Harmonic functions representation of Besov-Lipschitz functions on nested fractals

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This is an author produced version of a paper Citation for the published paper:

Katarzyna Pietruska-Paluba and Mats Bodin

Harmonic functions representation of Besov-Lipschitz functions on nested fractals Research Report in Mathematics, ISSN 1653-0810, No. 2, 2010

Published with permission from:

Umeå universitet

Harmonic functions representation of Besov-Lipschitz functions on nested

fractals

Katarzyna Pietruska-Pa luba ^∗ and Mats Bodin ^†

Research Report

in Mathematics

Harmonic functions representation of Besov-Lipschitz functions on nested fractals

Katarzyna Pietruska-Pa luba

and Mats Bodin

Abstract. R. S. Strichartz proposes a discrete definition of Besov spaces on self- similar fractals having a regular harmonic structure. In this paper, we characterize some of these H¨older-Zygmund and Besov-Lipschitz functions on nested fractals by means of the magnitude of the coefficients of the expansion of a function in a continuous piecewise harmonic base.

1. Introduction

The analysis on irregular sets such as the Sierpi´ nski gasket has been developed since the late 80’s, starting with the pioneering works [9, 13]. Also, the Brownian motion on fractals has been defined (see e.g. [1] for its definition, properties, and an exten- sive literature list). Both approaches yield the definition of the ‘Laplace operator’ A on fractals. Having correctly defined the Laplacian, one can say that a function is

‘harmonic’ when it is annihilated by the Laplacian. However, such a definition of a harmonic function is very general and its properties are difficult to investigate.

On nested fractals (Sierpi´ nski gasket-like), we are in a more favorable setting: their self-similar structure permits harmonic and ‘locally harmonic’ functions to be defined in an iterative manner. The Brownian motion, or more precisely the Dirichlet form associated with the Brownian motion, can be defined using discrete approximations as well. Such a discrete approach naturally yields the characteristics of the fractal, such as its Hausdorff dimension d

and its walk dimension d

. They are recovered from the mass-scaling, length-scaling, and resistance-scaling factors, see (2.2) and (2.3) below.

Once the Dirichlet form E is defined together with its domain, then one has the variational definition of a harmonic function with prescribed boundary values: it is the unique function which minimizes E(f, f ) among functions with given boundary values. Similar considerations can be carried out locally, which gives rise to the notion

*Supported by the Polis Ministry of Science and Higher Education grant no. N N201 397837 (years 2009-2012).

†This work was in part conducted while being employed as a fellow of the Marie Curie Transfer of Knowledge Programme SPADE2 (Deterministic and stochastic dynamics, fractals, turbulence) at the Institute of Mathematics of the Polish Academy of Sciences (IMPAN).

of an m−harmonic function (i.e. a ‘locally harmonic’, or ‘piecewise harmonic’ func- tion), with values given at points of the m−th approximation of the fractal. Precise definitions are given below, in Section 2.2.2.

Since the geometry of nested fractals is decided after just one step of the iteration, the properties of harmonic functions are much easier to analyze with this approach.

In particular, one can prove that harmonic functions are Lipschitz with respect to the so-called resistance metric on fractals, see [12].

Piecewise harmonic functions on nested fractals play a central role in analysis on fractals. For example, they enter the definition of the gradient and the derivative on fractals (see e.g. [13], [10], [17], [19]). We will use them to another goal: to provide a characterization of spaces of Lipschitz functions (H¨older-Zygmund, Besov-Lipschitz) on fractals. The spaces we will be working with are defined using intrinsic properties of the set K and do not really use the fact that they are embedded in R

. For their definition, we refer to [5], [18], and also to Section 2.2 below.

It is known (see e.g. [11]) that piecewise harmonic functions constitute a basis in the space of all continuous functions on a given nested fractal K : every continuous function f on K can be written as a (countable) series P

c

ψ

(·), with ψ

piecewise harmonic, and the series convergent uniformly and unconditionally. In this paper, we provide a characterization of H¨older-Zygmund and Besov-Lipschitz functions on K expressed in terms of the ‘size’ of their coefficients c

(Theorems 3.1, 3.2). Similar theorems for piecewise linear basis for certain subsets of R

(in particular for the Sierpi´ nski gasket) can be found in [7].

2. Preliminaries

We start with a description of the sets we will be working with, then we introduce the notion of harmonic and m−harmonic functions, and provide the definition of H¨older- Zygmund and Besov-Lipschitz spaces on nested fractals.

2.1. Nested fractals. Their definition goes back to T. Lindstrøm [14].

Suppose that φ

, ..., φ

, M ≥ 2, are similitudes of R

with common scaling factor L > 1 (i.e. φ

(x) =

U

(x) + v

, U

−isometries, v

−vectors in R

). There exists a unique nonempty compact set (see [3], [14]) K ⊂ R

such that

K = [

φ

(K). (2.1)

K is called the self-similar fractal generated by the family of similitudes φ

, ...φ

.

As all the similitudes φ

are contractions, each of them has a unique fixed point

x

(not necessarily distinct). Denote by F the collection of those fixed points. A

fixed point x ∈ F is called an essential fixed point of the system φ

, ...φ

if there

exists another fixed point y ∈ F and two different transformations φ

, φ

such that

φ

(x) = φ

(y). Denote by V

= {v

, ..., v

} the set of all essential fixed points of the system {φ

, ..., φ

}.

For A ⊂ R

, let Φ(A) = S

i=1

φ

(A) and Φ

= Φ ◦ · · · ◦ Φ | {z }

m times

. Then we define ‘the m-th level’ vertices as

V

= Ψ

(V

), m = 1, 2...

(V

)

is an increasing family of sets (it follows from the inclusion V

⊂ Φ(V

)).

The vertices that are added in the m−th step will be denoted by V

:= V

\V

, m = 1, 2, ... For consistency, we set V

= V

. Also, letting

V

= [

V

= [

V

.

we have that K = V

, the closure taken in the Euclidean topology of R

(see [3], p. 120).

We impose certain conditions concerning the maps φ

, ..., φ

. Condition 1. #V

≥ 2.

Condition 2 (Open Set Condition). There exists an open, bounded, nonempty set U ⊂ R

such that φ

(U ) ∩ φ

(U ) = ∅ for i 6= j and

Φ(U ) = [

φ

(U ) ⊂ U.

If the open set condition is satisfied, then the Hausdorff dimension of K is equal to (Th. 8.6 of [3])

d

= d

(K) = log M

log L ≤ N (2.2)

By µ we will denote the Bernoulli measure on K, i.e. the restriction of the x

−Haus- dorff measure to K, normalized so as to have µ(K) = 1. This measure is Ahlfors d

−regular, i.e. measures of balls with radius 0 < t ≤ diam K and centers in K are comparable with t

.

We collect the remaining notation that will be needed in the sequel in a single definition below. If w = (i

, ..., i

), then we will abbreviate φ

◦ · · · ◦ φ

to φ

.

Definition 2.1. Let m ≥ 0 be given.

• An m−simplex is any set of the form φ

(K), with w = (i

, ..., i

). (m-simplices are just scaled down copies of K).

• Collection of all m-simplices will be denoted by T

.

• For an m-simplex S = φ

(K), let V (S) = φ

(V

) be the set of its vertices.

• An m−cell is any of the sets φ

(V

).

• Two points x, y ∈ V

are called m−neighbors (denoted x ∼ y), if they belong to

a common m−cell.

Condition 3. Nesting. For each m ≥ 1, and S, T ∈ T

, S ∩ T = V (S) ∩ V (T ).

Condition 4. Connectivity. Define the graph structure E

on V

as follows: we say that (x, y) ∈ E

if x and y are 1-neighbors. Then we require the graph (V

, E

) to be connected.

Condition 5. Symmetry. For x, y ∈ V

. let R

be the reflection in the hyperplane bisecting the segment [x, y]. Then we stipulate that

∀

∀

∃

R

(φ

(V

) = φ

(V

).

Definition 2.2 (nested fractal). The self-similar fractal K, defined by (2.1) is called a nested fractal if conditions 1-5 are satisfied.

2.2. Functions on nested fractals. We will now describe notions necessary to define harmonic functions on nested fractals, as well as introduce the H¨older and Besov-Lipschitz spaces we will be working with.

2.2.1. The nondegenerate harmonic structure on K. The material in this section is classical and follows [1]. See also [11].

An r × r real matrix A = [a

]

is called a conductivity matrix on V

if:

(1) ∀x 6= y a

≥ 0, a

= a

; (2) ∀x, P

y

a

= 0.

We assume that A is irreducible, i.e. the graph hV

, E

i is connected, where E

is the graph structure on V

induced by A, i.e. V

× V

∋ (x, y) ∈ E

if and only if a

> 0.

Any conductivity matrix on V

gives rise to a Dirichlet form on C(V

) : for f ∈ C(V

),

E

(f, f ) = 1 2

X

x,y∈V⁽⁰⁾

a

(f (x) − f (y))

.

Two operations on Dirichlet forms are then defined, both relying on the geometry of K.

1. Reproduction. For f ∈ C(V

) we let E e

(f, f ) =

X

E

(f ◦ φ

, f ◦ φ

).

The mapping E

7→ e E

is called the reproduction map and is denoted by R.

2. Decimation. Given a symmetric form E on C(V

), denote its restriction to

C(V

), E

, as follows. Take f : V

→ R, then set

E|

(f, f ) = inf{E(g, g) : g : V

→ R and g|

= f }.

This mapping is called the decimation map and will be denoted by De.

Let G be the symmetry group of V

, i.e. the group of transformation generated by symmetries R

, x, y ∈ V

. Then we have ([14], [16]):

Theorem 2.1. 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

, invariant under the action of G, and such that

(De ◦ R)(E

) = 1

ρ E

. (2.3)

A is called the symmetric nondegenerate harmonic structure on K. It can be proven that all the nondiagonal entries of A are positive.

By analogy with the electrical circuit theory, ρ is called the resistance scaling factor of K. The number d

= d

(K)

=

> 1 is called the walk dimension of K.

2.2.2. The Dirichlet form and harmonic functions on K. Suppose now that A is given by Theorem 2.1. Define E

= E

, then let

E e

(f, f ) = X

w=(i1,...,im)

E

(f ◦ φ

, f ◦ φ

), and further

E

(f, f ) = ρ e E

(f, f ); E

(f, f ) = ρ

E e

(f, f ).

Multiplication by ρ in every step makes the sequence E

nondecreasing, so we obtain that: for every f : V

→ R, and every m = 0, 1, 2, ... one has

E

(f, f ) ≤ E

(f, f ).

Set e D = {f : V

→ R : sup

E

(f, f ) < ∞} and for f ∈ e D E(f, f ) = lim e

m→∞

E

(f, f ), (2.4)

and further D = {f ∈ C(K) : f |

∈ e D}, E(f, f ) = e E(f |

, f |

) for f ∈ D.

(E, D) is a regular Dirichlet form on L

(K, µ), which agrees with the group of local symmetries of K.

There are several possible ways of defining a harmonic (and an m−harmonic) func- tion on K. We choose the following one.

Definition 2.3. • Suppose f : V

→ 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|

= f.

• Similarly,when f : V

→ R is given (m ≥ 1), then h ∈ D(E) is called m−harmonic

with boundary values f , if E(h, h) minimizes the expression E(g, g) among all g ∈ D(E) such that g|

= f.

Denote by H

the space of all m−harmonic functions on K. For f ∈ C(K), let S

f be the m−harmonic function that agrees with f on V

.

Definition 2.4. Suppose ξ ∈ V

, and let m be the unique number such that ξ ∈ V

. Then ψ

is, by definition, the unique m−harmonic function whose values on V

are given by:

ψ

(η) =

 1 η = ξ, 0 η 6= ξ.

2.2.3. Harmonic representation of continuous functions on K. The collection of func- tions {ψ

: ξ ∈ V

= S

i=0

V

} constitutes a linear basis in H

; one can write S

f =

X

X

ξ∈V^(m)

c

ψ

, (2.5)

where the coefficients c

are given by c

= f (ξ) − S

f (ξ) (we set S

≡ 0).

It is not hard to see (cf. eg. [1], p. 99. proof of Th. 7.14, or [11]), that for f ∈ C(K), S

f → f uniformly as m → ∞. It follows that every continuous function on K admits an expansion

f (x) = X

i=0

X

ξ∈V⁽ⁱ⁾

c

ψ

(x), (2.6)

with the series uniformly convergent in K.

2.2.4. H¨ older functions on K. Following Strichatz (see [18], Def. 7.1.), we define the H¨older-Zygmund spaces on K as follows.

Definition 2.5. Suppose α ∈ (0, 1). Then f ∈ Λ

(K) if and only if f ∈ C(K) and

∃ M ∀ m ≥ 0 ∀ x ∼ y

|f (x) − f (y)| ≤ M L

.

It is known that (see [18], the discussion following Def. 7.1.), that functions from Λ

(K) are precisely those that satisfy |f (x) − f (y)| ≤ CR(x, y)

, where R(x, y) is the resistance metric on K. On the Sierpi´ nski gasket, the resistance metric is equivalent to

|x−y|

, in general this might not be true (|x− y| denotes the Euclidean distance).

2.2.5. Besov-Lipschitz functions on K. For the standard Sierpi´ nski gasket G in R

, recall that the L = 2, M = 3, d

=

, d

=

, ρ =

= L

. The spaces (Λ

)

(G) were defined by Strichartz in [18] for

< α ≤

+α

, where α

=

w−df

,

and consist of those continuous functions on the gasket (this is why there is a restriction

df

p

< α), for which

k((2

)

δ

(f ))

k

= k(ρ

δ

(f ))

k

< ∞, (2.7) where

δ

(f ) =

 





2

P

x^m∼y

|f (x) − f (y)|



p

if p < ∞, sup{|f (x) − f (y)| : x ∼ y}

if p = ∞.

The relation x

∼ y is the neighboring relation on the m−th approximation of the gasket V

, as described in Def. 2.1. In this particular case, x

∼ y if and only if x, y ∈ V

and |x − y| =

.

We can clearly extend this definition to nested fractals: supposing f ∈ C(V

), we put

δ

(f ) =

 



 L

P

x^m∼y

|f (x) − f (y)|



if p < ∞, sup{|f (x) − f (y)| : x ∼ y}

if p = ∞.

And we have:

Definition 2.6. Let 1 ≤ p, q ≤ ∞ are given and let α >

. We say that the function f ∈ L

(K, µ) ∩ C(K) belongs to the space (Λ

)

(K), if

kf k

:= k(L

δ

(f ))

k

< ∞.

The norm in this space is given by

kf k

= kf k

+ kf k

.

Remark 2.1. One can naturally extend the ”0-level” conductivities a

(given for x, y ∈ V

) to m−th level conductivities. Suppose x ∼ y, i.e. x and y belong to a

common m−cell φ

◦ · · · ◦ φ

(V

). Therefore we can uniquely determine two distinct vertices v

, v

∈ V

such that x = φ

◦ · · · ◦ φ

(v

) and y = φ

◦ · · · ◦ φ

(v

). We set

a

= a

.

It would be more natural to take into account these m−level conductivities in the definition of δ

, i.e. to replace |f (x) − f (y)|

by a

|f (x) − f (y)|

. But since the matrix A related to the nondegenerate harmonic structure has nonzero off-diagonal entries, it would not change the class of functions in (Λ

)

(K), and the norms would be equivalent.

3. Harmonic functions and Besov-Lipschitz spaces

Another possible basis in C(K), and historically the one first considered, is that con-

sisting of ”piecewise linear” functions on K. On the Sierpi´ nski gasket G for example,

one have an expansion of continuous function, similar to the one previously described

(2.6), in a series of piecewise linear functions: ”piecewise linear” functions η

will replace the harmonic functions ψ

in (2.6): for f ∈ C(G), we would have

f (x) = X

X

ξ∈V^(m)

d

η

(x),

with the series uniformly convergent. Here d

are coefficients defined analogously to c

. For details, see [7]. That paper dealt with certain subsets of R

, including e.g. the Sierpi´ nski gasket (and also the Sierpi´ nski carpet!), and function spaces Lip (β, p, q) on those sets. For their definition we refer e.g. to [8] or [5]. Since this definition is irrelevant for our purposes, we omit it.

The spaces Lip (β, p, q) are closely related to Strichartz’s (Λ

)

spaces: it is proven in [2] that (Λ

)

= Lip (β, p, q) for nested fractals (see [15] for a definition), where α and β are related through the relation β = α(d

− d

). The authors of [7]

have established a characterization of spaces Lip (β, p, q) on those subsets on R

that admit a proper triangulation: a function f ∈ C(K) belongs to Lip (β, p, q) if and only if its coefficients of its expansion in the piecewise linear basis (η

) are not too big (the precise condition is given in [7]). Similar considerations are carried over in [2] also for the H¨older-Zygmund spaces Λ

(K) (again, under appropriate restraints concerning the set K).

In what follows, we will give similar characterizations for the H¨older-Zygmund and Besov-Lipschitz spaces on K, employing the piecewise-harmonic basis.

Before we start, we formulate and prove two lemmas, concerning the harmonic functions on the fractal.

Lemma 3.1. For all x, y ∈ K, and i ≥ 0, we have that

#{ξ ∈ V

: ψ

(x) 6= ψ

(y)} ≤ 2r, where r is the number of essential fixed points.

Proof. If z ∈ ∆ ∈ T

, then ψ

(z) = 0 unless ξ ∈ V (∆), i.e. ξ is a vertex of the i-simplex ∆, which have r vertices. Hence #{ξ ∈ V

: ψ

(z) 6= 0} ≤ r for all z ∈ K.

We get that #{ξ ∈ V

: ψ

(x) 6= ψ

(y)} ≤ #{ξ ∈ V

: ψ

(x) 6= 0 or ψ

(y) 6= 0} ≤

≤ #{ξ ∈ V

: ψ

(x) 6= 0} + #{ξ ∈ V

: ψ

(y) 6= 0} ≤ 2r.  Lemma 3.2. Let f ∈ C(K) and ξ ∈ V

. Then

S

f (ξ) = f (v

)ψ

(ξ) + · · · + f (v

)ψ

(ξ). (3.1) Moreover, for all x ∈ K one has

ψ

(x) + · · · + ψ

(x) = 1. (3.2)

Proof of the lemma. The function φ(·) = f (v

)ψ

(·) + ... + f

ψ

(·) is harmonic,

and agrees with f on S

. Therefore S

f = φ.

For the second statement, observe that ψ

(x)+...+ψ

(x) is the harmonic extension of a function which is identically equal to 1 on V

. Because of its uniqueness, it must

be identically equal to 1 on K. 

By self-similarity, this statement carries over to subsequent steps. More precisely, we have:

Corollary 3.1. Suppose that ξ ∈ V

, let ∆ ∈ T

be the (m − 1)−simplex ξ belongs to, and let ˜ v

, ..., ˜ v

be its vertices. Then

S

f (ξ) = f (˜ v

)ψ

(ξ) + · · · + f (˜ v

)ψ

(ξ), with ψ

(x) + ... + ψ

(x) = 1 for all x ∈ ∆.

3.1. Characterization of H¨ older-Zygmund spaces. We start with the charac- terization of the spaces Λ

(K).

Theorem 3.1. Suppose the harmonic expansion of a function f ∈ C(K) is given by (2.6), take α ∈ (0, 1). Then f ∈ Λ

(K) if and only if there exists a constant M such that for ξ ∈ V

one has

|c

| ≤ M L

. (3.3) Proof. Suppose f ∈ Λ

(K). Let ξ ∈ V

and let ∆ be the (m − 1)−simplex ξ belongs to, ˜ v

, ..., ˜ v

– its vertices. Then, by Corollary 3.1

|c

| = |f (ξ) − S

f (ξ)|

= |f (ξ) − (f (˜ v

)ψ

(ξ) + · · · + f (˜ v

)ψ

(ξ))|

≤ ψ

(ξ)|f (ξ) − f (˜ v

)| + · · · + ψ

(ξ)|f (ξ) − f (˜ v

)|.

Points ξ and ˜ v

might, or might not, be m−neighbors. If they are not, we find a chain ξ = x

, ..., x

= ˜ v

such that x

∈ ∆ and x

∼ x

, 1 = 1, ..., n − 1. It is clear that n ≤ r, the number of all vertices of ∆. In either case, we have

|f (ξ)−f (˜ v

)| ≤ |f (x

)−f (x

)|+· · · +|f (x

)−f (x

)| ≤ r sup{|f (x)−f (y)| : x ∼ y},

so that

|c

| ≤ sup{|f (x) − f (y)| : x ∼ y} ≤ rM L

.

To get the opposite implication, suppose that the coefficients c

satisfy (3.3).

Take x

∼ y, then in particular x, y ∈ V

, so that f (x) = S

f (x) and f (y) = S

f (y). Therefore, using the assumption (3.3),

|f (x) − f (y)| = |S

f (x) − S

f (y)| = | X

i=0

X

ξ∈V⁽ⁱ⁾

c

(ψ

(x) − ψ

(y))|

≤ M X

i=0

L

X

ξ∈V⁽ⁱ⁾

|ψ

(x) − ψ

(y)|.

Because of the nesting, the self-similarity of the fractal, and the way the m−harmonic function arises, we infer that when x ∼ y, i ≤ m, ξ ∈ V

then |ψ

(x) − ψ

(y)| =

|ψ

(¯ x)−ψ

(¯ y)|, for some ¯ ξ ∈ V

(= V

) and ¯ x

∼ ¯ y. The function ψ

is 0−harmonic, i.e. harmonic, so by Proposition 5.12. of [12] it is Lipschitz with respect to the resis- tance metric R(·, ·) on K. Therefore

|ψ

(x) − ψ

(y)| ≤ A(¯ ξ)R(¯ x, ¯ y),

where ¯ ξ, ¯ x, ¯ y are as above. There are finitely many points ¯ ξ in V

, so that sup

A(¯ ξ) =: A < ∞. Since the points ¯ x, ¯ y are (m − i)−neighbors, there ex- ists w = (i

, ..., i

) such that ¯ x = ψ

(v

), ¯ y = ψ

(v

), for certain v

, v

∈ V

. By Theorem A.1. of [12], one has

R(¯ x, ¯ y) = R(ψ

(v

), ψ

(v

)) ≤ ρ

R(v

, v

) ≤ ρ

sup

v1,v2∈V⁽⁰⁾

R(v

, v

) = Cρ

, where ρ is the resistance scaling factor from Theorem 2.1 above. Note that ρ = L

. Collecting all of these, we see that

|ψ

(x) − ψ

(y)| ≤ A · C · L

.

According to Lemma 3.1, when x, y ∈ V

are given, there are at most 2r vertices ξ ∈ V

for which the difference |ψ

(x) − ψ

(y)| might be nonzero, so that finally

|f (x) − f (y)| ≤ 2rM AC X

L

L

≤ C(K, α)L

.

The constant in the last expression depends on on the geometry of K and on α (it

explodes as α → 1). The theorem is proven. 

3.2. Characterization of Besov-Lipschitz spaces. We now will give a character- ization of functions from (Λ

)

(K), similar to that from [7], but expressed in terms of their coefficients in the harmonic basis.

Theorem 3.2. Let K be a nested fractal, and f a continuous function on K having the harmonic expansion (2.6). Suppose 1 ≤ p, q ≤ ∞, α >

, and assume that ψ

∈ (Λ

)

(K) for some β > α, where v

is an arbitrarily chosen vertex from V

, and ψ

is a harmonic function from Definition 2.4. Then f ∈ (Λ

)

(K) if and only if S

(f ) < ∞, where (1 ≤ p, q < ∞)

S

(f ) :=



  X

L



 1

L

X

ξ∈V^(m)

|c

|





q/p



 

1/q

< ∞. (3.4)

Moreover, we have

1

C kf k

≤ S

(f ) ≤ Ckf k

, (3.5)

where C > 0 is a universal constant, not depending on f.

(When p = ∞ or q = ∞, then the usual modifications are needed.) Proof. For short, denote R

(f ) = 

1 L^mdf

P

ξ∈V^(m)

|c

|



, and e

(f ) = L

R

(f ) , for m = 0, 1, 2, ...

In this notation, one has S

(f ) = k(e

(f ))k

.

Part 1. Assume that f ∈ (Λ

)

(K), i.e. k(L

δ

(f ))k

< ∞ . We will show that

R

(f ) ≤ Cδ

(f ), (3.6)

with C > 0 not depending on m, f. This inequality will result in e

(f ) ≤ L

δ

(f )

and consequently we will get

S

(f ) < Ckf k

< ∞.

To prove (3.6), take ξ ∈ V

, and assume that ∆ ∈ T

is such that ξ ∈ V

∩ ∆.

There is only one such ∆ (this is because (m − 1)-cells meets through points from V

by nesting, and ξ / ∈ V

). Let w = (i

, ..., i

) be the address of the cell

∆, i.e. ∆ = φ

(K), and let ˜ v

, ..., ˜ v

be the vertices of ∆. It follows that there exist k ∈ {1, ..., M }, l ∈ {1, ..., r} such that

ξ = φ

(φ

(˜ v

)).

To estimate |c

|, we use Lemma 3.2, or rather its Corollary 3.1.

We can write

c

= f (ξ) − S

f (ξ) = f (ξ) − (f (˜ v

)ψ

(ξ) + · · · + f (˜ v

)ψ

(ξ))

= ψ

(ξ)(f (ξ) − f (˜ v

)) + ... + ψ

(ξ)(f (ξ) − f (˜ v

)).

Jensen’s inequality give

|c

|

≤ ψ

(ξ)|f (ξ) − f (˜ v

)|

+ · · · + ψ

(ξ)|f (ξ) − f (˜ v

)|

.

If ξ and ˜ v

belong to a common m−cell, we do nothing and leave |f (ξ) − f (˜ v

)| as it is.

Otherwise, we can find a chain ξ = x

, x

, ..., x

= ˜ v

such that x

∈ V

and x

∼ x

, j = 1, ..., n. The length of this chain is less than or equal to r. In this case, we have, using the elementary inequality (y

+ · · · + y

)

≤ n

(y

+ · · · + y

):

|f (ξ) − f (˜ v

)|

≤ (|f (ξ) − f (x

)| + · · · + |f (x

) − f (˜ v

)|)

≤ n

(|f (ξ) − f (x

)|

+ · · · |f (x

) − f (˜ v

)|

)

≤ r

X

x,y∈∆,x^m∼y

|f (x) − f (y)|

,

and hence

|c

|

≤ r

X

x,y∈∆,x^m∼y

|f (x) − f (y)|

.

Since the number of points from V

in any ∆ ∈ T

is fixed, summing up we arrive

at X

ξ∈V^(m)∩∆

|c

|

≤ C X

x^m∼y,x,y∈∆

|f (x) − f (y)|

,

where the constant C does not depend on f nor on m. Because of the nesting, the constant C is the same for all m ≥ 1 and all (m − 1)−cells ∆. This way we obtain

X

ξ∈V^(m)

|c

|

≤ X

∆∈Tm−1

X

ξ∈V^(m)∩∆

|c

|

≤ X

∆∈Tm−1

C X

x^m∼y,x,y∈∆

|f (x) − f (y)|

≤ C X

x^m∼y

|f (x) − f (y)|

.

Therefore (3.6) is proven.

Part 2. We now assume that f ∈ L

(K) is such that S

(f ) < +∞.

From the Minkowski inequality and the definition of δ

we have:

δ

(f ) = δ



 X

i=0

X

ξ∈V⁽ⁱ⁾

c

ψ



 ≤ X

δ



 X

ξ∈V⁽ⁱ⁾

c

ψ





= X

i=0



 1

L

X

x^m∼y

X

ξ∈V⁽ⁱ⁾

c

(ψ

(x) − ψ

(y))      

p





1 p

≤ X

i=0



 1

L

(2r)

X

ξ∈V⁽ⁱ⁾

|c

|

X

x^m∼y

|ψ

(x) − ψ

(y)|





1/p

, (3.7)

where the last inequality follows from the fact that, given x ∼ y, there are at most 2r

nonvanishing terms in the sum P

ξ∈V⁽ⁱ⁾

c

(ψ

(x) − ψ

(y)), for every i ≤ m (see Lemma 3.1).

Next, from the scaling and symmetry,we get, when i ≤ m and ξ ∈ V

are given:

X

x^m∼y

|ψ

(x) − ψ

(y)|

≤ r X

x^m−i∼ y

|ψ

(x) − ψ

(y)|

, (3.8)

for some ¯ ξ ∈ V

. We will estimate the last sum using the assumption ψ

∈ (Λ

)

. Indeed, since

sup

n

L

δ

(ψ

) := B ≤ ∞, it follows that X

x∼yⁿ

|ψ

(x) − ψ

(y)|

≤ B

L

L

. (3.9) Inserting (3.8) and (3.9) into (3.7) we see that

δ

(f ) ≤ CL

X

i=0



 L

L

X

ξ∈V⁽ⁱ⁾

|c

|





1/p

. (3.10)

Therefore, X

L

(δ

(f ))

≤ C X

L

X

i=0

a

!

, where we have denoted a

= L

P

ξ∈V⁽ⁱ⁾

|c

|

. Since (α − β) < 0, we can implement the classical discrete Hardy inequality:

X

ρ

( X

i=0

x

)

≤ K(ρ, q) X

ρ

x

,

valid for ρ ∈ (0, 1), q > 0, x

> 0 (with the convention: left-hand side infinite implies right-hand side infinite). We obtain:

X

L

(δ

(f ))

≤ K X

L

a

= K X

L



 1

L

X

ξ∈V^(m)

|c

|





q/p

= K X

L

R

(f )

= K[S

(f )]

.

We are done. 

We would like to discuss the importance of the assumption ψ

∈ (Λ

)

(K).

When p = 2, the answer is complete: all m−harmonic functions belong to the domain of the Brownian-motion induced Dirichlet form on K, which is known (see [15]) to be Lip(

, 2, ∞)(K) = (Λ

)

(K), with β =

w−df)

. Also, we know that the spaces Lip(γ, 2, ∞)(K) are trivial for γ >

, only containing constant functions; therefore Theorem 3.2 is valid for for

< α <

w−df)

. Since on nested fractals one always

has d

< d

≤ d

+ 1 (see [1], Remark 8.5 and [4], Theorem 8.8), this interval is nonempty for any nested fractal K.

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Katarzyna Pietruska-Pa luba Institute of Mathematics

University of Warsaw, ul. Banacha 2, 02-097 Warsaw, Poland E-mail address: kpp@mimuw.edu.pl

Mats Bodin

Department of Ecology and Environmental Science Ume˚a University, SE-901 87 Ume˚a, Sweden

E-mail address: mats.bodin@emg.umu.se