Mathematical Research Letters

Volume 22 (2015)

Number 1

Optimal exponents in weighted estimates without examples

Pages: 183 – 201

DOI: http://dx.doi.org/10.4310/MRL.2015.v22.n1.a10

Authors

Teresa Luque (School of Mathematics, University of Birmingham, Edgbaston, Birmingham, United Kingdom)

Carlos Pérez (Department of Mathematics, University of the Basque Country UPV/EHU, Leioa, Bizkaia; and IKERBASQUE, Basque Foundation for Science, Bilbao, Spain)

Ezequiel Rela (Departamento de Matem´atica, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Argentina)

Abstract

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\[{\lVert T \rVert}_{L^p (w)} \leq c[w]^\beta_{A_p} \quad w \in A_p \textrm{,}\]then the optimal lower bound for $\beta$ is closely related to the asymptotic behaviour of the unweighted $L^p$ norm ${\Vert T \rVert}_{L^p (\mathbb{R}^n)}$ as $p$ goes to $1$ and $+\infty$.

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, Calderón–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.

2010 Mathematics Subject Classification

Primary 42B25. Secondary 43A85.

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