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.
The relationship between the operator norms of fractional integral operators acting on weighted Lebesgue spaces and the constant of the weights is investigated. Sharp boundsare obtained for both the fractional integral operators and the associated fr
actional maximal functions. As an application improved Sobolev inequalities are obtained. Some of the techniques used include a sharp off-diagonal version of the extrapolation theorem of Rubio de Francia and characterizations of two-weight norm inequalities.
