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Fine properties of Besov functions $B^r_{q,\infty}$ in metric spaces

Paz Hashash, Arkady Poliakovsky

Abstract

Let $X$ be a metric space and $μ$ an $s$-regular Ahlfors measure. Let $Y$ be a metric space. We prove that for Besov functions $u \in B^r_{q,\infty}(X,μ;Y)$, every point is a {\it general average Lebesgue point} of $u$ outside a $σ$-finite set with respect to the Hausdorff measure $\mathcal{H}^{s - rq}$. The proof is based on density-type estimates involving Hausdorff measure. In addition, we prove that for functions $u$ in the fractional Sobolev space $W^{r,q}(X,μ;Y)$, almost every point with respect to $\mathcal{H}^{s - rq}$ is an {\it average Lebesgue point} of $u$. Finally, if $Y$ is also complete, we prove that for $u \in B^r_{q,\infty}(X,μ;Y)$, almost every point is a {\it Lebesgue point} outside a set of Hausdorff dimension at most $s - rq$.

Fine properties of Besov functions $B^r_{q,\infty}$ in metric spaces

Abstract

Let be a metric space and an -regular Ahlfors measure. Let be a metric space. We prove that for Besov functions , every point is a {\it general average Lebesgue point} of outside a -finite set with respect to the Hausdorff measure . The proof is based on density-type estimates involving Hausdorff measure. In addition, we prove that for functions in the fractional Sobolev space , almost every point with respect to is an {\it average Lebesgue point} of . Finally, if is also complete, we prove that for , almost every point is a {\it Lebesgue point} outside a set of Hausdorff dimension at most .
Paper Structure (8 sections, 38 theorems, 238 equations)

This paper contains 8 sections, 38 theorems, 238 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Density-type lemmas involving Carathéodory's construction
  4. Fine properties of general functions with respect to the Carathéodory resulting measure $\psi$
  5. The set of points that are not general average Lebesgue points of functions in $B^r_{q,\infty}$ is $\sigma$-finite with respect to the Hausdorff measure $\mathcal{H}^{s-rq}$
  6. The exceptional set of non-average Lebesgue points of $W^{r,q}$-functions has zero $(s-rq)$-dimensional Hausdorff measure
  7. The logarithmic Hausdorff measure
  8. Hausdorff dimension of the set of Lebesgue points for $B^r_{q,\infty}$-functions

Key Result

Theorem 1.4

Let $X$ be a metric space, and let $\mu$ be an Ahlfors $s$-regular measure on $X$ for some $s\in (0,\infty)$. Let $0\leq r\leq 1$, $0<q<\infty$ be such that $rq\leq s$. Let $Y$ be a metric space. Let $u\in B^r_{q,\infty}(X,\mu;Y)$. Then, there exists a $\mathcal{H}^{s-rq}$$\sigma$-finite set $D\subs

Theorems & Definitions (83)

  • Definition 1.1: Lebesgue points, Average Lebesgue points, and general average Lebesgue points
  • Definition 1.2: Besov space $B^r_{q,\infty}(X,\mu;Y)$
  • Remark 1.1
  • Definition 1.3: Ahlfors $s$-regularity
  • Theorem 1.4: General average Lebesgue points of $B^r_{q,\infty}$
  • Definition 1.5: Fractional Sobolev space $W^{r,q}(X,\mu;Y)$
  • Theorem 1.6: Average Lebesgue points of $W^{r,q}$
  • Theorem 1.7: Lebesgue points of $B^r_{q,\infty}$
  • Remark 1.2
  • Lemma 2.1: Vitali covering lemma
  • ...and 73 more