Parametric Marcinkiewicz integrals on weighted Hardy and weak Hardy spaces

TL;DR

This paper investigates the boundedness of parametric Marcinkiewicz integrals on weighted Hardy and weak Hardy spaces using atomic decomposition, under Lipschitz conditions on the kernel.

Contribution

It establishes boundedness results for these integrals on weighted Hardy and weak Hardy spaces, extending previous understanding with atomic decomposition techniques.

Findings

Boundedness of $oldsymbol{}$ on weighted Hardy spaces

Boundedness of $oldsymbol{}$ on weak Hardy spaces

Results depend on Lipschitz condition of the kernel

Abstract

Let $0<\rho<n$ and $\mu^{\rho}_{\Omega}$ be the parametric Marcinkiewicz integral. In this paper, by using the atomic decomposition theory of weighted Hardy and weak Hardy spaces, we will obtain the boundedness properties of $\mu^{\rho}_\Omega$ on these spaces, under the Lipschitz condition imposed on the kernel $\Omega$.