In the past decades, "weighted inequalities" have been a very attractive realm in singular Integral theory. One basic problem concerning them consists in determining conditions for a given operator (e. g. Hilbert transform) to be bounded in $L^p(w)$ with an appropriate weight $w$. To solve such problems, probabilistic methods or the dyadic technique --- which is a game of cubes --- has been used. In this talk, we will present some recent results about weighted bounds for "multilinear square functions" and certain singular integral operators such as linear and multilinear Fourier multipliers and the Riesz transforms associated to Schrödinger operators on $\mathbb{R}^n$.
References:
[1] T. A. Bui, M. Hormozi, Weighted bounds for multilinear square functions, Potential Analysis 46 (1), 135-148
[2] T. A. Bui, J. M. Conde-Alonso, X. T. Duong, M. Hormozi, Weighted bounds for multilinear operators with non-smooth kernels, Studia Mathematica 236, 245-269
[3] W. Damian, M. Hormozi, K. Li, New bounds for bilinear Calderon-Zygmund operators and applications, 35 pages, Accepted to Revista Matematica Iberoamericana