Definition 1

Let $k\in \mathscr{N}^{p,q}(X\times Y)$. One says that the mapping $u_k$ is the integral operator with kernel $k$ from $L^p(X)$ into $L^{q'}(Y)$.

For $k\in \mathscr{N}^{p,q}(X\times Y)$ and $f\in \mathscr{L}^p$(X), one will sometimes also denote by $u_k(f)$ the image under $u_k$ of the class of $f$ in $L^p(X)$.