Schur multipliers in Schatten-von Neumann classes

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We establish a rather unexpected and simple criterion for the boundedness of Schur multipliers on Schatten p-classes which solves a conjecture proposed by Mikael de la Salle. Given 1<p<¿, we give a full matrix (nonToeplitz/nontrigonometric) amplification of the Hörmander-Mikhlin multiplier theorem, which admits lower fractional differentiability orders as well. It trivially includes Arazy's conjecture for Schur multipliers and extends it to Holder divided differences. It also leads to new Littlewood-Paley characterizations of p-norms and strong applications in harmonic analysis for nilpotent and high rank simple Lie group algebras.