On differentiation of integrals in Lebesgue spaces

TL;DR

This work studies differentiation of integrals in Lebesgue spaces on the infinite-dimensional torus $\mathbb{T}^\omega$ by constructing axis-aligned rectangle Busemann--Feller bases that realize sharp $L^p$ differentiation ranges. For a fixed $p_0$, it builds bases $\mathcal{B}_\geq$ and $\mathcal{B}_>$ so that $\mathcal{B}_\geq$ differentiates $L^p(\mathbb{T}^\omega)$ iff $p \ge p_0$ and $\mathcal{B}_>$ differentiates iff $p > p_0$, extending Hayes-type results to product spaces with unbounded rectangle dimensions. The paper proves a complete six-form classification for the possible ranges diff$(\mathcal{B})$ on complete spaces (and a four-form variant for spaces that are countable unions of finite-measure opens), and shows that every form arises via explicit constructions; it also links differentiation to weak-type maximal operators and discusses measurability issues and corollaries on the unit interval. By leveraging maximal-operator techniques and novel bases on $\mathbb{T}^\omega$, the authors provide a unified framework for the possible ranges of differentiation across arbitrary bases and metric-measure spaces.

Abstract

We study the problem of differentiation of integrals for certain bases in the infinite-dimensional torus $\mathbb{T}^ω$. In particular, for every $p_0 \in [1,\infty)$, we construct a basis $\mathcal{B}$ which differentiates $L^p(\mathbb{T}^ω)$ 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.

Figures (7)

• Figure 1: The space $\mathcal{X}$ from Example \ref{['E1']}.
• Figure 2: The space $\mathcal{X}$ from Example \ref{['E2']} and Example \ref{['E3']}.
• Figure 3: The space $\mathcal{X}$ from Example \ref{['E4']} (dot thickness $\leftrightsquigarrow$ size $2^{-r(i,j)}$).
• Figure 4: The first eight elements $V_m$ in $\mathbb{T}^{3} \times \mathbb{T}^{3, \omega}$.
• Figure 5: The $(\frac{1}{4},2)$-configuration around $(0,\frac{1}{2})^2 \times \mathbb{T}^{2,\omega}$.

Theorems & Definitions (28)

• Theorem 1.1
• Corollary 1.2
• Theorem 1.3
• Theorem 1.4: CF75
• Theorem 1.5: $p=\infty$ in dP36, $p \in (1,\infty)$ in HP55Ha76
• Proposition 2.1
• proof
• Remark 2.2
• Example 2.3
• Remark 2.4