The two-weight inequality for the Hilbert transform with general measures

Abstract

The two-weight inequality for the Hilbert transform is characterized for an arbitrary pair of positive Radon measures $\sigma$ and $w$ on $\mathbb{R}$. In particular, the possibility of common point masses is allowed, lifting a restriction from the recent solution of the two-weight problem by Lacey, Sawyer, Shen and Uriarte-Tuero. Our characterization is in terms of Sawyer-type testing conditions and a variant of the two-weight $A_2$ condition, where $\sigma$ and $w$ are integrated over complementary intervals only. A key novely of the proof is a two-weight inequality for the Poisson integral with `holes'.

The last version of the article submitted to the publisher (but without journal style file)

References in corpus (1)

• Two Weight Inequality for the Hilbert Transform: A Real Variable Characterization, II

Cited by in corpus (3)

• Dyadic harmonic analysis and weighted inequalities: the sparse revolution
• Bilinear Calderón-Zygmund theory on product spaces
• Parabolic Muckenhoupt Weights Characterized by Parabolic Fractional Maximal and Integral Operators with Time Lag