<p>This paper deals with Lebesgue points and studies properties of the set of Lebesgue points for various classes of functions. We consider continuous functions, L¹ functions and Sobolev functions. In the case of uniformly continuous functions and Hölder continuous functions we develop a characterization in terms of Lebesgue points. For Sobolev functions we study the dimension of the set of non-Lebesgue points.</p>

This paper deals with Lebesgue points and studies properties of the set of Lebesgue points for various classes of functions. We consider continuous functions, L1 functions and Sobolev functions. In the case of uniformly continuous functions and Hölder continuous functions we develop a characterization in terms of Lebesgue points. For Sobolev functions we study the dimension of the set of non-Lebesgue points.

Abstract: This thesis explores the application of isoperimetric functions to gain weak functional inequalities involving Dirichlet forms. The connection between such weak functional inequalities and bounds on the convergence speed of the corresponding Markov semi-group is established. Three examples of discrete graphs and the corresponding Dirichlet forms are discussed.

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 assumptions 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 verification 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 Muckenhoupt A1 weight, and use this to construct a counterexample establishing the sharpness of the parameter in the above mentioned result from [6].

In this thesis we study radial weights on Rn. We study two radial weights with different exponent sets. We show that they are both 1-admissible by utilizing a previously shown sufficient condition, for radial weights to be 1-admissible, together with some results connecting exponent sets and Ap weights. Furthermore applying a similar method on a more general radial weight, we manage to improve the previously shown sufficient condition for radial weights to be 1-admissible. Finally we show for one of these two weights that even though it is 1-admissible, whether or not it belongs to some class Ap depends both on the value of p and on the dimension n. Additionally, both of these weights as well as another simple weight are, at least in some dimensions n, not A1 even though they are 1-admissible.

Kapaciteter kan vara till stor nytta, bland annat då partiella differentialekvationer ska lösas. Kapaciteter är dock i många fall väldigt svåra att beräkna exakt, speciellt i viktade rum. Vad som istället kan göras är att försöka uppskatta kapaciteterna, vilket för ringar runt en fix punkt kan utföras med hjälp av fyra olika exponentmängder, \underline{Q}_0, \underline{S}_0, \overline{S}_0 och \overline{Q}_0, som beskriver hur vikten beter sig i närheten av denna punkt och i viss mån ger rummets lokala dimension. För att kunna dra nytta av exponentmängderna är det bra att veta vilka kombinationer av dessa som kan förekomma. För att få fram nya kombinationer använder vi olika sätt att mäta volym av klot med varierande radier. Dessa mått är definierade genom olika vikter. Det har tidigare funnits ett fåtal exempel på hur olika kombinationer av exponentmängderna kan se ut. Variationerna består av hur avstånden är i förhållande till varandra och om ändpunkterna tillhör mängderna eller inte. I denna rapport har vi tagit fram ytterligare fem nya kombinationer av mängderna, bland annat en där \underline{Q}_0 är öppen. / Capacities can be of great benefit, for instance when solving partial differential equations. In most cases, capacities can be difficult to calculate exactly, in particular on weighted spaces. In these cases, it can be sufficient with an estimation of the capacity instead. For annuli around a given point, the estimation can be done using four exponent sets \underline{Q}_0, \underline{S}_0, \overline{S}_0 and \overline{Q}_0, which describe how the weight behaves in a neighbourhood of that point and in some sense define the local dimension of the space. To be able to use the exponent sets, it is useful to know which combinations of them can exist. For this we use various measures, which are a way to measure volumes of balls with varying radii in Rn. These measures are defined by different weights. Earlier, there existed a few examples giving different combinations of exponent sets. The variations consist in their relationship to each other and if their endpoints belong to the set or not. In this thesis we present five new combinations of the exponent sets, amongst them one where \underline{Q}_0 is open.

Neste trabalho consideramos um processo de Markov a tempo contínuo com espaço de estados finito em um meio aleatório, a saber, a dinâmica de Metropolis para o Modelo de Energia Aleatória Generalizado (GREM) com um número de níveis finito e discutimos o comportamento do seu tempo de chegada ao equilíbrio, o qual é dado pelo inverso da lacuna espectral de sua matriz de probabilidades de transição. No principal resultado desta tese provamos que o quociente entre o volume do sistema e o logaritmo do inverso da lacuna é quase sempre limitado, por cima, por uma função da temperatura, que também é a que descreve a energia livre do GREM sob o regime de temperaturas baixas. Como um estudo adicional, também é discutido um correspondente limitante inferior em um caso particular do GREM com 2 níveis. / In this work we consider a finite state continuous-time Markov process in a random environment, namely, the Metropolis dynamics for the Generalized Random Energy Model (GREM) with a finite number of levels, and we discuss the behavior of its reaching time to equilibrium which is given by inverse of the spectral gap of its transition probability matrix. On the main result of this thesis, we prove the division between the system volume and the logarithm of the inverse of the gap is almost surely upper bounded by a function of the temperature that it is also the function that describe the free energy of the GREM at low temperature. As an additional study, it is also discuss the corresponding limiting lower in a particular case of the 2-level GREM.

In this thesis we discuss possible ways to give quantitative measurement for two spaces not being quasi-isometric. From this quantitative point of view, we reconsider the definition of quasi-isometries and propose a notion of ''quasi-isometric distortion growth'' between two metric spaces. We revise our article [32] where an optimal upper-bound for Morse Lemma is given, together with the dual variant which we call Anti-Morse Lemma, and their applications.Next, we focus on lower bounds on quasi-isometric distortion growth for hyperbolic metric spaces. In this class, $L^p$-cohomology spaces provides useful quasi-isometry invariants and Poincaré constants of balls are their quantitative incarnation. We study how Poincaré constants are transported by quasi-isometries. For this, we introduce the notion of a cross-kernel. We calculate Poincaré constants for locally homogeneous metrics of the form $dt^2+\sum_ie^{2\mu_it}dx_i^2$, and give a lower bound on quasi-isometric distortion growth among such spaces.This allows us to give examples of different quasi-isometric distortion growths, including a sublinear one (logarithmic).

Neste trabalho consideramos um processo de Markov a tempo contínuo com espaço de estados finito em um meio aleatório, a saber, a dinâmica de Metropolis para o Modelo de Energia Aleatória Generalizado (GREM) com um número de níveis finito e discutimos o comportamento do seu tempo de chegada ao equilíbrio, o qual é dado pelo inverso da lacuna espectral de sua matriz de probabilidades de transição. No principal resultado desta tese provamos que o quociente entre o volume do sistema e o logaritmo do inverso da lacuna é quase sempre limitado, por cima, por uma função da temperatura, que também é a que descreve a energia livre do GREM sob o regime de temperaturas baixas. Como um estudo adicional, também é discutido um correspondente limitante inferior em um caso particular do GREM com 2 níveis. / In this work we consider a finite state continuous-time Markov process in a random environment, namely, the Metropolis dynamics for the Generalized Random Energy Model (GREM) with a finite number of levels, and we discuss the behavior of its reaching time to equilibrium which is given by inverse of the spectral gap of its transition probability matrix. On the main result of this thesis, we prove the division between the system volume and the logarithm of the inverse of the gap is almost surely upper bounded by a function of the temperature that it is also the function that describe the free energy of the GREM at low temperature. As an additional study, it is also discuss the corresponding limiting lower in a particular case of the 2-level GREM.

In the study of the weighted p-Laplace equation, it is often important to acquire good estimates of capacities. One useful tool for finding such estimates in metric spaces is exponent sets, which are sets describing the local dimensionality of the measure associated with the space. In this thesis, we limit ourselves to the weighted Rn space, where we investigate the relationship between exponent sets and Muckenhoupt Ap-weights - a certain class of well behaved functions. Additionally, we restrict our scope to radial weights, that is, weights w(x) that only depend on |x|. First, we determine conditions on α such that |x|α ∈ Ap(μ) for doubling measures μ on Rn. From those results, we develop weight exponent sets - a tool for making Ap-classifications of general radial weights, under certain conditions. Finally, we apply our techniques to the weight |x|α(log 1/|x|)β. We find that the weight belongs to Ap(μ) if α ∈ (-q, (p-1)q), where q = sup Q(μ) is a constant associated with the dimensionality of μ. The Ap-conditions in this thesis are found to be sharp.