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Transactions of the American Mathematical Society

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Weighted local estimates for singular integral operators


Authors: Jonathan Poelhuis and Alberto Torchinsky
Journal: Trans. Amer. Math. Soc. 367 (2015), 7957-7998
MSC (2010): Primary 42B20, 42B25
DOI: https://doi.org/10.1090/S0002-9947-2015-06459-9
Published electronically: February 19, 2015
MathSciNet review: 3391906
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Abstract: A local median decomposition is used to prove that a weighted mean of a function is controlled locally by the weighted mean of its local sharp maximal function. Together with the estimate $ M^{\sharp }_{0,s}(Tf)(x) \le c\,Mf(x)$ for Calderón-Zygmund singular integral operators, this allows us to express the local weighted control of $ Tf$ by $ Mf$. Similar estimates hold for $ T$ replaced by singular integrals with kernels satisfying Hörmander-type conditions or integral operators with homogeneous kernels, and $ M$ replaced by an appropriate maximal function $ M_T$. Using sharper bounds in the local median decomposition we prove two-weight, $ L^p_v-L^q_w$ estimates for the singular integral operators described above for $ 1<p\le q<\infty $ and a range of $ q$. The local nature of the estimates leads to results involving weighted generalized Orlicz-Campanato and Orlicz-Morrey spaces.


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Additional Information

Jonathan Poelhuis
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Email: jpoelhui@indiana.edu

Alberto Torchinsky
Affiliation: Department of Mathematics, Indiana University, Bloomington, Indiana 47405
Email: torchins@indiana.edu

DOI: https://doi.org/10.1090/S0002-9947-2015-06459-9
Received by editor(s): August 21, 2013
Published electronically: February 19, 2015
Dedicated: In remembrance of Björn Jawerth (1952-2013) who believed in local sharp maximal functions
Article copyright: © Copyright 2015 American Mathematical Society