1,p σ (K) ⊂ KS 1,p σ (K), for all p ≥ 1 and σ > 0 (Theorem 1.1. of [START_REF] Hu | A note on Haj lasz-Sobolev spaces on fractals[END_REF]). Moreover, this theorem asserts that one has the inclusion KS 1,p σ (K) ⊂ M 1,p σ ′ (K), for all 0 < σ ′ < σ. It is not known whether the inclusion KS 1,p σ (K) ⊂ M 1,p σ (K) holds true on nested fractals, even if we assume that Property (P) holds.

≤ C K L -kσp ∆ * * k-k 0 (x)
Recall that for p ≥ 1, the 'weak' L p , or the Marcinkiewicz space L p w (K, µ) consists of those measurable functions f for which sup t>0 {t p µ{x : |f (x)| > t}} < +∞.

We can consider the 'weak' Hajlasz-Sobolev spaces. Definition 4.4. Let p ≥ 1 and σ > 0. One says that f ∈ L p (K, µ) belongs to the weak Hajlasz-Sobolev space (M 1,p σ ) w (K), if there exists g ∈ L p w (K, µ) such that (4.1) holds true.

We have the following. Proposition 4.5. Suppose that the nested fractal K satisfies Property (P). Assume p ≥ 1, σ > 0. Then one has:

(1) P 1,p σ (K) ⊂ (M 1,p σ ) w (K) ⊂ M 1,p ′ σ (K), with any 1 ≤ p ′ < p (the last inclusion requires p > 1).

(2) When p = 2, σ = d w /2, then M 1,2 σ (K) ⊂ P 1,2 σ (K).

Proof.

(1) Once we have proven estimates for fractal Riesz-potentials, this result is immediate. Let f ∈ P 1,p σ (K), and let (f, f ) satisfy the (1, p, σ) Poincaré inequality. The function g (corresponding to the upper gradient), needed in the definition of Haljasz-Sobolev spaces, will be a fractal variant of the Hardy-Littlewood maximal function: for x ∈ K \ V (∞) we set

g(x) = (M f )(x) def = sup m≥1 1 µ(∆ * m (x)) ∆ * m (x) f p dν 1/p .
It is obvious that for any n ≥ 1

J p ( f , n, x) ≤ CL -nσ g(x), (4.16)
with some universal constant C > 0. Recall the estimate (4.5):

|f (x)f (y)| ≤ C J p ( f , ind (x, y)k 0 , x) + J p ( f , ind (x, y)k 0 , y) (k 0 was a universal index depending only on the geometry of the fractal), so that further, taking into account the relation (2.4) |f (x)f (y)| ≤ CL -σind(x,y) (g(x) + g(y)) ≤ Cρ(x, y) σ (g(x) + g(y)).

The argument that proves g ∈ L p w (K, ν) is also classical. Fix t > 0 and suppose that g(x) > t for some x ∈ K \ V (∞) . By the definition of g, there exists m = m(x) such that µ(∆ * m (x)) ≤

1 t p ∆ * m (x)
f p dν. (4.17)

Consider the covering of the set A(t) = {x ∈ K : g(x) > t} by balls B(x, 2L -m(x) ),

x ∈ A(t) \ V (∞) . By the 5r-covering lemma there is a countable subcollection of these balls, B i = B(x i , ρ i ), with ρ i = 2L -m(xi) , such that the B i 's are pairwise disjoint, yet A(t) ⊂ i B(x i , 5ρ i ). Due to (3.16), the sets ∆ * m(xi) (x i ) are disjoint. Then, by the doubling property of µ, µ({x : g(x) > t}

≤ µ i B(x i , 5ρ i ) ≤ C i µ(B(x i , ρ i )) ≤ C i µ(B(x i , αL -m(xi) )) ≤ C i µ(∆ * m(xi) ) (4.17) ≤ C t p i ∆ * m(x i ) f p dν ≤ C t p K f p dν.
Since µ(K) < ∞, we have L p w (K, µ) ⊂ L p ′ (K, µ) for p ′ < p. This way (1) is proven. Assertion (2) follows from Hu's inclusion M 1,2 σ (K) ⊂ KS 1,p σ (K) and Proposition 4.2 (2) above.

Definition 2 . 1 .

 21 Let m ≥ 1.

  , N, y)dµ(y) ≤ C L -N σp K g p dν. (4.13) Proof. For y ∈ ∆ * N (x) and k ≥ N one has ∆ * k (y) ⊂ ∆ * N (y) ⊂ ∆ * * N (x) and therefore

g

  p dν dµ(x)≤ CL -k(d+σp) K g p dν.(4.15)Summing up (4.14) and (4.15) over k ≥ m we get that the right-hand side of (4once m ≥ k 0 . The proposition follows.

Acknowledgements. The work of K.P.P. was supported by the Polish Ministry of Science grant no. N N201 397837 (years 2009-2012). Part of this research was conducted while K.P.P. was visiting Univerité Blaise Pascal, Clermont-Ferrand. The author wants to thank UBP for its hospitality. A.S. was partially supported by the MNiSW grant N N201 373136.

Appendix

We will now prove the statement from Remark 1. Set α 0 = inf{dist (A, B) : A, B ∈ T 2 , A ∩ B = ∅} and α = Lα 0 .

More precisely, we will be proving the following. Proposition 5.1. Let K be the nested fractal associated with similitudes {φ i } r i=1 with contraction factor L. Suppose that the φ i 's share their unitary parts, i.e. there is an isometry

The key argument in the proof is provided by the following lemma.

Lemma 5.2. Let n ≥ 1. Suppose A, B are two neighbouring n-simplices, and let

Proof. We proceed by induction on n. Clearly, the statement is true for n = 1.

Suppose that the statement is true for 1, ..., n -1. Let A, B ∈ T n , A 1 , B 1 ∈ T n+1 be as in the statement; let (i 1 , ..., i n ) be the address of A and (w 1 , ., , , w n ) -the address of B. Define k 0 = min{l :

Those simplices satisfy the assumptions for nk 0 + 1 ≤ (n -1)) and the statement follows. Now, suppose that k 0 = 1. We have

Let v be a junction point of A and B. Because of the inclusions above, v ∈ K i1i2 ∩ K w1w2 ⊂ K i1 ∩ K w1 as well.

We will now show that K i1i2 ∪ K w1w2 is similar to K i1 ∪ K w1 . More precisely, we will see that

where S = 1 L U is the similitude such that (1) , there exist z 1 , z 2 ∈ V (0) and mappings φ j1 , φ j2 such that z l is the fixed point of φ j l , l = 1, 2, and v = φ i1 (z 1 ) = φ w1 (z 2 ). Further, since v ∈ K i1i2 , there exist another essential fixed point u, such that v = φ i1i2 (u). Then we have φ i1 (z 1 ) = φ i1i2 (u) and so z 1 = φ i2 (u). In particular, z 1 ∈ K i2 ∩ K j1 . By Proposition IV.13 of [START_REF] Lindstrøm | Brownian motion on nested fractals[END_REF], any element in V (0) belongs to exactly one n-cell for each n. It follows that j 1 = i 2 . The same argument for K w1w2 gives j 2 = w 2 .

Since S is linear, we have

On the other hand, since z 1 = φ i2 (z 1 ), we get

Identical arguments hold for the pair K w1 and K w1w2 and the proof of (5.1) is complete. Now,

are two pairs of (n -1)-and (n -2)-simplices satisfying the assumptions, hence dist (A ′ 1 , B ′ 1 ) ≥ α L n-1 , and thus dist (A 1 , B 1 ) ≥ α L n . This completes the proof.

Proof of Proposition 5.1. We proceed by induction on n.

If n = 1 and y / ∈ ∆ * 1 (x) \ ∆ * 2 (x), then the 2-simplices ∆ 2 (x) and ∆ 2 (y) are disjoint. Thus, ρ(x, y) ≥ dist (∆ 2 (x), ∆ 2 (y)) ≥ α 0 = α/L.

Suppose now that the statement is true for 1, 2, ..., n -1, and take y ∈ ∆ * n (x) \ ∆ * n+1 (x). Then the sets ∆ n+1 (x) and ∆ n+1 (y) are disjoint, whereas ∆ n (x) and ∆ n (y) are not. There are two possibilities: either ∆ n (x) = ∆ n (y), or they are adjacent n-simplices. Let (i 1 , ..., i n ) be the address of ∆ n (x) and (w 1 , ..., w n ) be the address of ∆ n (y).

If ∆ n (x) = ∆ n (y), we consider points x ′ = φ -1 i1...in-1 (x), y ′ = φ -1 i1...in-1 (y). Then ∆ 2 (x ′ ) and ∆ 2 (y ′ ) are disjoint 2-simplices, so from the assumption we get ρ(x ′ , y ′ ) ≥ α L , thus ρ(x, y) ≥ α L n . If ∆ n (x) and ∆ n (y) are adjacent n-simplices, we apply lemma 5.2 to A = ∆ n (x), B = ∆ n (y), A 1 = ∆ n+1 (x), B 1 = ∆ n+1 (y).


Mark Swinbank

Alastair Edge

Johan Richard

Ian Smail

Carlos De Breuck

Andreas Lundgren

Giorgio Siringo

Axel Weiss

Andrew Harris

Andrew Baker

Steve Longmore

Rob Ivison

The Properties of Star-forming Regions within a Galaxy at Redshift 2 Astronomical Science

The discovery and subsequent followup of one of the brightest sub-mm galaxies discovered so far is presented. First identified with the LABOCA instrument on APEX in May 2009, this galaxy lies at z = 2.32 and its brightness of 106 mJy at 870 µm is due to the gravitational magnification caused by a massive intervening galaxy cluster. Followup observations with APEX SABOCA have been used to constrain the farinfrared spectral energy distribution and hence measure the star formation rate, and Swedish Heterodyne Facility Instrument observations help constrain the excitation of the cold molecular gas. Furthermore, high resolution follow-up with the sub-mm array resolves the star-forming regions on scales of just 100 parsecs. These results allow study of galaxy formation and evolution at a level of detail never before possible and provide a glimpse of the exciting possibilities for future studies of galaxies at these early times, particularly with ALMA.

Massive galaxies in the early Universe have been shown to be forming stars at surprisingly high rates. The most prominent examples are sub-millimetre galaxies (SMGs) whose star formation rates can exceed 1000 M A year -1 (Chapman et al., 2003;Coppin et al., 2008). With this intense star formation rate, a massive galaxy (comparable to the stellar mass of a local elliptical galaxy) can be built in just 100 million years (Tacconi et al., 2008). As such, the sub-mm galaxy population may represent the formation epoch of today's massive elliptical galaxies (Lilly et al., 1999;Smail et al., 1997). This is, however, a theoretically provocative conclusion as it is at odds with theoretical models. Indeed, sophisticated galaxy formation models have had to alter their prescriptions for starbursts radically, invoking exotic physics such as strong variations in the stellar initial mass function in order to account for the SMG population (Baugh et al., 2004). Although controversial, there is growing evidence that the increased interstellar medium (ISM) pressure within the warm, dense gas in local ultra-luminous infrared galaxies (ULIRGs) results in an increased Jeans mass (Perez-Torres et al., 2009). In order to test these controversial prescriptions, direct observational constraints on the properties of individual star-forming regions within high redshift galaxies are necessary. Such observations are technologically challenging, requiring the increased collecting area and sensitivity of the next generation tele scopes, such as the Extremely Large Tele scopes currently being planned and the Atacama Large Millimeter/Submillimeter Array (ALMA) under construction. Moreover, since the most intensely star-forming galaxies are also the most obscured, they are optically faint and difficult to identify and study on sub-kiloparsec (sub-kpc) scales.

Gravitational lensing

The most promising route for investigating the properties of high redshift galaxies on sub-kpc scales is to use massive galaxy clusters as natural lenses. Galaxy clusters magnify the images of distant galaxies that serendipitously lie behind them. This natural magnification causes background galaxies to be strongly amplified and stretched, providing us with the opportunity of studying young and intrinsically faint galaxies with a spatial resolution that cannot be attained via conventional observations. The natural amplification caused by the galaxy cluster has two effects: (i) the image of the background galaxy is magnified at a fixed surface brightness (i.e. the total brightness is increased); and, (ii) the galaxy is not only amplified, but it is also stretched, making it possible to spatially resolve components of the galaxy from the ground.

LABOCA imaging of distant clusters

During recent Atacama Pathfinder Experiment (APEX) 870-µm observations of a massive galaxy cluster (MACS 2135-0102) we serendipitously discovered an extremely bright sub-mm galaxy with an 870-µm flux of 106 mJy (see Figure 1, left). This is three times brighter than any other known high-redshift star-forming galaxy and even brighter than the Cloverleaf and APM08279 quasars. Due to the high significance of the source, we were able to centroid the sub-mm emission to an accuracy of less than 0.5 arcseconds and identify the mid-infrared counterpart with a bright source detected by the Spitzer Space Telescope Infra-Red Array Camera (IRAC) and Multiband Imaging Photometer (MIPS); see Figure 1 right. Due to the brightness, we were able to obtain a redshift measurement for the galaxy of z = 2.3259 via the blind detection of CO(1-0) using the Zpectrometer on the Green Bank Telescope (GBT), confirming that the galaxy is high redshift and strongly gravitationally lensed. Indeed, using a detailed mass model of the galaxy cluster we derive an amplification factor of 32.5 ± 4.5. Intrinsically therefore the 870-µm flux is ~ 3 mJy, typical of the sub-mm background, and, on account of the amplification, an ideal target for study. The CO(1-0) emission line flux can also be used to estimate the cold molecular gas mass; after correcting for lensing amplification we derive a gas mass of 2 × 10 10 M A .

The Messenger 139 -March 2010 SABOCA With confirmation that this is a highly amplified background galaxy, observations at 350 µm using the SABOCA camera (see article by Siringo et al.,p. 20) were carried out to parameterise the spectral energy distribution (SED); see Figure 2. With a measured 350-µm flux of 530 mJy we derive an intrinsic far-infrared luminosity of 1.2 × 10 12 L A . This suggests a star formation rate of 210 M A /yr. If this star formation rate has been maintained, then it takes just ~ 120 Myr to build the observed stellar mass of 4 × 10 10 M A and the remaining gas depletion timescale is a further 70 Myr. Together, this suggests that this intense star formation episode may be the first major growth phase of this galaxy.

Molecular gas properties

To further constrain the properties of the cold molecular gas, observations of the high-J CO emission (up to J = 7) were obtained with a combination of Institut de Radioastronomie Millimétrique Plateau de Bure Interferometer (IRAM PdBI; CO(3-2) up to CO(7-6)) and APEX Swedish Heterodyne Facility Instrument (SHFI) (CO(7-6)).

In Figure 3a we show the CO(1-0) and CO(7-6) line profiles obtained with the IRAM Eight MIxer Receiver (EMIR) and APEX/SHFI respectively. All the CO spectra show multiple peaks that we decompose into a "red" component and a "blue" component which are separated by 300 ± 24 km/s. Since the ratio of the high/ low-J emission lines reflects the excitation of the molecular gas, we extract and model the CO spectral line energy distribution (SLED) of both "red" and "blue" components using large velocity gradient (LVG) models (e.g., Weiss et al., 2005). Both components are best fit with two excitation models (with temperatures ~ 40 K and densities of approximately 10 3 and 10 4 cm -3 ; see Figure 3b. The LVG modelling also includes the "equivalent size" of the CO-emitting region as a free parameter, and we are therefore able to estimate the size of each component indirectly. The SLED modelling suggests that both the "red" and "blue" components appear to have a compact core with radius ~ 150 pc surrounded by a more extended, diffuse region with radius ~ 500 pc. Although these equivalent sizes come with some uncertainty, the brightest CO-emitting regions appear similar to those seen locally as starbursts with a dense, active star-forming core surrounded by a diffuse gas disc, perhaps providing the first direct evidence for a diffuse disc in an SMG.

Resolving the star-forming regions

To resolve the sub-mm emission directly, Submillimeter Array (SMA) observations were used to image the galaxy's 870-µm (345 GHz) continuum emission with a 0.2-arcsecond synthesised beam and hence investigate the sub-mm morphology. The map resolves the galaxy into eight discrete components over ~ 4 arcseconds in projection (Figure 4). These components represent two mirror images of the galaxy each comprising four separate emission regions. Reconstructing the source-plane image using the lensing model, the galaxy comprises four bright star-forming regions in the source plane (indicated as A, B, C and D in Figure 4), that are separated by 1.5 kpc in projection. At the most highly amplified position, the source-plane resolution reaches ~ 90 pc which is comparable to the characteristic size of giant molecular clouds in the Milky Way (Scoville et al., 1989).

We compare the sizes and luminosities of the star-forming regions within SMMJ2135-0102 to giant molecular clouds (GMCs) in the local Universe (Scoville et al., 1989;Snell et al., 2002;Caldwell et al., 1996) in Figure 5. With-in typical GMCs, constant energy density leads to a correlation between size and 260-µm luminosity, such that luminosity is proportional to radius cubed. However, within the dense cores of GMCs and young H ii regions, massive stars dominate the emission and produce luminosity densities a factor ~ 100 times higher than in typical GMCs (Hill et al., 2005) and shown by the upper dashed line in Figure 5. The star-forming regions within SMMJ2135-0102 are ~ 100 pc across, which is 100 times larger than dense GMC cores, but as Figure 5 shows their luminosities are approximately 100 times higher than expected for typical starforming regions. Indeed, the luminosity densities of the star-forming regions within SMMJ2135-0102 are comparable Swinbank M. et al., The Properties of Star-forming Regions within a Galaxy at z = 2 Astronomical Science Figure 3a (above). Spectra of the 12 CO(1-0) and 12 CO(7-6) emission from SMMJ2135-0102 obtained with GBT/Zpec (left) and APEX/SFHI (right). The CO emission shows at least two velocity components separated by 300 km/s.

Figure 3b (below).

The spectral line energy distributions (SLEDs) for the "blue" and "red" velocity components seen in CO emission. to dense GMC cores, but with luminosities 10 7 times larger. Thus, it is likely that each of the star-forming regions in SMMJ2135-0102 comprises ~ 10 7 dense GMC cores.

The luminosity (and therefore star formation) density of the star-forming regions within SMMJ2135-0102 are also similar to those found in the highest density regions of the local starburst galaxy Arp220 (shown by blue triangles in Figure 5), although they are scaled up by a factor 10 in both size and luminosity (Sakamoto et al. 2008). Thus, the energetics of the starforming regions within SMMJ2135-0102 are unlike anything found in the presentday Universe, yet the relations between size and luminosity are similar to local, dense GMC cores, suggesting that the underlying physics of the star-forming processes is similar. Overall, these results suggest that the recipes developed to understand star-forming processes in the Milky Way and local galaxies can be used to model the star formation processes in these high-redshift galaxies.

Outlook

Overall, these results provide unique insight into the physics of star formation within a galaxy at z ~ 2 on scales that would otherwise only be achieved with the increased light grasp and resolution of the next generation of facilities, such as ALMA. Indeed, due to the lensing effect we are able to provide an effective glimpse of two of the three science drivers for ALMA: to provide images at an angular resolution of 0.1 arcsecond of the dust continuum from distant galaxies; detect spectral CO emission in star-forming galaxies out z = 3 and measure their redshifts; albeit only for a single object. Thus, due to the fortuitous discovery with LABOCA and subsequent APEX/SHFI, SMA and GBT/Zpectrometer follow-up of this galaxy, we are able to provide unique insights into the key science that will be routine once ALMA reaches full science operations.

Figure 1 .Figure 2 .

 12 Figure 1. True colour HST VI-band image of the massive galaxy cluster MACSJ2135-0102 (z cl = 0.325) is shown (left). The white contours denote the APEX/ LABOCA 870-µm contours from the galaxy (contours denote 5, 10, 15, 20, 25, 30σ), identifying an SMG with flux 106 mJy. The solid red lines denote the z = 2.326 radial and tangential critical curves from the best-fit lens model. The true colour IRAC 3.6, 4.5, 8.0-µm image of the cluster core is shown (right). The contours denote the intensity of the 350-µm map and show 5, 10, 15, 20σ. The red lines again show the lens model critical curves.

Figure 5 .

 5 Figure 5. Correlation between size and luminosity for star-forming regions in SMMJ2135-0102 compared to those in the Milky Way and local galaxies. Black squares denote the size and 260-µm luminosities of giant molecular clouds in the Milky Way and the local group. The lower dashed line shows a fit to this relation (with slope fixed at L 260 µ ∝ r 3 ; i.e. constant energy density). The solid red points denote the sizes and luminosities of the star-forming regions in SMMJ2135-0102. The sizes and luminosities of dense cores of galactic GMCs in Henize 2-10 and M82 are plotted (open diamonds) as well as the two dominant star-forming regions within the local ultraluminous infrared galaxy Arp220 (blue triangles).

Figure 4a .

 4a Figure 4a. The SMA map of SMMJ2135-0102 showing the image-plane morphology of the lensed galaxy at 870 µm. The galaxy comprises eight individual components, separated by up to 4 arcseconds in projection. The z = 2.326 radial critical curve (red line) is overlaid. The beam size is shown (lower right).

Figure 4b .

 4b Figure 4b. The spatial distribution of the four components in the image plane from the lens model are shown. Each of the components (A, B, C and D), which are mirrored about the lensing critical curve, are identified.