Adrián González-Pérez

Hörmander-Mikhlin conditions for Schur multipliers

An operator T acting on a Hilbert space is said to be in the Schatten p-class if its singular values are absolutely p-summable. An important problem that has received increasing attention in the last two decades is whether, given a symbol m(j,k), cell-wise matrix multiplication by m preserves the Schatten p-class. These cell-wise multiplicators, called Schur multipliers, were instrumental in the solution of Krein's conjecture and had been studied in connection with approximation properties of groups and L^p-spaces. Here, we present a recent result that provides sufficient conditions on m, in terms of its off-diagonal smoothness, to ensure its boundedness in S_p. This criterion, inspired by classical results on pseudodifferential operators, provides as a consequence a fractional improvement to the (previously known) solution to the Krein conjecture.
This is a joint work with Jose Conde, Javier Parcet and Eduardo Tablate.