Weighted Calder\'on-Hardy spaces

TL;DR

This paper studies weighted Calderón-Hardy spaces on Euclidean space, establishing conditions under which the iterated Laplace operator acts as a bijection between these spaces and weighted Hardy spaces, extending harmonic analysis tools.

Contribution

It introduces weighted Calderón-Hardy spaces and proves the bijectivity of the iterated Laplace operator for certain parameters and weights, advancing the understanding of weighted harmonic analysis.

Findings

The iterated Laplace operator $ riangle^m$ is bijective between specific weighted Calderón-Hardy and Hardy spaces.

The paper characterizes conditions on weights and parameters for the operator's bijectivity.

Establishes new links between weighted Calderón-Hardy spaces and classical Hardy spaces.

Abstract

Let $0 < p \leq 1 < q < \infty$ and $\gamma >0$. In this note we discuss the weighted Calder\'on-Hardy spaces on $\mathbb{R}^{n}$, $\mathcal{H}^{p}_{q, \gamma}(\mathbb{R}^{n}, w)$. For $\gamma = 2m$, $m \in \mathbb{N}$, and $n (2m + n/q)^{-1} < p \leq 1$, we show that for certain power weights $w$ the iterated Laplace operator $\Delta^{m}$ is a bijective mapping from $\mathcal{H}^{p}_{q, 2m}(\mathbb{R}^{n},w)$ onto $H^{p}(\mathbb{R}^{n}, w)$.