The Hyt\"onen-Vuorinen L^{p} conjecture for the Hilbert transform, with an extended energy side condition, when (4/3)<p<4 and the measures share no point masses

Abstract

In the case (4/3)<p<4, and assuming a pair of locally finite positive Borel measures on the real line have no common point masses, we prove variants of two conjectures of T. Hyt\"onen and E. Vuorinen from 2018 on two weight testing theorems for the Hilbert transform on weighted L^{p} spaces, but with extended energy side conditions. Namely, assuming the extended energy conditions, the two weight norm inequality holds (1) if and only if the global quadratic interval testing conditions hold, (2) if and only if the local quadratic interval testing, the quadratic Muckenhoupt, and the quadratic weak boundedness conditions all hold. We also give a slight improvement of the second conjecture in this setting by replacing the quadratic Muckenhoupt conditions with two smaller conditions.