On differentiation of integrals in Lebesgue spaces

TL;DR

This paper constructs specific bases in Lebesgue spaces on infinite-dimensional tori to classify differentiation properties across various $L^p$ spaces, extending classical results and providing a comprehensive framework for complete metric measure spaces.

Contribution

It introduces a novel basis of rectangles in infinite-dimensional tori that differentiates $L^p$ spaces based on $p$, and classifies all possible differentiation ranges in complete metric measure spaces.

Findings

Constructed bases differentiate $L^p$ spaces for $p o p_0$

Classified all differentiation ranges in complete spaces

Extended classical differentiation theorems to infinite dimensions

Abstract

We study the problem of differentiation of integrals for certain bases in the infinite-dimensional torus $\mathbb{T}^\omega$. In particular, for every $p_0 \in [1,\infty)$, we construct a basis $\mathcal{B}$ which differentiates $L^p(\mathbb{T}^\omega)$ if and only if $p \geq p_0$, thus reproving classical theorems of Hayes in $\mathbb{R}$. The main novelty is that our $\mathcal{B}$ is a Busemann--Feller basis consisting of rectangles (of arbitrarily large dimensions) with sides parallel to the coordinate axes. Our construction gives us the opportunity to classify all possible ranges of differentiation for general complete spaces. Namely, let $\mathcal{B}$ be a basis in a metric measure space $\mathcal{X}$. If $\mathcal{X}$ is complete, then the set $\{ p \in [1,\infty] : \mathcal{B} \text{ differentiates } L^p(\mathcal{X}) \}$ takes one of the six forms\[ \emptyset, \, \{\infty\}, \, [p_0,\infty], \, (p_0,\infty], \, [p_0,\infty), \, (p_0,\infty) \quad \text{for some} \quad p_0 \in [1,\infty). \] Conversely, for every $p_0 \in [1,\infty)$ and each of the six cases above, we construct a complete space $\mathcal{X}$ and a basis $\mathcal{B}$ illustrating the corresponding range of differentiation.