The T1 theorem for the Hilbert transform fails when p is not 2

TL;DR

This paper demonstrates that the T1 theorem for the Hilbert transform, which holds at p=2, does not extend to other p values in the range (1, infinity).

Contribution

The authors prove the failure of the T1 theorem for the Hilbert transform on L^p spaces when p is not equal to 2.

Findings

T1 theorem holds at p=2

T1 theorem fails for p ≠ 2

Hilbert transform behavior differs across p-values

Abstract

Given p between 1 and infinity, but not 2, we show that the T1 theorem for the Hilbert transform fails for L^{p}, despite holding for p equal to 2