Singular value estimates for certain convolution-product operators (Q1378924)

Let \(S_{f\varphi}\) be the integral operator on \(L^2({\mathbb{R}}^d)\) with kernel \(f(x)\overline{\varphi(y-x)}\). Under suitable hypotheses on the supports of \(f\) and \(\varphi\), the author proves the compactness of \(S_{f\varphi}\), derives upper and lower bounds for its singular values, and gives necessary and sufficient conditions for \(S_{f\varphi}\) to belong to a certain Schatten ideal \(S^p\) with \(0<p\leq+\infty\). In fact, \(S_{f\varphi}\in S^p\) if and only if a suitable mixed norm of the Fourier transform of \(\varphi\) is finite.