We establish a rather unexpected and simple criterion for the boundedness of Schur multipliers S M on Schatten p-classes which solves a conjecture proposed by Mikael de la Salle. Given 1 < p < ∞, a simple form our main result reads for R n × R n matrices as followsIn this form, it is a full matrix (nonToeplitz/nontrigonometric) amplification of the Hörmander-Mikhlin multiplier theorem, which admits lower fractional differentiability orders σ > n 2 as well. It trivially includes Arazy's conjecture for Sp-multipliers and extends it to α-divided differences. It also leads to new Littlewood-Paley characterizations of Sp-norms and strong applications in harmonic analysis for nilpotent and high rank simple Lie group algebras.