Luque Martínez, TeresaPérez Moreno, CarlosRela, Ezequiel2016-06-292016-06-292015Pérez Moreno, C. y Rela, E. (2015). Optimal exponents in weighted estimates without examples. Mathematical Research Letters, 22 (1), 183-201.1073-2780http://hdl.handle.net/11441/42933t. 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.application/pdfengAttribution-NonCommercial-NoDerivatives 4.0 Internacionalhttp://creativecommons.org/licenses/by-nc-nd/4.0/Muckenhoupt weightsCalderón-Zygmund operatorsMaximal functionsinfo:eu-repo/semantics/articleinfo:eu-repo/semantics/openAccesshttps://doi.org/10.4310/MRL.2015.v22.n1.a10