Boundedness of parameterized Marcinkiewicz integrals with variable kernels on Hardy spaces and weak Hardy spaces

TL;DR

This paper proves the boundedness of parameterized Marcinkiewicz integrals with variable kernels on Hardy and weak Hardy spaces using atomic decomposition, under Dini or Hörmander conditions on the kernel.

Contribution

It establishes boundedness results for these integrals on Hardy spaces using atomic decomposition and specific kernel conditions, advancing understanding of their behavior.

Findings

Boundedness on Hardy spaces established

Boundedness on weak Hardy spaces established

Results depend on Dini or Hörmander conditions

Abstract

In this paper, by using the atomic decomposition theory of Hardy space and weak Hardy space, the author establishes the boundedness of parameterized Marcinkiewicz integral with variable kernel on these spaces, under the Dini condition or H\"{o}rmander condition imposed on kernel.