Schur multipliers in Schatten-von Neumann classes

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 < infinity, a simple form of our main result for R-n x R-n matrices reads as follows: ||S-M : S-p -> S-p||(cb) less than or similar to p(2)/p-1 Sigma(|gamma|less than or similar to[n/2]+1)|||x y|(|gamma|) {|partial derivative M-gamma(x)(x, y) | + |partial derivative M-gamma(y)(x , y)|}||(proportional to). In this form, it is a full matrix (nonToeplitz/nontrigonometric) amplification of the Hormander-Mikhlin multiplier theorem, which admits lower fractional differentiability orders sigma > n/2 as well. It trivially includes Arazy's conjecture for S-p-multipliers and extends it to alpha-divided differences. It also leads to new Littlewood-Paley characterizations of S-p-norms and strong applications in harmonic analysis for nilpotent and high rank simple Lie group algebras.