Optimal exponents in weighted estimates without examples

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

Revised and corrected version. To appear in Math. Res. Lett