Sharp Two Weight Inequalities for Commutators of Riemann-Liouville and Weyl Fractional Integral Operators |
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Authors: | Ana L. Bernardis María Lorente |
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Affiliation: | (1) IMAL-CONICET, Güemes 3450, 3000 Santa Fe, Argentina;(2) Departamento de Análisis Matemático, Facultad de Ciencias, Universidad de Málaga, Málaga, Spain |
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Abstract: | Let b be a BMO function, and the commutator of order k for the Weyl fractional integral. In this paper we prove weighted strong type (p, p) inequalities (p > 1) and weighted endpoint estimates (p = 1) for the operator and for the pairs of weights of the type (w, ), where w is any weight and is a suitable one-sided maximal operator. We also prove that, for weights, the operator is controlled in the L p (w) norm by a composition of the one-sided fractional maximal operator and the one-sided Hardy-Littlewood maximal operator iterated k times. These results improve those obtained by an immediate application of the corresponding two-sided results and provide a different way to obtain known results about the operators . The same results can be obtained for the commutator of order k for the Riemann-Liouville fractional integral This research has been partially supported by Spanish goverment Grant MTM2005-8350-C03-02. The first author was also supported by CONICET, ANPCyT and CAI+D-UNL. The second author was also supported by Junta de Andalucía Grant FQM 354. |
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Keywords: | KeywordHeading" >. Weighted inequalities Riemann-Liouville and Weyl fractional integrals commutators |
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