A multi-parameter Hardy type inequality

TL;DR

This paper introduces a multi-parameter Hardy type inequality derived from weak factorization of product H^1 spaces and explores the dual relationship involving Fourier transforms and product BMO-Sobolev spaces.

Contribution

It presents a novel multi-parameter Hardy inequality and links Fourier transforms of bounded functions to product BMO-Sobolev spaces, expanding classical harmonic analysis results.

Findings

Established a multi-parameter Hardy inequality.

Connected Fourier transforms of bounded functions to product BMO-Sobolev spaces.

Provided new insights into multi-parameter harmonic analysis.

Abstract

This note contains two simple observations. First, by the weak factorization of product $H^1$ (Ferguson--Lacey, Lacey--Terwilleger), we obtain a multi-parameter analogue of Hardy's inequality. Second, as a dual statement, the Fourier transform of an essentially bounded function belongs to a certain product $\mathrm{BMO}$-Sobolev space.