An estimate for Schur multipliers in \(S_ p\)-classes (Q1336397)

It is proved that \(\| A \circ X \|_ p \leq \| A \|_{\ell_ \infty (\ell_ r)} \| X \|_ p\) if \(r\) is at least 2, \(| 1/p - 1/2 | \leq 1/r\), where \(A \circ X\) is the entrywise product of the infinite matrices \(A\) and \(X\), so that \(A\) is viewed as a Schur multiplier, and this bounds its norm.