Optimal exponents in weighted estimates

(See http://www.math.sciences.univ-nantes.fr/~viola-j/Seminaire/OptimalDec15.pdf)

We are interested in proving the optimality of weighted inequalities from L^p to either L^p or weak L^p for a certain operator T and w an A_p weight. In the first part of the seminar, we show that whenever the former bound is true, then necessarily the exponent on the A_p constant of the weight satisfies a lower bound which is a function of the asymptotic behaviour of the unweighted L^p operator norm as p goes to 1 and +∞. By combining these results with known weighted inequalities, we derive the sharpness of such exponent, without building any specific example, for maximal, Calderón–Zygmund and fractional integral operators. The main underlying idea of this result comes from extrapolation theory and the Rubio de Francia algorithm. (joint with Carlos Pérez and Ezequiel Rela)

In the second part, we focus on the case where T is a multiparameter operator, in particular, the strong maximal function, and w is a strong-A_p weight. Multiparameter optimal weighted inequalities have not been developed, as there exists a serious obstruction in carrying over the well-known achievements of classical weighted theory to the multiparameter setting. As we will see, this is somehow a manifestation of the failure of the Besicovitch covering argument. We will try to present the obstacles we need to deal with and the partial results we have been able to prove.