Title: Weighted norm inequalities for rough singular integral operators

Speaker: Luz Roncal (BCAM - Basque Center for Applied Mathematics, Spain)

Date: 24 April 2017

Time: 4pm

Venue: LH-1, Mathematics Department

The study of weighted inequalities in Classical Harmonic Analysis started in 70’s, when B. Muckenhoupt characterised in 1972 the weights $w$ for which the Hardy–Littlewood maximal function is bounded in $L^p(w)$. At that time the question about how the operator depended on the constant associated with $w$, which we denote by $[w]_{A_p}$, was not considered (i.e., quantitative estimates) were not investigated.

From the beginning of 2000’s, a great activity has been carried out in order to obtain the sharp dependence for singular integral operators, reaching the solution of the so-called $A_2$ conjecture by T. P. H"ytonen.

In this talk we consider operators with homogeneous singular kernels, on which we assume smoothness conditions that are weaker than the standard ones (this is why they are called rough). The first qualitative weighted estimates are due to J. Duoandikoetxea and J. L. Rubio de Francia. For the norm of these operators in the space $L^2(w)$ we obtain a quantitative estimate which is quadratic in the constant $[w]_{A_2}$.

The results are based on a classical decomposition of the rough operators as a sum of other operators with a smoother kernel, for which a quantitative reelaboration of a dyadic decomposition proposed by M. T. Lacey is applied.

We will overview as well the most recent advances, mainly associated with quantitative estimates for these rough singular integrals. In particular, Coifman–Fefferman type inequalities (which are new even in their qualitative version), weighted $A_p$-$A_{\infty}$ inequalities and a quantitative version of weak $(1,1)$ estimates will be shown.