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.
In this survey we collect and discuss some recent results on the so called Furstenberg set problem, which in its classical form concerns the estimates of the Hausdorff dimension of planar sets containing, for any direction, a subset of an interval po
itning in that direction of some prescribed dimension. This problem is closely related to the Kakeya needle problem. In this work we approach this problem from a more general point of view, in terms of generalized Hausdorff measures associated to dimension functions. We generalize the known results in terms of logarithmic gaps and obtain analogues to the classical estimates. Moreover, these analogues allow us to extend our results to the zero dimensional endpoint. We also obtain results about the dimension of a variation of Furstenberg sets defined for a fractal set of directions. We prove analogous inequalities reflecting the interplay between the size of the set of directions and the size of each fiber. This problem is also studied in the general scenario of Hausdorff measures.
