Capacity estimates and Poincaré inequalities for the weighted bow-tie

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the weighted bow-tie

Department of Mathematics, Linköping University Andreas Christensen

LiTH-MAT-EX2017/07SE

Credits: 30 hp Level: A

Supervisor: Anders Björn,

Department of Mathematics, Linköping University Examiner: Jana Björn,

Department of Mathematics, Linköping University Linköping: June 2017

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Abstract

We give a short introduction to various concepts related to the theory of p-harmonic functions on Rn_{, and some modern generalizations of these concepts} to general metric spaces. The article Björn-Björn-Lehrbäck [6] serves as the starting point of our discussion. In [6], among other things, estimates of the variational capacity for thin annuli in metric spaces are given under the as-sumptions of a Poincaré inequality and an annular decay property. Most of the parameters in the various results of the article are proven to be sharp by counterexamples at the end of the article. The main result of this thesis is the verication of the sharpness of a parameter.

At the center of our discussion will be a concrete metric subspace of weighted Rn, namely the so-called weighted bow-tie, where the weight function is assumed to be radial. A similar space was used in [6] to verify the sharpness of several parameters. We show that under the assumption that the variational p-capacity is nonzero for any ball centered at the origin, the p-Poincaré inequality holds in Rn _{if and only if it holds on the corresponding bow-tie}

Finally, we consider a concrete weight function, show that it is a Mucken-houpt A1 weight, and use this to construct a counterexample establishing the sharpness of the parameter in the above mentioned result from [6].

Keywords:

Bow-tie, Capacity, Metric space, Muckenhoupt A1-weight, Poincaré in-equality, Upper gradient, Weight function.

URL for electronic version:

http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-138111

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Sammanfattning

Vi ger en introduktion till ett urval av begrepp relaterade till p-harmoniska funktioner på Rn_{, och moderna generaliseringar av dessa begrepp till} allmän-na metriska rum. Artikeln Björn-Björn-Lehrbäck [6] är utgångspunkten för vår diskussion. I [6] ges bland annat uppskattningar av variationskapaciteten för tunna ringar i metriska rum, förutsatt att rummet uppfyller en Poincaréolikhet och att måttet av ringar avtar på ett visst sätt. De esta parametrar i de olika resultat som ges i artikeln visas i slutet vara skarpa med hjälp av motexempel. Det huvudsakliga resultatet av det här arbetet är en veriering av skarpheten hos en parameter i artikeln.

Den diskussion vi för kretsar kring ett konkret metriskt delrum till viktade Rn, den så kallade ugan, där viktfunktionen antas vara radiell. Ett liknande rum användes i [6] för ett antal av de redan verierade parametrarna. Vi visar bland annat att en p-Poincaréolikhet är uppfylld Rn _{om och endast om den är} uppfylld i motsvarande delrum detta slag.

Slutligen betraktar vi en konkret viktfunktion och visar att den är en Muc-kenhoupt A1-vikt. Vi använder sedan detta för att konstruera ett motexempel som visar skarpheten hos parametern i det ovan nämnda resultatet från [6]. Nyckelord:

Kapacitet, Metriskt rum, Muckenhoupt A1-vikt, Poincaréolikhet, Vikt-funktion, Övre gradient.

URL för elektronisk version:

http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-138111

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Acknowledgements

First, I would like to thank my supervisor, Anders Björn, for introducing me to an exciting subject and for showing me how to approach a problem I would otherwise not even have known where to begin solving. Throughout the project you have provided great help whenever needed and shown patience with me.

I would also like to thank my examiner, Jana Björn, for carefully scrutinizing the text and helping me nd, and remedy, everything from grammatical errors to more serious problems in the mathematical arguments still lingering in the later versions of the text.

My opponent, Olle Abrahamsson, also has my gratitude for reading through the nal draft and providing valuable suggestions. I thank all of you for having taken time out of your already busy lives to help me improve my work.

Finally, I would like to thank my family and friends for encouraging words, and for putting up with the slightly asocial behavior sometimes required of a person studying mathematics.

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Notation

Below is a list of often used symbols for quick reference. Note that most of these concepts will be more thoroughly dened in the rst three chapters of the thesis.

X A metric space.

B An open ball of unspecied center and radius. B(x, r) The open ball centered at x with radius r. Br The open ball centered at 0 ∈ Rn with radius r. Sn−1 _{The unit sphere in R}n_.

intA The interior of the set A. ∂A The boundary of the set A. A The closure of the set A. µ A general measure.

mn The n-dimensional Lebesgue measure.

Ef dµ The mean value integral of f on E, i.e. Ef dµ = 1 µ(E) Ef dµ. w A weight function. γ A curve. Γ A family of curves.

. a . b if there is C > 0 such that a ≤ Cb. & a & b if b . a.

' a ' b if a . b and a & b.

Conventions

We will use the terms increasing and decreasing in the strict sense, writing nondecreasing and nonincreasing respectively for the nonstrict counterparts. We employ the symbol ⊂ for nonstrict set inclusion and write ( for strict inclusion.

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Introduction

1.1 Background

The main subject of this thesis is the concept of capacity, which has a very natural interpretation in electromagnetic eld theory. As the reader probably knows, a capacitor in physics is, in its simplest form, two conductors separated by some simple dielectric medium. When a voltage is applied between the conductors, they become oppositely charged.

Consider the problem of determining the electric eld E = ∇u, where u(x) is the electric potential at a point x between two ideal conductors that are kept at dierent potentials, and separated by a simple medium (meaning that the charge density between the conductors vanishes). As the conductors are assumed to be ideal, u is constant there. Suppose that the potential at the conductors is assumed to be known, and that it is zero at the outer conductor and unity at the inner conductor. Let K and Ω be 3-dimensional, connected, open subsets of R3 _{such that K ⊂ Ω. The capacitor is modeled as the pair of 2-dimensional} surfaces ∂K and ∂Ω (the inner and outer conductors, respectively).

It follows from one of Maxwell's equations for electrostatics in a simple medium, ∇ · E = 0, that ∆u = ∇ · ∇u = 0 everywhere in Ω \ K. Hence the problem of nding the eld E therein reduces to nding the unique1 har-monic function u such that u = 0 on ∂Ω and u = 1 on ∂K. It can be shown2 that this is equivalent to minimizing the integral

1 2

Ω

|∇u|2_dx _(1.1)

1_{A proof of the uniqueness can be found in Cheng [10, pp. 157-159]} 2_{See Uppman [21, pp. 1-2] for a proof.}

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over all u ∈ C2_{(Ω) ∩ C(R}3₎_{such that u = 1 in K and u = 0 in R}3_{\ Ω}_.

From a physics perspective, (1.1) is the capacity of the capacitor, and mea-sures the energy stored in it. From this point of view, the integral should be viewed as a limit of a sum where the sum is taken over all individual electrons that move from one plate to the other at discharge, and the quantity summed is the charge of an electron multiplied by the voltage drop between the plates at the start (see Cheng [10, pp. 133-138] for the details). The factor 1

2 can intuitively be explained by the fact that when the capacitor is discharged and the energy is released by connecting a load between the conductors, the po-tential dierence (the voltage) decreases while more and more charges move from one plate to the other. Since the voltage depends linearly on the charge dierence, the quantity to be summed decreases linearly from the starting value to 0, averaging of course half the starting value.

The factor 1

2 in (1.1), which does not play an important role in the mathe-matical context, is omitted in the more general denition of capacity below, see (1.2).

1.2 p-harmonic functions

From a mathematics perspective, minimizing the integral in (1.1) yields the so-called variational 2-capacity of the condenser (K, Ω). More generally, the variational p-capacity of a condenser in Rn _{is given by}

capp(K, Ω) = inf

Ω

|∇u|pdx, (1.2)

where the inmum is taken over all u ∈ C∞_{(Ω) ∩ C(R}n₎_{such that u = 1 on K} and u = 0 on Rn_{\ Ω}_{. The variational integral on the right-hand side of (1.2)} has a corresponding Euler-Lagrange equation,

∆pu = ∇ · |∇u|p−2∇u = 0, p > 1, (1.3) which is known as the p-Laplace equation. Solutions to this equation are called p-harmonic. In the case p = 2 this is the ordinary Laplace equation and we can assume that the solution is smooth, but for p 6= 2 there is no such guarantee and we must consider the directional derivatives (the components of the gradient) to be weak derivatives, to allow for nonsmooth functions to be solutions. The space of functions where one looks for solutions is the so-called Sobolev space, which we dene later.

The Sobolev space makes use of directional (weak) derivatives in its deni-tion. As we have seen, so do the denitions of p-harmonic functions and the

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variational capacity. To be able to dene these concepts on more general spaces where directions are not even dened, we must therefore dene some quantity that can replace the gradient. This is done by noting that in (1.2), the direc-tion of the gradient is never considered, only its magnitude. It turns out that one can replace the modulus of the gradient by a positive function which does not require directions to be dened, and still coincides with the modulus of the gradient when the gradient is dened. This is done with the help of upper gra-dients, and it is the key to extending the analysis of p-harmonic functions from Rn to general metric spaces.

1.3 Capacity estimates

Calculating the variational p-capacity of a condenser is in general a dicult task. The capacities of some special cases of condensers such as annuli in unweighted Rnhave however been calculated, see Heinonen-Kilpeläinen-Martio [14, pp. 35-36].

In more general spaces, there is no explicit formula for the capacity of an annulus. Estimates have however been made for various cases, and recently for annuli in the general metric space setting in Björn-Björn-Lehrbäck [6], [7], under some assumptions on the space. Most of these assumptions have been proven necessary, in the same article. In this thesis we aim to verify the sharpness of one parameter by a counterexample.

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Chapter 2

Preliminaries

In this chapter we present some fundamental denitions and results from the theory of metric spaces, functional analysis and integration theory. Naturally, these subjects contain more topics than is feasible to include here, and we focus on introducing and dening concepts necessary to follow the main part of the thesis.

2.1 Metric spaces

In ordinary calculus, functions on Rn _{are considered. We have a natural way} of dening the distance between any points x, y ∈ Rn _{by what we can call the} euclidean distance function, d, given by d(x, y) = |x−y|. Some of the denitions and theorems in this thesis are however valid in a more general setting, namely so-called metric spaces, of which a standard denition is given below. Note that the nature of the elements of X is left entirely unspecied.

Denition 2.1. A metric space is a pair (X, d) where X is a set and d is called a metric, dened on X × X. The metric satises the following conditions for all x, y, z ∈ X:

• d(x, y) is real-valued, nite and nonnegative. • d(x, y) = 0if and only if x = y.

• d(x, y) = d(y, x).

• d(x, y) ≤ d(x, z) + d(z, y).

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When talking about X we will take it to mean the corresponding metric space (i.e. we assume X = (X, d)), unless we specically state that we are only talking about the set X.

We will let B denote an open ball in a metric space. When we want to be more specic, we will let B(x, r) denote the open ball centered at the point x ∈ X with radius r, i.e.

B(x, r) = {y ∈ X : d(x, y) < r}.

It is common to write Brfor a ball with radius r when the center point is known or unimportant. In this thesis, we will always take Br to mean the ball in Rn with radius r centered at the origin, i.e. Br= B(0, r) ⊂ Rn.

When considering Rn_{in relation to some space X that is created from some} subset of Rn_{, which we will do a lot, we will write B for balls in R}n _{and, if} there is risk for confusion, B ∩ X for the part of the (same) ball that intersects X. Note that B ∩ X is only a ball in X, by denition, if the center of the ball lies in X. A ball B(x, ε) is often called an ε-neighborhood of x.

Denition 2.2. A set A ⊂ X is open if for every x ∈ A there exists an ε > 0 such that B(x, ε) ⊂ A.

We remind the reader that the complement of a set A ⊂ X, written Ac_, consists of all the points that do not belong to the set, i.e. Ac_{= {x ∈ X : x /}_{∈ A}}_. Denition 2.3. A set A ⊂ X is closed if Ac _{is open.}

We will also need the notion of connectedness in later arguments. A nonempty set that is open and connected, is referred to as a domain.

Denition 2.4. A metric space X is connected if there do not exist two open, disjoint, nonempty subsets of X such that X is the union of these sets. A subset of X is connected if it, viewed as a metric subspace of X, is connected.

If there is a way of ordering the elements of a set A, and there exists a smallest element x and/or a largest element y we write x = min A and/or y = max A. Note however that for an arbitrary set, there do not always exist such elements, even if there is some way to order the elements. This is for example easily seen for an open interval (a, b) = {x ∈ R : a < x < b}, by contradiction. Recall that for any subset A of the real line that is bounded from above or below, we can however always nd a least upper bound or greatest lower bound, respectively1_. These bounds are called the supremum and inmum respectively, of the set A.

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Denition 2.5. The inmum of a set A ⊂ R, written inf A has the following properties

1. inf A ≤ x for every x ∈ A.

2. If l ≤ x for every x ∈ A, then inf A ≥ l.

The supremum of a set A ⊂ R, written sup A has the following properties 1. sup A ≥ x for every x ∈ A.

2. If u ≥ x for every x ∈ A, then sup A ≤ u.

We also write inf A = −∞ if A is not bounded from below and sup A = ∞ if Ais not bounded from above. Obviously, for the empty set a lower bound can be chosen arbitrarily large and an upper bound arbitrarily small. It therefore makes sense, and is sometimes useful, to employ the convention that inf ∅ = ∞ and sup ∅ = −∞.

Using the inmum, one denes the distance between points and sets in the following way.

Denition 2.6. Let X be a metric space. The distance between a point x ∈ X and a set A ⊂ X is given by

dist(x, A) = inf

a∈Ad(x, a).

In some cases, it is useful to specify an upper bound of the metric on the space X.

Denition 2.7. The diameter of a metric space X = (X, d) is dened by diam(X) = sup

x,y∈X d(x, y).

2.1.1 R

n

_{and the bow-tie}

Central to this thesis will be the following subsets of Rn_: X+= {(x1, x2, ... , xn) : xj≥ 0, j = 1, ... , n},

X−= {(x1, x2, ... , xn) : xj ≤ 0, j = 1, ... , n} and

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Figure 2.1. X = X+∪ X− as a subset of R2.

The set X is referred to as the bow-tie, because of its resemblance in the case n = 2. Later on, we will equip these sets with a metric and a measure function µ, yielding a measure space. When considering these spaces, it is understood that they depend on n. Whenever X, X− or X+ are treated in relation to Rn, they are assumed to be of the same dimension n. Furthermore, Rn _{and subsets} thereof, are always assumed to be equipped with the euclidean metric.

A sphere in Rn_{centered at 0 with radius ρ will, being an n − 1 dimensional} surface, be denoted by Sn−1_(ρ)_{. That is,}

Sn−1(ρ) = {x ∈ Rn: |x| = ρ}. The unit sphere will be denoted by Sn−1_{, i.e.}

Sn−1= Sn−1(1).

2.1.2 Completeness

To dene completeness of metric spaces, we need the following basic concepts. Denition 2.8. A sequence (xn) in a metric space X is convergent, if there exists an x ∈ X, such that for every ε > 0 there exists an N = N(ε) satisfying d(x, xn) < εwhenever n > N. We write xn→ x.

Denition 2.9. A sequence (xn) in a metric space X is Cauchy if there for every ε > 0 exists an N = N(ε) such that d(xn, xm) < εwhenever m, n > N.

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The fact that a convergent sequence is always Cauchy is easily seen, since xn→ ximplies that for any ε > 0 there exists an N such that

d(xi, x) < ε 2 whenever i > N. Hence d(xn, xm) ≤ d(xn, x) + d(xm, x) < ε whenever n, m > N.

The converse is not true in general. Spaces for which it is true are called complete.

Denition 2.10. A space X is complete if every Cauchy sequence in X is convergent in X.

It is a well known fact that R is complete2_{, and from this fact one obtains} the following for all n ∈ N.

Theorem 2.11. Rn _{is complete.} Proof. See Kreyszig [17, p. 33]

Lemma 2.12. Any closed subspace Y of a complete metric space X is complete. Proof. Let (xn) be an arbitrary Cauchy sequence in Y . Since X is complete, this implies that xn → x ∈ X. Assume that x 6∈ Y . Since Yc is open, this implies that there exists an ε-neighborhood Vε of x such that Vε ⊂ Yc. Since xn→ x, this in turn implies that there exists an N such that xn ∈ Vε⊂ Yc for all n > N, which of course contradicts the fact that (xn) is a sequence in Y . Thus it must be the case that x ∈ Y , and since (xn)was an arbitrary Cauchy sequence in Y , the statement follows.

Corollary 2.13. The spaces X+, X− and X of Section 2.1.1 are complete. Proof. The spaces are all closed subspaces of Rn_{, which is perhaps most easily} seen by the fact that their complements (in Rn_{) are open; for any of these} sets, a point in the complement always has an ε-neighborhood contained in the complement. Since Rn _{is complete by Theorem 2.11, Lemma 2.12 yields the} result.

2_{For the proof of the completeness of R and which assumptions are needed, and further}

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2.2 Mappings on R

n

_{and general metric spaces}

In this section we briey discuss mappings on metric spaces, and we will follow the notation of Kreyszig [17], which is standard, and write T x for the image of xwith respect to the mapping T .

2.2.1 Continuous mappings

Continuous mappings are the natural generalization of continuous functions to metric spaces. They are dened as follows.

Denition 2.14. Let X and Y be metric spaces equipped with metrics dXand dY respectively. The mapping T : X → Y is continuous at x ∈ X if for every ε > 0there exists a δ > 0 such that dY(T x, T y) < εwhenever dX(x, y) < δ.

Among the continuous mappings are Lipschitz mappings, which are obtained by imposing a stronger type of continuity.

Denition 2.15. A mapping T : (X, dX) → (Y, dY)is L-Lipschitz, L ≥ 0, if for every a, b ∈ X,

dY(T (a), T (b)) ≤ LdX(a, b). (2.1) If L ≤ 1, then T is a contraction. If L = 1 and we have equality in (2.1) for all a, b ∈ X, then T is isometric. If the constant L is of no importance, then the mapping is typically referred to simply as a Lipschitz mapping or Lipschitz function.

Let us look at some mappings which will be of interest later, and let them serve as an illustrative example of these concepts.

Example 2.16. Let X+ be as in Section 2.1.1 and dene T : Rn → X+ by T (x1, ... , xn) = (|x1|, ... , |xn|). Let x, y ∈ Rn be arbitrary. Then

dX+(T x, T y) = p

(|x1| − |y1|)2+ ... + (|xn| − |yn|)2 ≤p(x1− y1)2+ ... + (xn− yn)2 = dRn(x, y).

Thus, we see that T is 1-Lipschitz and hence a contraction. It is however not isometric (consider for example the points x = (−1, −1) and y = (1, 1) in R2_). Now letXe be some octant of Rn. That is, let

e

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for some (j1, ... , jn) ∈ J = {(j1, ... , jn) : ji∈ {−1, 1}, i = 1, ... , n}. Dene T : Xe + → eX by T (xe 1, ... , xn) = (j1x1, ... , jnxn). Take x, y ∈ X+ arbitrarily. Then d e X( eT x, eT y) = p (j1(x1− y1))2+ ... + (jn(xn− yn))2 =p(x1− y1)2+ ... + (xn− yn)2 = dX+(x, y). Thus,Teis isometric.

The denition of continuous functions and a rather short argument yields the following useful fact.

Theorem 2.17. Let X and Y be metric spaces. A mapping T : X → Y is continuous if and only if T−1_{(Ω) ⊂ Y} _{is open for every open Ω ⊂ X.}

Proof. For a proof, see Kreyszig [17, pp. 20-21].

Mappings dened on vector spaces are referred to as operators. A linear operator is, of course, an operator that is linear. Linear operators from Rn _to Rn_{may be represented by n×n matrices, and one can use this fact to explicitly} show that for any two points a, b ∈ Rn_{with kak = kbk, there exists an isometric} mapping R : Rn _{→ R}n _{such that Ra = b. We carry out an explicit verication} of this to some perhaps rather trivial fact. This distraction is however relegated to Appendix A.

2.3 Measures and integration

Although we aim to be rigorous, we will, as in previous sections of this chapter, settle with dening concepts necessary for the further development of the text. Readers interested in an extensive treatment of more general measure and in-tegration theory are referred to a text covering the fundamentals of the subject in detail, for example Folland [12] or Bass [4].

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2.3.1 Measures

Denition 2.18. Let X 6= ∅ be a set. If M is a nonempty family of subsets of X which has the following properties:

M1, M2, ... ∈ M =⇒ ∞ [ i=1 Mi∈ M, M ∈ M =⇒ Mc∈ M, then M is called a σ-algebra of sets in X.

Denition 2.19. Let (X, d) be a metric space. The smallest σ-algebra of sets in X containing every open subset of X is called the Borel σ-algebra, and is denoted by BX. The members of BX are called the Borel sets of X.

Denition 2.20. A measurable space is a pair (X, M) where M is a σ-algebra on the set X. The members of M are called measurable sets.

Denition 2.21. Let M be a σ-algebra. A function µ : M → [0, ∞] such that µ(∅) = 0and M1, M2, ... ∈ M =⇒ µ ∞ [ i=1 Mi = ∞ X i=1 µ (Mi)

whenever the sets M1, M2, ...are pairwise disjoint, is called a measure on X. Denition 2.22. A measure space is a triplet (X, M, µ) where M is a σ-algebra of sets in X and µ is a measure dened on M.

Denition 2.23. A complete measure is a measure with the property that for every set A such that there is a measurable set B with A ⊂ B and µ(B) = 0, A is also measurable.

Denition 2.24. A property is said to hold almost everywhere, abbreviated a.e., if it holds everywhere except for a set of measure zero.

From a measure space, one can construct a complete measure space. In Sjödin [20], it is done in the following way.

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Denition 2.25. Let (X, M, µ) be a measure space. The completion of (X, M, µ) is the triplet (X, M, µ), where

M = {A ⊂ X : there are E, F ∈ M such that E ⊂ A, A \ E ⊂ F, µ(F ) = 0}, (2.2) and µ is dened by

µ(A) = µ(E) for every A ∈ M where E is as in (2.2).

Proposition 2.26. The completion (X, M, µ) of a measure space (X, M, µ) is in fact a measure space, and the measure µ is complete with µ(A) = µ(A) for all A ∈ M.

Proof. See Appendix C.

We can now start to dene the Borel regular measures, which are also the only type of measures we will be using in this thesis.

Denition 2.27. A measure µ : BX→ [0, ∞], is called a positive Borel measure on X.

Denition 2.28. The completion µ of a positive Borel measure µ is said to be Borel regular on X if for every A ⊂ X there exists B ∈ BX such that A ⊂ B and µ(A) = µ(B).

A measure space equipped with a metric is a metric measure space, and henceforth we will always assume that a space X is equipped with some metric and a positive Borel regular measure. A special case of a Borel regular measure is the Lebesgue measure mn, dened on Rn. Dening the Lebesgue measure is done by the construction of a so-called outer measure, and the process is some-what involved. We will simply state the following, and refer readers interested in the proof of these claims to, for example, Bass [4, Chapter 4].

Remark 2.29. There is a Borel regular measure dened on Rn _{called the} Lebesgue measure which we denote by mn. For any rectangle R in Rn, i.e. a set for which there exist real numbers ai≤ bi such that

R = {x ∈ Rn: ai ≤ xi≤ bi for all i = 1, 2, ... , n}, mn has the property that

mn(R) = N Y

i=1

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In certain proofs in a metric measure space setting, some additional assump-tions regarding the properties of the measure at hand are needed. We conclude this subsection with the denition of one such property.

Denition 2.30. The measure µ is doubling if there is a common constant C > 0for all x ∈ X and r > 0, such that

µ(B(x, 2r)) ≤ Cµ(B(x, r)).

2.3.2 Integration

When integrating in Rn _{or subsets thereof, we will use the Lebesgue integral.} Since the Lebesgue measure is only dened on Rn_{, a more general Borel regular} measure will be denoted by µ in the general metric space setting. In the special case of Rn_{, which will after all be of most interest, we will typically consider a} so-called weighted measure, i.e. a measure given by dµ = w(x) dx where w is a weight function and dx denotes integration with respect to (the n-dimensional) Lebesgue measure.

Denition 2.31. Let (X, M) be a measurable space. A function f : X → [−∞, ∞]is said to be a measurable function if for all a ∈ R,

f−1 (α, +∞] ∈ M.

Example 2.32. Let X = (X, BX) and f : X → R be continuous. Since (α, +∞] is open for any a ∈ R and f is continuous, f−1((α, +∞]) is open by Theorem 2.17. Thus, f−1 _{(α, +∞] ∈ B}

X, and hence f is measurable.

There are denitions for the inmum and supremeum of measurable real valued functions, analogous to the ones for sets on the real line. Since we will use it later, we give the denition of the essential inmum. The essential supremum is dened in a similar way.

Denition 2.33. Let f : X → R be measurable. The essential inmum of f on a set E ⊂ X is given by

ess inf

E f = sup{a ∈ R : µ({x ∈ E : f (x) < a}) = 0}.

Later on, we will consider Borel functions, which are dened as follows. Denition 2.34. A function f : X → R is called a Borel function if f−1_(E)_is a Borel set for every open E ⊂ R.

Proposition 2.35. For every measurable f : X → [−∞, ∞] there exists a Borel function g such that f = g a.e. on X.

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Proof. See Färm [11, p. 14] for a proof.

Now, let us take a look at how integration can be dened using so called simple functions.

Denition 2.36. The characteristic function of a set A is the function χA(x) =

(

1, x ∈ A, 0, x /∈ A.

Denition 2.37. A function s dened on a measurable space is a simple func-tion if there is an n ∈ N, measurable sets A1, A2, ... , An and real numbers α1, α2, ... , αn such that s = n X i=1 αiχAi.

Denition 2.38. The integral of a nonnegative simple function as given in Denition 2.37 on a measurable set E ⊂ X is given by

E s dµ = n X i=1 αiµ(E ∩ Ai). In particular, we have µ(E) = E dµ.

Denition 2.39. The integral of a nonnegative measurable function f on a measurable set E ⊂ X is given by

E f dµ = sup E s dµ

where the supremum is taken over all simple functions s such that 0 ≤ s ≤ f. The integral of any measurable function f can easily be dened by decom-posing f into its positive and negative parts f = f+− f−, both of which are easily checked to be measurable, yielding

E f dµ = E f+dµ − E f−dµ.

Note that it is enough that at least one of the two integrals on the right-hand side above is nite for the integral on the left-hand side to be well dened with a value in [−∞, ∞]. In order for the integral to exist as a real value however, both must be nite, exposing one of very few drawbacks with replacing the Riemann integral with the Lebesgue integral.

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Denition 2.40. A measurable function is said to be integrable on a measurable

set E ⊂ X if

E

|f | dµ < ∞.

On the other hand, the Lebesgue integral behaves in many ways like the Riemann integral, for example if the Riemann integral of a bounded function on a bounded interval exists, then so does the Lebesgue integral of that function with the same value. There is also an equivalent form of the fundamental theorem of calculus, valid for the Lebesgue integral.

2.3.3 Function spaces

Denition 2.41. For any number p ≥ 13_{and measurable E ⊂ X (in particular,} we may have E = X), we dene the space of p-integrable functions as

Lp(E; µ) = f : f is measurable and E |f |p_{dµ < ∞} . This is a normed vector space, with the norm

kf k_Lp_(E;µ)= E |f |p_dµ 1/p .

Denition 2.42. For any number p ≥ 1 and measurable E ⊂ Rn_{, the space of} locally p-integrable functions, Lp

loc(E), is the set of measurable functions that, for every x ∈ E, are p-integrable on E ∩ B(x, r) for some r > 0.

Denition 2.43. The support of f, denoted by supp f, is the complement of the largest open set where f = 0. A function has bounded support if supp f is a bounded set.

Denition 2.44. For any open E ⊂ Rn _{we dene C}∞ 0 (E)by

C₀∞(E) = {f : f has bounded support in E and f ∈ C∞(E)} . Denition 2.45. Let E ⊂ Rn _{be open. The function g ∈ L}1

loc(E) is a weak derivative of f ∈ L1

loc(E)with respect to xi in E, written g = Dif, if E φ(x)g(x) dx = − E f (x)∂φ(x) ∂xi dx for all φ ∈ C∞ 0 (E).

3_Lp_{-spaces are sometimes dened for 0 < p < 1 as well. They are however not normed}

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We dene weak gradients analogously to the usual gradient, i.e. ∇f = (D1f, D2f, ... , Dnf ).

Now we may dene the Sobolev space, that we mentioned in the introduction. Denition 2.46. Let E ⊂ Rn _{be open. The Sobolev space W}1,p_(E)_{is given} by W1,p(E) =nf : f _L_p_(E)+ |∇f | _L_p_(E)< ∞ o .

2.3.4 Integrating radial functions

To nish this section we present a few easy results that will be useful when integrating functions over balls in Rn_{, with respect to the Lebesgue measure} (or some radial weighted measure).

Proposition 2.47. Suppose f is a locally integrable radial function on Rn_. That is, suppose f(x) = g(|x|) a.e. for some g ∈ L1

loc([0, +∞)). Then, Br f (x) dx = mn−1(Sn−1) r 0 g(ρ)ρn−1dρ for any r > 0.

Proof. Fix r > 0. Note that on one hand we have mn(Br) =

|x|≤r

dx. The coordinate transformation x = ru with Jacobian

∂(x1, x2, ... , xn) ∂(u1, u2, ... , un) = det      r 0 ... 0 0 r ... 0 ... ... ... ... 0 0 ... r      = rn yields mn(Br) = |u|≤1 rndu = mn(B1)rn. (2.3) On the other hand,

mn(Br) = r

0

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Dierentiating the right-hand sides of (2.3) and (2.4) and equating, we have nmn(B1)rn−1= mn−1(Sn−1(r)), (2.5) from which it is clear that mn−1(Sn−1(ρ)) = mn−1(Sn−1)ρn−1. Using this, and the fact that f does not vary on a sphere centered at the origin (in reverse order), we get Br f (x) dx = r 0 g(ρ)mn−1(Sn−1(ρ)) dρ = mn−1(Sn−1) r 0 g(ρ)ρn−1dρ, completing the proof.

In order to avoid having to introduce an auxiliary one-variable function g as in Proposition 2.47 above we will abuse notation slightly. Since a radial function only depends on |x|, we will write it as a one-variable function even though strictly speaking it, being dened on an n-dimensional space, is a function of n variables. To further simplify notation, we will let ωn−1 denote the Lebesgue measure of the unit sphere in Rn_{. It is not hard to see that the proof of} Proposition 2.47 works just as well for the bow-tie or X+ as for Rn. Thus, we have the following corollary.

Corollary 2.48. Suppose f is a radial and locally integrable function on Rn and let X and X+ be as in Section 2.1.1. Then, for any r > 0,

Br f (|x|) dx = ωn−1 r 0 f (ρ)ρn−1dρ, (2.6) Br∩X f (|x|) dx = 21−nωn−1 r 0 f (ρ)ρn−1dρ, (2.7) Br∩X+ f (|x|) dx = 2−nωn−1 r 0 f (ρ)ρn−1dρ. (2.8) If the function to be integrated is allowed not to be radial, the last step of the proof of Proposition 2.47 does not work. We can however obtain a similar result by dening the polar coordinates ρ and θ by

ρ = |x|, θ = x |x|

for x 6= 0, as in Folland [12, p. 78]. Then there exists a surface measure σ(θ) on the unit sphere, and we have the following result.

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Proposition 2.49. Suppose f is a locally integrable function on X, where X is any octant, the bow-tie from Section 2.1.1, or the whole of Rn_{. Then there} exists a Borel measure σ(θ) on Sn−1_{, such that}

Br∩X f dx = r 0 Sn−1_∩X f (ρθ)ρn−1dσ(θ) dρ for any r > 0.

Proof. See Folland [12, Theorem 2.49] for the case X = Rn_{, and note that the} same proof also works for the other cases if Sn−1 _{is replaced by S}n−1_{∩ X}_.

2.4 Curves

Denition 2.50. Let X be a metric space. A curve in X is a continuous mapping

γ : I → X,

where I is an interval. If the curve has nite length (see Denition 2.52), it is rectiable.

In this thesis, we will only be interested in rectiable curves dened on compact intervals.

Denition 2.51. A partition P of a closed interval [a, b] is a nite ordered set of points from [a, b] including the points a and b. That is, P may be written on the form P = {t0, ... , tn}were

a = t0< t1< ... < tn = b.

We denote the collection of all possible partitions of the interval [a, b] by P[a,b]. Denition 2.52. We dene the length of a curve γ : [a, b] → X as

length(γ) = sup P ∈P[a,b]

X

ti∈P \{t0}

d(γ(ti), γ(ti−1)). (2.9)

For convenience, we will sometimes write lγ instead of length(γ).

Remark 2.53. We may modify P[a,b] when determining the length of γ : [a, b] → X. Suppose we have some partition P ∈ P[a,b]. If we add a (new) point z from the interval [a, b] between two consecutive points x, y in P , the corresponding sum in (2.9) will be unchanged or larger, since by the denition

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of a metric d(γ(x), γ(y)) ≤ d(γ(x), γ(z)) + d(γ(z), γ(y)). Further, if we add a nite amount of points to a partition it is still nite, hence we may disre-gard the partition we started out with and replace it by the modied one, since we are interested in the supremum of the sum (2.9). Thus, if we add a nite amount of points to each partition of P[a,b] by some rule, yielding a new collec-tionP ⊂ Pe [a,b], we may consider only the partitions inPe when determining the length of γ.

For a curve γ dened on an interval I = [a, b], a subcurve is given by the restriction γ|I0 where I0 ⊂ I is an interval. We dene the length function s_γ of the curve γ by

sγ(t) =length(γ|[a,t]) (2.10) Our aim is now to nd a parametrization that may be used for all curves, so that no ambiguity arises when integrating along curves that dier from each other only by their parametrization. We will use the same denition as in Heinonen et al. [16, p. 124], and dene the arc length parametrization γs: [0,length(γ)] → X of γ : [a, b] → X by

γs(t) = γ(s−1γ (t)), (2.11) where the inverse is given by

s−1_γ (t) := sup{s : sγ(s) = t} = max{s : sγ(s) = t}. (2.12) Note that the one sided inverse is used, since the length function may not be in-creasing, and thus may not be invertible in the ordinary sense. The last equality in 2.12 follows by continuity. Furthermore, it is shown in [16, Proposition 5.1.8] that γsis 1-Lipschitz. Now, we may dene line integration in the following way. Denition 2.54. Let γ : [a, b] → X be a curve and g : X → [0, ∞] be a Borel function. Then, the line integral of g over γ is given by

γ g ds = length(γ) 0 g(γs(t)) dt. (2.13)

We note, as in [16], that since γs is continuous g ◦ γs is a Borel function on [0,length(γ)] and thus the integral is well dened. Hence, integration along a curve reduces to an ordinary one-dimensional Lebesgue integral, whose proper-ties (such as linearity) the line integral thereby inherits. This will be helpful later on when we prove statements about upper gradients, as will the following lemmas.

Lemma 2.55. Let X and Y be metric spaces. Assume that g : Y → [0, ∞] is a Borel function, γ : [a, b] → X is a curve and that the mapping T : X → Y is

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continuous. Dene ˆg by ˆg(x) = g(T x) and eγ by eγ = T γ. Assume further that sγ = s_eγ and length(γ) = length(eγ). Then,

γ ˆ g ds = e γ g ds.

Proof. Since T is continuous, ˆg is a Borel function on X, so the integral is well dened. Furthermore, the assumptions yield

γ ˆ g ds = lγ 0 ˆ g(γs(t)) dt = lγ 0 ˆ g(γ(s−1_γ (t)) dt = lγe 0 g(_eγ(s−1 e γ (t)) dt = lγe 0 g(_eγs(t)) dt = e γ g ds.

Lemma 2.56. Let γ : [0, lγ] → X be 1-Lipschitz and let g : [0, lγ] → X be a Borel function. Then

γ g ds ≤ lγ 0 g(γ(t)) dt. Proof. See [16, Lemma 5.1.14].

Lemma 2.57. Let X and Y be metric spaces. Assume that g : Y → [0, ∞] is a Borel function, γ : [0, lγ] → X is an arc length parameterized curve and that the mapping T : X → Y is a contraction. Dene ˆg by ˆg(x) = g(T x) and e γ : [0, lγ] → Y by eγ = T γ. Then, γ ˆ g ds ≥ e γ g ds.

Proof. Note rst that γ = γs as γ is arc length parameterized. Since T is a contraction, we have

dY(γ(a),e eγ(b)) = dY(T γs(a), T γs(b)) ≤ dX(γs(a), γs(b)) ≤ |a − b| for any a, b ∈ [0, lγ]. Thus,γ : [0, l_e γ] → Y is 1-Lipschitz, and hence Lemma 2.56 yields e γ g ds ≤ lγ 0 g(γ(t)) dt ≤_e lγ 0 (g ◦ T γs)(t) dt = lγ 0 ˆ g(γs(t)) dt = γ ˆ g ds.

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2.4.1 Curve modulus

In later chapters we will study concepts that are dened in terms of curves. It is then sometimes possible to exclude certain curves from consideration, if it can be shown that they are insignicant in determining the quantity at hand. We nish o the chapter with a brief look at curve modulus, which in a sense measures the signicance of a family of curves.

Denition 2.58. Let Γ be a family of curves in a metric space X. The p-modulus of Γ is dened by Modp(Γ) = inf g X gpdµ, (2.14)

where the inmum is taken over all positive Borel functions such that_γg ds ≥ 1 for all γ ∈ Γ.

It will be helpful to establish notation that distinguishes dierent curve families, by what set of points in the space the curves in a family intersects. Let Γ(X) denote the family of all curves in a metric space X. Then, we write

ΓE= {γ ∈ Γ(X) : γ−1(E) 6= ∅}. (2.15) We will later on be interested in knowing when the p-modulus of all curves passing through the origin in weighted Rn_{is zero, for a specic weight function} w. To determine this we will use the following fact.

Lemma 2.59. Let p ≥ 1, n > 1 and dene g(x) = ( ₁ d(x,0), when d(x, 0) < ε, 0, elsewhere, (2.16) for ε > 0. Then Modp(Γ{0}) ≤ X gpdµ (2.17)

in Rn _{equipped with any measure µ.} Proof. See Färm [11, Example 3.3.4].

Färm [11, Example 3.3.4] goes on to show that the p-modulus, p < n, of the family of curves passing through the origin in unweighted Rn_{, n > 1, is zero.} In a similar way, we get the following result.

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Lemma 2.60. Let p ≥ 1, n > 1 and w(x) = |x|α_φ(x)β_{, where φ(x) =} max{1, − log |x|}. Suppose that α + n > p and β ∈ R or α + n = p and β < −1. Then

Modp(Γ{0}) = 0 (2.18)

in (Rn_{, d, µ)}_{, where dµ = w(x) dx.}

Proof. Since w and gεgiven by (2.16) are radial, we have Rn gp_εdµ = Bε 1 d(x, 0)pw(x) dx = ωn−1 ε 0 ρα+n−p−1φ(ρ)βdρ (2.19) by Proposition 2.47. Thus by Lemma 2.59 and Propostion B.1 we get

Modp(Γ{0}) .      εα+n−p_φ(ε)β_, _{α + n > p,} _{b ∈ R,} φ(ε)β+1, α + n = p, β < −1, ∞, _else.

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Chapter 3

p

-harmonic functions on

metric spaces

In this chapter, we generalize some concepts related to p-harmonic functions to general metric spaces, by introducing the upper gradient.

3.1 Upper gradients

Upper gradients were rst dened in Heinonen-Koskela [15], and there they were referred to as very weak gradients.

Denition 3.1. A Borel function g : X → [0, ∞] is an upper gradient of a function f : X → [−∞, ∞] if for all nonconstant rectiable curves γ : [0, lγ] → X,

|f (γ(0)) − f (γ(lγ))| ≤

γ g ds.

Remark 3.2. We follow the convention of Björn-Björn [5], that ∞ − ∞ = ∞ and (−∞)−(−∞) = −∞, which implies thatγg ds = ∞whenever f is innite in any of the endpoints of the curve.

Lemma 3.3. Assume that g1 and g2 are upper gradients of u1 and u2 respec-tively. Then g := g1+ g2 is an upper gradient of u := u1+ u2.

Proof. This follows from the linearity of the Lebesgue integral and the triangle

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inequality: γ g ds = γ g1ds + γ g2ds ≥ |u1(γ(0)) − u1(γ(lγ))| + |u2(γ(0)) − u2(γ(lγ))| ≥ |u1(γ(0)) + u2(γ(0)) − u1(γ(lγ)) − u2(γ(lγ))| = |u(γ(0)) − u(γ(lγ))|.

3.2 Newtonian spaces

With our denition of upper gradients, we can now dene the Sobolev spaces of Rn _{on an arbitrary metric space X. This is done by dening the Newtonian} space on X, which was rst done in the articles Shanmugalingam [18], [19]. Denition 3.4. The Newtonian space on X is

N1,p(X; µ) = {f : X → [−∞, ∞] : ||f ||N1,p_(x)< ∞}, where ||f ||N1,p_(X;µ)= X |f |p_{dµ + inf} g X gpdµ 1/p . Remark 3.5. The space N1,p

loc(X; µ) is dened analogously with the space of locally p-integrable functions, i.e. f ∈ N1,p

loc(X; µ)if there for every x ∈ X exists r > 0such that f ∈ N1,p_{(B(x, r; µ)).}

3.3 Poincaré inequalities

Denition 3.6. A space X supports a (q, p)-Poincaré inequality if there exist constants C > 0 and λ ≥ 1 such that for all balls B ⊂ X, all f ∈ L1_(X)_{, and} all upper gradients g of f,

B |f − fB|qdµ 1/q ≤ Cr λB gpdµ 1/p , (3.1) where fB:=

Bf dµ. If X supports a (1, p)-Poincaré inequality, we simply say that X supports a p-Poincaré inequality.

We call a space that supports a p-Poincaré inequality a p-Poincaré space. If furthermore the measure is doubling, we call the space a doubling p-Poincaré space.

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Remark 3.7. Let q > 1 and p be such that 1 p +

1

q = 1. It then follows from Hölder's inequality that

B |f − fB| dµ ≤ 1 µ(B) B dµ 1/p B |f − fB|qdµ 1/q = µ(B)p1−1 B |f − fB|qdµ 1/q = B |f − fB|qdµ 1/q ,

showing that X supports a p-Poincaré inequality if it supports a (q, p)-Poincaré inequality for any q ≥ 1. Similarly, if X supports a p-Poincaré inequality, then it supports a ˜p-Poincaré inequality for all ˜p ≥ p

Denition 3.8. When considering Rn_{equipped with a radial weighted measure} µgiven by dµ(x) = w(x) dx, we say that the weight function w is p-admissible1 if the measure is globally doubling and the space supports a global p-Poincaré inequality.

The following lemma will be very useful later on, when proving statements related to Poincaré inequalities.

Lemma 3.9. If f ∈ L1_(X)_{, 1 ≤ q < ∞, a ∈ R and E is a measurable set with} positive measure, then

E |f − fE|qdµ 1/q ≤ 2 E |f − a|q_dµ 1/q , Proof. See Björn-Björn [5, Lemma 4.17].

In the next section we will introduce a class of weight functions that are p-admissible. First, we will however nish o this section by briey discussing quasiconvexity, quoting the denition from Björn-Björn [5], and its relation to the Poincaré inequality.

Denition 3.10. A metric space X is L-quasiconvex, L ≥ 1, if for all x, y ∈ X there is a (possibly constant) curve γ : [0, lγ] → Xsuch that γ(0) = x, γ(lγ) = y and lγ ≤ Ld(x, y). A metric space is quasiconvex if it is L-quasiconvex for some L, and geodesic if it is 1-quasiconvex.

1_{While the original denition of p-admissibility included additional criteria, all but the two}

mentioned in this denition have been proven to be superuous. In Heinonen-Kilpeläinen-Martio [14, Chapter 20] proofs are presented, along with references to their origins.

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Example 3.11. Rn _{is geodesic, since for every two points x, y ∈ R}n _{there is a} curve γ : [0, lγ] → Rn between them (the straight line) with lγ = d(x, y). X− and X+ are also geodesic, for the same reason.

Example 3.12. The space X = X−∪ X+ is √

2-quasiconvex. Consider any two points x, y ∈ X. If both points lie in either X− or X+, we have a curve γ : [0, lγ] → X between them (the straight line) with lγ = d(x, y). Assume therefore that x ∈ X− and y ∈ X+. Then there is no straight line connecting them contained in X in general. They can however of course be connected by the curve consisting of two straight lines meeting at the origin. Let lx and ly denote the length of the respective lines from x and y to the origin. Together with the straight line between x and y (which is only contained in X if x and y, viewed as position vectors in Rn_{, are parallell) with length d = d(x, y) they} form a triangle.

Now let θ ∈ [0, π] denote the angle between the lines at the origin. By the denition of the angle between two vectors we have

|x||y| cos θ = x · y.

Further, since x ∈ X− and y ∈ X+ we have x · y ≤ 0, so it must be the case that θ ∈ [π/2, π]. Furthermore, we have

d2= l2y+ lx2− 2lxlycos θ. (3.2) Suppose we know the lengths lx and ly, and that they are arbitrary. We now want to nd the angle θ that requires the largest L in the L-quasiconvexity condition, i.e. we want to maximize

lγ d(x, y) =

lx+ ly

d . (3.3)

with respect to the angle θ. But clearly, given that we already know about the lengths lxand ly, we only need to minimize d(θ) for θ ∈ [π/2, π]. From (3.2), it is clear then that θ = π/2. Now the lengths lx and ly were arbitrary, hence we may assume that the triangle is right-angled at the origin.

Now, we want to nd the relation between lx and ly that maximizes (3.3). Since we may assume a right angle at the origin, this is equivalent to maximizing

lx+ ly q

l2 y+ l2x

. (3.4)

Let lx be arbitrary (and xed) and consider the function f : [0, ∞) → [0, ∞) given by f (ly) = lx+ ly q l2 y+ l2x .

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Dierentiating yields df dly = q l2 y+ lx2− (lx+ ly) ly √ l2 y+l2x l2 y+ l2x .

When checking the sign, we only need to consider the numerator since the denominator is always positive. Note that when ly< lxwe have

(lx+ ly) ly q l2 y+ l2x < l 2 y+ l2x q l2 y+ l2x =ql2 y+ l2x,

and hence f is increasing on [0, lx]. Similarly, for ly> lxwe have (lx+ ly) ly q l2 y+ l2x > l 2 y+ l2x q l2 y+ l2x =ql2 y+ l2x,

and hence f is decreasing on [lx, ∞). Thus, by continuity, f attains its maximum when ly= lx, and we have

max ly,θ lγ d(x, y) = maxly f (ly) = lx+ lx pl2 x+ lx2 =√2. But this is obviously independent of lx, which was arbitrary, thus

L = max lx,ly,θ lγ d(x, y) = √ 2.

Although not essential when constructing proofs, the main reason why we are interested in quasiconvexity is the following fact.

Proposition 3.13. Assume that µ is doubling and that X is L-quasiconvex and satises a p-Poincaré inequality with some dilation constant. Then X supports a p-Poincaré inequality with dilation constant L.

Proof. See Björn-Björn [5, Corollary 4.40].

3.3.1 Muckenhoupt A

p

-weights

We dene the Muckenhoupt classes A1, Ap and A∞ as follows. Denition 3.14. Let 0 < w ∈ L1

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(i) w is in the Muckenhoupt class Ap if there exists a constant C > 0 such that sup B⊂Rn B w(x) dx B w(x)1/(1−p)dx p−1 ≤ C.

(ii) w is in the Muckenhoupt class A1 if there exists a constant C > 0 such that for all balls B ⊂ Rn_,

B

w(x) dx ≤ C ess inf

B w.

(iii) The Muckenhoupt class A∞is given by A∞=

[

p>1 Ap.

We call a function w ∈ Ap an Ap-weight. If q ≥ p, then it follows that Ap⊂ Aq, see Heinonen-Kipeläinen-Martio [14, Chapter 15]. For Ap-weights, we also have the following, very useful, theorem.

Theorem 3.15. Let p ≥ 1 and w be an Ap-weight. Then w is p-admissible. Proof. See Heinonen-Kilpeläinen-Martio [14, Theorem 15.21] for the case p > 1 and Björn [8, Theorem 4] for the case p = 1.

3.4 Capacity

Denition 3.16. The Sobolev p-capacity of an arbitrary set E ⊂ X is Cp(E) = inf

u kuk p

N1,p(X;µ),

where the inmum is taken over all u ∈ N1,p_{(X; µ)}_{such that u ≥ 1 on E.} The following two results will be useful later on. In fact, using them will be key tricks when we construct the counterexample we mentioned in the intro-duction.

Lemma 3.17. Let X be a p-Poincaré space and assume that Ω ⊂ X is open and connected. If F ⊂ Ω is relatively closed with Cp(F ) = 0, then Ω \ F is connected.

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Proof. See Björn-Björn [5, Lemma 4.6].

Proposition 3.18. Let X be a metric space and E ⊂ X. Then Cp(E) = 0 if and only if µ(E) = Modp(ΓE) = 0.

3.4.1 Variational capacity

Denition 3.19. Let Ω ⊂ X be open. The variational p-capacity of E ⊂ Ω with respect to Ω is

capp(E, Ω) = inf_u

Ω gpdµ,

where the inmum is taken over all u ∈ N1,p_(X) _{such that u = 1 in E and} u = 0on X \ Ω, and all upper gradients g of u.

Remark 3.20. If there is risk for confusion, we will add a superscript to Cp and capp to clarify with respect to which space the capacity is intended, i.e. we write CX

p and capXp for the capacities in X.

In [5, Chapter 6] several properties of the variational p-capacity are given. For instance it is shown that, without any assumptions on the space, if E1 ⊂ E2⊂ Ω, then capp(E1, Ω) ≤capp(E2, Ω). It is furthermore shown that if X is a complete doubling p-Poincaré space, then capp is an outer capacity, in the sense that for any compact E ⊂ Ω it holds that

capp(E, Ω) = inf Gopen E⊂G⊂Ω

capp(G, Ω).

Remark 3.21. The above mentioned prop-erties will be useful later, when we will cal-culate the capacity of a single point x0 with respect to some bounded set Ω in a com-plete doubling p-Poincaré space X. Since any open G containing x0also contains some ε-neighborhood around x0 (that is, it con-tains an open ball Bε = B(x0, ε) for some ε > 0) and capp(Bε, Ω) ≤ capp(G, Ω), we see that

capp({x0}, Ω) = inf B(x0,r)⊂Ω

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When the space considered is constructed from a subset of Rn _equipped with a radial measure, one may consider only radial functions with correspond-ing (radial) upper gradients when determincorrespond-ing the variational p-capacity of an annulus centered at the origin. We clarify the details in a lemma, for later reference.

Lemma 3.22. Let X be Rn_{, an octant, or the bow-tie. Furthermore, let 0 <} w ∈ L1

loc(X; µ) be a radial weight on X ⊂ Rn, n ≥ 2, and 0 < r < R. Then, the variational p-capacity of Br∩ X with respect to BR∩ X is given by

capX p(Br, BR) = inf u BR gpdµ,

where dµ = w dx, and the inmum is taken over all radial functions u ∈ N1,p_{(X; µ)} _{such that u = 1 in B}

r∩ X and u = 0 on X \ (BR ∩ X), and all upper gradients g of u.

Proof. The whole statement follows from Denition 3.19 except for the assertion that we may consider only radial functions. To see this, let u ∈ N1,p_{(X; µ)}_be arbitrary and let g be any upper gradient of u. Then, by Proposition 2.49 and the Fubini-Tonelli theorem (see for example Folland [12, Theorem 2.37]),

BR∩X gpdµ = Sn−1_∩X R 0 gp(ρθ)w(ρθ)ρn−1dρ dσ(θ). Now write V (θ) = R 0 gp(ρθ)w(ρθ)ρn−1dρ

and suppose V0is the mean value of V (θ) for θ ∈ Sn−1∩ X. Then, there exists ˆ

θ ∈ Sn−1_{∩ X} _{such that V (ˆθ) ≤ V}

0. Consider the mapping T : X → X dened pointwise by (ρ, θ) 7→ (ρ, ˆθ). This mapping is easily seen to be a contraction since for any two points in X, their images under T obviously lie closer to each other than the points themselves do to each other, see Figure 3.1.

Now let γ : [0, lγ] → Xbe an arbitrary curve, and dene the radial functions ˆ

uand ˆg by ˆu(ρ) = u(ρ, ˆθ) and ˆg(ρ) = g(ρ, ˆθ), and the curve_eγ : [0, lγ] → X by e

γ = T γ. Then, using Lemma 2.57 we have γ ˆ g ds ≥ e γ

g ds ≥ |u(_eγ(0)) − u(_eγ(lγ))| = |ˆu(γ(0)) − ˆu(γ(lγ))|,

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Figure 3.1. The mapping T maps the points a = (ρ1, θ1)and b = (ρ2, θ2)to the points T a = (ρ1, ˆθ)and T b = (ρ2, ˆθ), which lie on the same radial ray from the origin. we have BR ˆ gpdµ = Sn−1 V (ˆθ) dσ(θ) ≤ Sn−1 V0dσ(θ) = Sn−1 V (θ) dσ(θ) = BR gpdµ. (3.5) Since u ≤ 1 in BRand u = 0 on X\BR, the same is true for û. This together with (3.5) makes it obvious that û ∈ N1,p_{(X; µ)}_{, and hence for every u ∈ N}1,p_{(X; µ)} and upper gradient g of u, there exists a radial function û ∈ N1,p_{(X; µ)} _with radial upper gradient ˆg such that

BR∩X ˆ gpdµ ≤ BR∩X gpdµ, establishing the truth of the assertion.

The following results related to the variational capacity will be useful later on.

Proposition 3.23. Let 0 < w ∈ L1

loc(Rn) be a radial weight on Rn, n ≥ 2. Assume that the corresponding measure dµ = w dx supports a p-Poincaré inequality for some p > 1. Let f(r) = µ(Br), where Br= B(0, r) ⊂ Rn. Then

capp(Br, BR) = R r (f0)1/(1−p)dρ 1−p .

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Proof. See Björn-Björn-Lehrbäck [7, Proposition 10.8].

Lemma 3.24. Let X be a metric space and Ω ⊂ X be open and bounded. Assume that E ⊂ Ω and Cp(E) = 0. Then capp(E, Ω) = 0.

Proof. See Björn-Björn [5, Lemma 6.14]. The converse needs additional assumptions.

Lemma 3.25. Let X be a doubling p-Poincaré space and Ω ⊂ X be open and bounded. Assume that E ⊂ Ω and Cp(X \ Ω) > 0. Then capp(E, Ω) = 0 implies that Cp(E) = 0.

Theorem 3.26. Assume that X supports a p-Poincaré inequality. For u ∈ N_loc1,p(X; µ), let S = {x ∈ X : u(x) = 0}. Then for all balls B = B(x0, r),

2B |u|p dµ . _cap 1 p(B ∩ S, 2B) 2λB g_updµ, where λ is the dilation constant in the Poincaré inequality.

Proof. This is Maz0_{ya's inequality, see Björn-Björn [5, Theorem 6.21].}

3.4.2 Capacity estimates

Capacity estimates are central to our discussion, and here we present some results for annuli from the article Björn-Björn-Lehrbäck [6]. We will later use Lemma 3.27 to show that the 1-Poincaré inequality in Theorem 3.29 cannot be replaced by a q-Poincaré inequality for all q > 1.

Lemma 3.27. If 0 < r < R, then capp(Br, BR) ≤

µ(BR\ Br) (R − r)p . Proof. See Björn-Björn-Lehrbäck [6, Lemma 3.1].

In some cases, such as in the proof of Theorem 3.29, it is necessary to require that more restrictive conditions for the measure considered are met, than the doubling property.

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Denition 3.28. The measure µ has the 1-annular decay (1-AD) property at x ∈ X, if there is a constant C such that for all radii 0 < r < R we have

µ (B(x, R) \ B(x, r)) ≤ C1 − r R

µ(B(x, R)).

If there is a common constant C such that the above holds for all x ∈ X and radii 0 < r < R, then µ has the global 1-AD property.

The following theorem is the main result of the article Björn-Björn-Lehrbäck [6], and the aim of this thesis is to verify the sharpness of the requirement that X supports a 1-Poincaré inequality. This is done in Example 6.13.

Theorem 3.29. Assume that X supports a 1-Poincaré inequality and that µ has the global 1-AD property. Then

capp(B(x, r), B(x, R)) ' 1 − r R 1−pµ(B(x, R)) Rp (3.6) for 0 < R

2 ≤ r < R ≤ diam X3 , where the implicit constant is independent of x. Proof. See Björn-Björn-Lehrbäck [6, Theorem 1.3].

Remark 3.30. Note that the versions of the doubling property and Poincaré inequality that we employ, are commonly referred to as global. Since we never will deal with pointwise versions of these concepts, we omit this adjective. We will however need to distinguish the pointwise annular decay property from its global counterpart.

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Chapter 4

Properties of the weighted

bow-tie

In this chapter, we consider X, X−and X+as dened in Section 2.1.1, equipped with the Euclidean metric and a weighted measure µ given by dµ = w dx, where w is radial. The space X is, as briey mentioned in the introduction, referred to as the weighted bow-tie.

4.1 Preliminary discussion

In order to make the proofs in upcoming sections somewhat easier to overview, we will make some preliminary observations that we will later refer to. We begin by noting that any octant in Rn _{is bounded by a set of (n − 1)-dimensional} planes. Clearly, there are n such planes, and a ball B in Rn _{may intersect any} combination of these planes. To make calculations easier, we will often seek to nd some (dilation) constant λ ≥ 1, such that there exists some ball Be that contains B and is symmetrical with respect to all planes (bounding octants) that it intersects, while at the same time contained in λB. Too see that we can do this, we rst prove the following fact.

Lemma 4.1. Dene the planes

σj= {(x1, ... , xn) : xj = 0}, j = 1, ... , n, (4.1) and let B be a ball in Rn _{that intersects every plane in some family}

Σm= {σjk : k ∈ {1, ... , m}}

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of m ≤ n such planes. Then there exists a constant C = C(n) and a ball e

Bm centered at some point on the intersection of the planes in Σm, such that B ⊂ eBm⊂ CB.

Proof. The indices are just labels, so we may assume without loss of generality that

Σm= {σj: j ∈ {1, ... , m}}.

Since B intersects all planes in Σm, we can pick m points x(1), x(2), ... , x(m)∈ B that can be written in the form

x(1)= (0, ξ₂(1), ξ₃(1), ... , ξ_m(1), ξ_m+1(1) , ... , ξ(1)_n ), x(2)= (ξ₁(2), 0, ξ₃(2), ... , ξ_m(2), ξ_m+1(2) , ... , ξ(2)_n ),

...

x(m)= (ξ₁(m), ξ(m)₂ ... , ξ(m)_m−1, 0, ξ_m+1(m) , ... , ξ(m)n ),

i.e. all points lie on dierent planes in Σm. Denote the radius of B by r. Since all the points lie in B, it must for all i ∈ {2, 3, ... , m} be the case that

2r > d(x(1), x(i)) ≥ |ξ(1)_i |,

since the i-th element of x(i) _{is zero. Thus, the m rst elements of x}(1) _have modulus less than 2r. Now let

x0= (0, 0, ... , 0, ξ (1) m+1, ξ (1) m+2, ... , ξ (1) n ).

Clearly, x0 is at the intersection of all planes in Σm. Furthermore, d(x(1), x0) = q (ξ(1)₂ )2_{+ (ξ}(1) 3 )2+ ... + (ξ (1) m )2< 2 √ nr,

and since any point x ∈ B lies at a distance less than 2r from x(1)_{, we obtain} d(x, x0) ≤ d(x, x(1)) + d(x(1), x0) ≤ 2

√

nr + 2r < 4√nr, so that B ⊂Bem where Bem = B(x0, 4

√

nr). Finally, since the center of B lies in Bem, any point inBemis at a distance less than 8

√

nr from the center of B. Thus, with C = 8√nwe have B ⊂Bem⊂ CB.

It may be the case that the larger ballBemintersects some, say l, additional planes σj, not contained in Σm, thus destroying the symmetry. In that case, CB of course also intersects those additional planes, and maybe even say k additional

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ones on top of that, and we then use Lemma 4.1 again on the ball CB to conclude that there exists a ballBem+l+k centered at some point in the intersection of all planes in some family Σm+l+k, with CB ⊂Bem+l+k ⊂ C2B. We can continue in this manner, constructing larger balls until we get a symmetric situation. In the extreme case, the ball B only intersects one plane and we do not get symmetry until all planes have been included in Σ, and in each step only one new plane is intersected by the ballBethat is symmetric about the intersecting planes. Thus, we can nd λ ≤ Cn _{in the following corollary, which we have now proven.} Corollary 4.2. For any ball B in Rn _{there exists a constant λ = λ(n) and a} ball Be that is symmetric with respect to all the planes bounding octants that it intersects, such that B ⊂B ⊂ λB.e

4.2 Upper gradients

Lemma 4.3. Let u be dened on X+ and dene ˆu by ˆ

u(x1, ... , xn) = u(|x1|, ... , |xn|). If g is an upper gradient of u, then ˆg given by

ˆ

g(x1, ... , xn) = g(|x1|, ... , |xn|) is an upper gradient of ˆu.

Proof. Let γ : [a, b] → Rn _{be a rectiable curve, i.e. γ(t) = (x}

1(t), ... , xn(t)). We can modify every partition in P[a,b], by individually adding nitely many points (possibly none) from the interval [a, b] to the partitions of P[a,b], yielding some new collectionPe. Since each partition inPe is still nite, it is clear that

e

P ⊂ P[a,b]. Also, when writing down the curve length, we may consider only partitions ofPe by the reasoning in remark 2.53.

When constructing Pe, our aim is to add points to each partition P ∈ P[a,b] in such a way that for any P = {te 0, t1, ... , tk} ∈ eP and ti ∈ eP, the nonzero components of γ(ti) = (x1(ti), ... , xn(ti))and γ(ti−1) = (x1(ti−1), ... , xn(ti−1)) have the same sign (i.e. points on the curve corresponding to two consecutive elements of the partition lie in the same octant; we consider points lying on a plane bounding an octant as belonging to all octants the plane boarders at that point, since we only care about the sign of nonzero elements).

Thus, to each partition P ∈ P[a,b], we want to add some nite amount of points from the interval corresponding to when the curve is intersecting planes bounding octants. However, we cannot add all such points, since the amount

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of intersections of this type may be innite (even uncountable), so we need to be selective. Let P ∈ P[a,b] be arbitrary and consider any ti ∈ P \ {t0} such that γ(ti−1)and γ(ti) lie in dierent octants. As mentioned, there may be an innite amount of points where γ(τ) crosses over from one octant to another as τ increases between ti−1 and ti. Surely, however, there must be some last point τ(1)_{on (t}

i−1, ti], corresponding to a point on a plane bordering the octant γ(ti−1)resides in, after which γ does not return to said octant or else γ would never get to the point γ(ti). Now, it may be that γ(τ(1))does not lie in the same octant as γ(ti). By the same reasoning as before however, as τ increases γ(τ) must in that case at some point τ(2) _{leave this octant (for good, on (τ}(1)_{, t}

i]) as well. Continuing in this way, in the extreme case, we need to visit every octant before every point remaining on the curve on this interval lies in the same octant as γ(ti), thus adding 2n− 1 points before reaching γ(ti). Hence, in total, we only need to add a nite amount of points, since we (at most) need to add 2n₋₁ points between every two consecutive points in the nite partition P in order to construct a partitionPewith the desired properties.

Now we apply this modication to all partitions of P[a,b]to constructPe. As established, when determining the length of γ|[a,b] we may take the supremum over the partitions inPe. That is, we may write

length(γ|[a,b]) = sup k X

i=1

d(γ(ti), γ(ti−1)),

where the supremum is taken over all sequences of points a = t0 ≤ t1 ≤ ... ≤ tk = b such that for all i = 1, 2, ... , k the nonzero components of γ(ti) = (x1(ti), ... , xn(ti)) and γ(ti−1) = (x1(ti−1), ... , xn(ti−1)) have the same sign. Now dene eγ : [a, b] → X+ by eγ(t) = (|x1(t)|, ... , |xn(t)|). The curveγe is the result of rotations and/or reections of each line segment of γ to the rst octant, see Figure 4.1 for an illustration of the situation in R2 _{. Since γ and}

e

γ are de-ned on the same interval [a, b], we may take the supremum over all partitions in Pe when calculating their respective lengths. It thus follows that they have the same length, because for anyP ∈ ee P and ti∈ eP \ {t0}we have

d(γ(ti), γ(ti−1)) = p

(x1(ti) − x1(ti−1))2+ ... + (xn(ti) − xn(ti−1))2 =p(|x1(ti)| − |x1(ti−1)|)2+ ... + (|xn(ti)| − |xn(ti−1)|)2 = d(_eγ(ti),γ(te i−1)),

since all nonzero xk(ti)and xk(ti−1), k ∈ {1, 2, ... , n}, have the same sign. It follows that sγ = s_eγ and length(γ) = length(eγ). Finally, using Lemma 2.55 with

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Figure 4.1. Some curve γ : [0, lγ] → R2 (left) and the corresponding curve ˜γ (right). T : Rn_{→ X} + given by T (x1, ... , xn) = (|x1|, ... , |xn|), we have γ ˆ g ds = e γ

g ds ≥ |u(_eγ(0)) − u(_eγ(lγ))| = |ˆu(γ(0)) − ˆu(γ(lγ))|.

Lemma 4.4. Let J = {(j1, ... , jn) : ji∈ {−1, 1}, i = 1, 2, ... , n}and let u be de-ned on X+. Dene û by û(x1, ... , xn) = u(j1x1, ... , jnxn)for some (j1, ... , jn) ∈ J. If g is an upper gradient of u, then ˆg given by ˆg(x1, ... , xn) = g(j1x1, ... , jnxn) is an upper gradient of û.

Proof. Let γ : [a, b] → Rn _{be an arbitrary (rectiable) curve, i.e. γ(t) =} (x1(t), ... , xn(t)). Dene _eγ by γ(t) = (j_e 1x1(t), ... , jnxn(t)) on the same inter-val. For any partition P of [a, b] and k ∈ {1, 2, ... , n} we have, for ti∈ P \ {t0}, that (xk(ti) − xk(ti−1))2= (jkxk(ti) − jkxk(ti−1))2, implying that γ and_eγhave the same length. It follows that sγ = sγ_e and length(γ) = length(eγ). Using Lemma 2.55 with T given by T (x1, ... , xn) = (j1x1, ... , jnxn), we have

γ ˆ g ds = e γ

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Lemma 4.5. Suppose g is an upper gradient of u. Let J = {(j1, ... , jn) : ji ∈ {−1, 1}, i = 1, 2, ... , n}and ˆu and ˆg be given by

ˆ u(x1, ... , xn) = X J u(j1x1, ... , jnxn) and ˆ g(x1, ... , xn) = X J g(j1x1, ... , jnxn) on X+. Then ˆg is an upper gradient of ˆu.

Proof. This follows immediately from repeated use of Lemmas 3.3 and 4.4. Lemma 4.6. Suppose g is an upper gradient of u. Let R : Rn _{→ R}n _{be an} isometry, and dene û by û(x) = u(Rx) and ˆg by ˆg(x) = g(Rx). Then ˆg is an upper gradient of û.

Proof. Let γ : [a, b] → Rn _{be an arbitrary (rectiable) curve, and let} e γ = Rγ, dened on the same interval. From the distance preserving property of an isometry it is obvious that γ and eγ have the same length. Using Lemma 2.55 with T given by T = R, we have

γ ˆ g ds = e γ

g ds ≥ |u(_eγ(0)) − u(_eγ(lγ))| = |ˆu(γ(0)) − ˆu(γ(lγ))|.

4.3 Poincaré inequalities and variational capacity

We are now equipped to prove some relations between the bow-tie and Rn_{, with} regards to Poincaré inequalities and capacity. Remember that we assume that µis a radial weighted measure. In this entire section, we also assume that µ is doubling.

In the proofs in this section, we will often consider functions u : X → R where X consists of an octant in Rn_{, or a combination of octants (such as the} bow tie or the whole of Rn_{). We will then typically dene a function ˆu : Y → R} by ˆu(x) = u(T x) where T : Y → X and Y is some combination of octants in Rn. In all cases we will consider, it is straight forward to show that

Y |ˆu|pdµ 1/p ' X |u|p_dµ 1/p ,

where the implicit constant only depends on n and p, which implies that ˆu is p-integrable on Y if and only if u is p-p-integrable on X. For example, any function

Capacity estimates and Poincaré inequalities for the weighted bow-tie

the weighted bow-tie

Abstract

Sammanfattning

Acknowledgements

Notation

Conventions

Contents

Chapter 1

Introduction

1.1 Background

1.2 p-harmonic functions

1.3 Capacity estimates

Chapter 2

Preliminaries

2.1 Metric spaces

2.1.1 R

and the bow-tie

2.1.2 Completeness

2.2 Mappings on R

and general metric spaces

2.2.1 Continuous mappings

2.3 Measures and integration

2.3.1 Measures

2.3.2 Integration

2.3.3 Function spaces

2.3.4 Integrating radial functions

2.4 Curves

2.4.1 Curve modulus

Chapter 3

p

-harmonic functions on

metric spaces

3.1 Upper gradients

3.2 Newtonian spaces

3.3 Poincaré inequalities

3.3.1 Muckenhoupt A

-weights

3.4 Capacity

3.4.1 Variational capacity

3.4.2 Capacity estimates

Chapter 4

Properties of the weighted

bow-tie

4.1 Preliminary discussion

4.2 Upper gradients

4.3 Poincaré inequalities and variational capacity

_{and the bow-tie}

_{and general metric spaces}