Extrapolation from A∞ weights and applications |
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Authors: | D Cruz-Uribe JM Martell C Pérez |
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Institution: | a Department of Mathematics, Trinity College, Hartford, CT 06106-3100, USA b Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain c Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, 41080 Sevilla, Spain |
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Abstract: | We generalize the Ap extrapolation theorem of Rubio de Francia to A∞ weights in the context of Muckenhoupt bases. Our result has several important features. First, it can be used to prove weak endpoint inequalities starting from strong-type inequalities, something which is impossible using the classical result. Second, it provides an alternative to the technique of good-λ inequalities for proving Lp norm inequalities relating operators. Third, it yields vector-valued inequalities without having to use the theory of Banach space valued operators. We give a number of applications to maximal functions, singular integrals, potential operators, commutators, multilinear Calderón-Zygmund operators, and multiparameter fractional integrals. In particular, we give new proofs, which completely avoid the good-λ inequalities, of Coifman's inequality relating singular integrals and the maximal operator, of the Fefferman-Stein inequality relating the maximal operator and the sharp maximal operator, and the Muckenhoupt-Wheeden inequality relating the fractional integral operator and the fractional maximal operator. |
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Keywords: | 42B20 42B25 |
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