A multi-parameter Hardy type inequality

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.

The original proof of the Ferguson--Lacey theorem contains a serious error. Although my paper describes how to circumvent the use of the Ferguson--Lacey result, the associated revisions to make this a proper and correct paper are beyond my current schedule. Until I've had time to make a proper revision, I will withdraw the paper