Boundedness of some operators on weighted amalgam spaces

TL;DR

This paper introduces weighted amalgam spaces and investigates the boundedness of classical harmonic analysis operators on these spaces, providing new weak-type results using extrapolation techniques.

Contribution

It defines weighted amalgam spaces and establishes boundedness properties for key harmonic analysis operators, extending known results to these new spaces.

Findings

Boundedness of Hardy--Littlewood maximal operator on weighted amalgam spaces.

Weak-type estimates for Calderón--Zygmund operators in these spaces.

Extension of classical operator bounds to weighted amalgam spaces.

Abstract

Let $t\in(0,\infty)$, $p\in(1,\infty)$, $q\in[1,\infty]$, $w\in A_p$ and $v\in A_q$. We introduce the weighted amalgam space $(L^p,L^q)_t(\mathbb R^n)$ and show some properties of it. Some estimates on these spaces for the classical operators in harmonic analysis, such as the Hardy--Littlewood maximal operator, the Calder\'on--Zygmund operator, the Riesz potential, singular integral operators with the rough kernel, the Marcinkiewicz integral, the Bochner-Riesz operator, the Littlewood-Paley $g$ function and the intrinsic square function, are considered. Our main method is extrapolation. We obtain some new weak results for these operators on weighted amalgam spaces.