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