A Marcinkiewicz Testing Criterion for Schur Multipliers

We prove a Marcinkiewicz testing condition for the boundedness of Schur multipliers on the Schatten $p$-classes. This generalizes a previous work of J. Bourgain for Toeplitz type Schur multipliers. As a corollary, we obtain a new unconditional decomposition for the Schatten $p$-classes ($1<p<\mathrm{\infty }$).