A characterization of two weight norm inequalities for maximal singular integrals with one doubling measure
Abstract
We characterize two-weight inequalities for certain maximal truncations of the Hilbert transform in terms of testing conditions on simpler functions. For 1<p<2 and two positive Borel measures u, v on R, we assume that u is doubling, and we consider maximal truncations T_# of the Hilbert transform. The norm estimate || T(f u) ||_{L^p(v)} < C || f ||_{L^p(u)} is characterized in terms of an A_p condition on the weights and two testing conditions. The first is the norm condition above, but the function f varies over bounded functions supported on a cube. The second is a dual weak-type condition, for arbitrary functions. This result should be compared to the result of Nazarov, Treil and Volberg, arXiv:math/0702758. Additional results are obtained for 2<p<\infty, and for the weak type inequality.
Keywords
Cite
@article{arxiv.0807.0246,
title = {A characterization of two weight norm inequalities for maximal singular integrals with one doubling measure},
author = {M. T. Lacey and E. T. Sawyer and I. Uriarte-Tuero},
journal= {arXiv preprint arXiv:0807.0246},
year = {2015}
}
Comments
64 pages. Submitted to A&PDE. Minor revisions, based on a final referees report