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Optimal exponents in weighted estimates without examples


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Author: Luque Martínez, Teresa
Pérez Moreno, Carlos
Rela, Ezequiel
Department: Universidad de Sevilla. Departamento de Análisis Matemático
Date: 2015
Published in: Mathematical Research Letters, 22 (1), 183-201.
Document type: Article
Abstract: t. We present a general approach for proving the optimality of the exponents on weighted estimates. We show that if an operator T satisfies a bound like kT kLp(w) ≤ c [w] β Ap w ∈ Ap, then the optimal lower bound for β is closely related to the asymptotic behaviour of the unweighted L p norm kT kLp(Rn) as p goes to 1 and +∞, which is related to Yano’s classical extrapolation theorem. By combining these results with the known weighted inequalities, we derive the sharpness of the exponents, without building any specific example, for a wide class of operators including maximal-type, Calder´on–Zygmund and fractional operators. In particular, we obtain a lower bound for the best possible exponent for Bochner-Riesz multipliers. We also present a new result concerning a continuum family of maximal operators on the scale of logarithmic Orlicz functions. Further, our method allows to consider in a unified way maximal operators defined over very general Muckenhoupt bases.
Cite: Pérez Moreno, C. y Rela, E. (2015). Optimal exponents in weighted estimates without examples. Mathematical Research Letters, 22 (1), 183-201.
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DOI: 10.4310/MRL.2015.v22.n1.a10

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