Basic functional properties of certain scale of rearrangement-invariant spaces

TL;DR

This paper investigates the properties of a new class of rearrangement-invariant spaces, exploring their relationships with classical spaces, and introduces a novel one-parameter family of function spaces bridging Lebesgue and Zygmund classes.

Contribution

It introduces and analyzes the basic properties, embeddings, and associate structures of the spaces $X^{ angle \alpha angle}$, revealing a new one-parameter family connecting Lebesgue and Zygmund spaces.

Findings

Spaces relate to Sobolev embeddings with Ahlfors regular measures

Characterization of associate structures in certain cases

Discovery of a new one-parameter path between Lebesgue and Zygmund spaces

Abstract

Let $X$ be a rearrangement-invariant space over a non-atomic $\sigma$-finite measure space $(\mathscr{R},\mu)$ and let $\alpha\in(0,\infty)$. We define the functional \begin{equation*} \|f\|_{X^{\langle \alpha \rangle}} = \|((|f|^\alpha)^{**})^{\frac{1}{\alpha}}\|_{\overline{X}(0,\mu(\mathscr{R}))}, \end{equation*} in which $f$ is a $\mu$-measurable scalar function defined on $(\mathscr{R},\mu)$ and $\overline{X}(0,\mu(\mathscr{R}))$ is the representation space of $X$. We denote by $X^{\langle \alpha \rangle}$ the collection of all almost everywhere finite functions $f$ such that $\|f\|_{X^{\langle \alpha \rangle}}$ is finite. These spaces recently surfaced in connection of optimality of target function spaces in general Sobolev embeddings involving upper Ahlfors regular measures. We present a variety of results on these spaces including their basic functional properties, their relations to customary function spaces and mutual embeddings and, in a particular situation, a characterization of their associate structures. We discover a new one-parameter path of function spaces leading from a Lebesgue space to a Zygmund class and we compare it to the classical one.