A characterization of two weight norm inequalities for maximal singular integrals with one doubling measure

By Michael T. Lacey, Eric T. Sawyer, Ignacio Uriarte-Tuero. 2008

54 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/8922e5e765dac883dee8d34f9848f6ab616d85e2

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In Applied Mathematics

Citations in the corpus

Systems of dyadic cubes in a doubling metric space (2010, methodology): “B). These adjacent systems have been exploited, e.g., in the work of C. Muscalu, T. Tao and C. Thiele [22, 21] on multi-linear operators, and very recently by M. Lacey, E. Sawyer and I. Uriarte-Tuero [15] on two-weight inequalities. We are not aware of the precise original occurrence of this latter ”
Sharp weighted estimates for classical operators (2010, background): “(See Nazarov, Treil and Volberg [39, 53, 40] and the recent preprints by Lacey, Sawyer and Uriarte-Tuero [29, 30].”