Calderón-Zygmund Capacities and Operators on Nonhomogeneous Spaces

Volberg, Alexander

doi:10.1090/cbms/100

Cited by 118 publications

(124 citation statements)

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“…Two weight inequalities for dyadic operators are relatively advanced, but their continuous versions are much harder. Innovative work of Nazarov-Treil-Volberg on non-homogenous harmonic analysis [9,10,14] renewed attention on these questions, and in particular the paper [10], giving sufficient conditions for a two weight inequality for singular integrals, has proven to be quite influential, in its use of the so-called pivotal condition.…”

Section: /2mentioning

confidence: 99%

Two weight norm inequalities for the $g$ function

Lacey

2014

Mathematical Research Letters

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Section: /2mentioning

confidence: 99%

Two weight norm inequalities for the $g$ function

Lacey

2014

Mathematical Research Letters

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“…We expect that this condition can be strengthened to a "Poisson two-weight A p condition". See [Nazarov et al 2010;Volberg 2003]. …”

Section: Theorem 12 (Maximal Function Inequalities)mentioning

confidence: 99%

“…We comment in more detail on prior results below, including the innovative work of Nazarov, Treil and Volberg [1999;2008;2010;2003].…”

Section: Introductionmentioning

confidence: 99%

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

Lacey

Sawyer

Uriarte-Tuero

2012

APDE

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Let σ and ω be positive Borel measures on R with σ doubling. Suppose first that 1 < p ≤ 2. We characterize boundedness of certain maximal truncations of the Hilbert transform T from L p (σ) to L p (ω) in terms of the strengthened A p condition for all pairwise disjoint decompositions Q = ∪ ∞ r=1 I r of the dyadic interval Q into dyadic intervals I r . We prove that analogues of these conditions are sufficient for boundedness of certain maximal singular integrals in R n when σ is doubling and 1 < p < ∞. Finally, we characterize the weak-type two weight inequality for certain maximal singular integrals T in R n when 1 < p < ∞, without the doubling assumption on σ, in terms of analogues of the second testing condition and the A p condition.

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“…The latter, especially, played a useful role in Tolsa's solution of the Painlevé problem and the related Vitushkin conjecture concerning the semi-additivity of analytic capacity. Finally, much of this theory has been extended to higher dimensions by Volberg [90].…”

Section: Analytic Capacitymentioning

confidence: 99%