Beyond Calderòn-Zygmund theory

speaker: Pascal Auscher (Université de Paris XI)

abstract: In these talks, I will explain some new developments in the $L^p$ theory for "singular non-integral operators" by myself and others. This goes beyond the classical work of pionners, Calderòn-Zygmund, Hörmander, Fefferman-Stein, Coifman, Weiss and others. The objective is to obtain $L^p$ bounds in absence of bounds for kernels. They are replaced by decay measured in some $B(L^p,L^q)$ norm. The $L^p$ bounds are obtained for arbitrary ranges of $p$. I also discuss the extension to weigthed norm inequalities. In the case of $-div(A \nabla)$ elliptic operators, this technology gives optimal results for $H^\infty$ functional calculi, boundedness of some Littlewood-Paley-Stein type functional, boundedness of the Riesz transforms.... This also applies to some questions related to Riemannian manifolds.

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