When does an integral operator belong to the Schatten - von Neumann class in terms of its kernel?

Question

It is well known that an integral operator $X: L^2(\mu)\to L^2(\nu)$ with kernel $k(x, y)$ belongs to the Schatten-von Neumann class $\mathfrak S_2$ if and only if $\int |k(x, y)|^2\, d\mu(x)\, d\nu(y)<\infty$, that is, the kernel $k$ belongs to $L^2(\mu\times\nu)$. Do there exist necessary or sufficient conditions of the form $k\in L^{p}(\mu\times\nu)$ for the property $X\in\mathfrak S_{p'}$ if $p, p'\ne 2$?