Weak Hardy Spaces Associated with Ball Quasi-Banach Function Spaces on Spaces of Homogeneous Type: Decompositions, Real Interpolation, and Calder\'{o}n--Zygmund Operators

TL;DR

This paper introduces a new class of weak Hardy spaces on spaces of homogeneous type associated with ball quasi-Banach function spaces, providing characterizations, interpolation results, and boundedness of Calderón--Zygmund operators, overcoming key geometric and measure-theoretic challenges.

Contribution

The authors develop a novel framework for weak Hardy spaces on spaces of homogeneous type, utilizing advanced techniques like the Aoki–Rolewicz theorem and dyadic systems to handle absence of triangle inequality and reverse doubling.

Findings

Characterization of $WH_X(X)$ via maximal functions and atoms

Establishment of real interpolation results for $WH_X(X)$

Boundedness of Calderón--Zygmund operators in the critical case

Abstract

Let $(\mathbb{X},d,\mu)$ be a space of homogeneous type in the sense of R. R. Coifman and G. Weiss, and $X(\mathbb{X})$ a ball quasi-Banach function space on $\mathbb{X}$. In this article, the authors introduce the weak Hardy space $WH_X(\mathbb{X})$ associated with $X(\mathbb{X})$ via the grand maximal function, and characterize $WH_X(\mathbb{X})$ by other maximal functions and atoms. The authors then apply these characterizations to obtain the real interpolation and the boundedness of Calder\'{o}n--Zygmund operators in the critical case. The main novelties of this article exist in that the authors use the Aoki--Rolewicz theorem and both the dyadic system and the exponential decay of approximations of the identity on $\mathbb{X}$, which closely connect with the geometrical properties of $\mathbb{X}$, to overcome the difficulties caused by the absence of both the triangle inequality of $\|\cdot\|_{X(\mathbb{X})}$ and the reverse doubling assumption of the measure $\mu$ under consideration, and also use the relation between the convexification of $X(\mathbb{X})$ and the weak space $WX(\mathbb{X})$ associated with $X(\mathbb{X})$ to prove that the infinite summation of atoms converges in the space of distributions on $\mathbb{X}$. Moreover, all these results have a wide range of generality and, particularly, even when they are applied to the weighted Lebesgue space, the Orlicz space, and the variable Lebesgue space, the obtained results are also new and, actually, some of them are new even on RD-spaces (namely, spaces of homogeneous type satisfying the additional reverse doubling condition).