Sharp Estimates for $p$-Adic Hardy, Hardy-Littlewood-P\'olya Operators and Commutators

TL;DR

This paper establishes sharp bounds for $p$-adic Hardy and Hardy-Littlewood-Pólya operators on weighted $L^q$ spaces and proves the boundedness of their commutators with BMO functions on Herz spaces.

Contribution

It provides the first sharp estimates for these operators in the $p$-adic setting and analyzes the boundedness of their commutators with BMO functions.

Findings

Sharp bounds for $p$-adic Hardy operators on weighted $L^q$ spaces.

Boundedness of commutators with BMO functions on Herz spaces.

Extension of classical results to the $p$-adic context.

Abstract

In this paper we get the sharp estimates of the $p$-adic Hardy and Hardy-Littlewood-P\'olya operators on $L^q(|x|^{\alpha}_pdx)$. Also, we prove that the commutators generated by the $p$-adic Hardy operators (Hardy-Littlewood-P\'olya operators) and the central BMO functions are bounded on $L^q(|x|^{\alpha}_pdx)$, more generally, on Herz spaces.