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 ∥ T ∥ Lp(w) ≤ c [w]βAp w ε Ap, then the optimal lower bound for β is closely related to the asymptotic behaviour of the unweighted Lp norm ∥ T ∥ Lp(Rn) as p goes to 1 and +∞. 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 maximaltype, Caldeŕon-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.