Representation of linear operator between $L^p$ spaces.

Question

I was wondering where I could find a reference to the a characterization of continuous linear operators: $$T:L^p(X,\mu)\to L^q(Y,\eta)$$

of the form $T(f)(y)=\int_{X} k(x,y)f(x)d\mu$ for some $k$ satisfying some properties.

Answer 1

Not every bounded linear operator from $L^p$ to $L^q$ has an integral kernel. Schachermeyer (Integral Operators on $L^p$ Spaces, Indiana Uni. Math. J.) gave the following characterization of integral operators:

Let $p\in[1,\infty)$ and $q\in [0,\infty]$. A linear operator $T\colon L^p(X,\mu)\to L^q(Y,\nu)$ has an integral kernel if and only if it maps order intervals to equi-measurable sets.

An order interval is a set of the form $\{f\in L^p(X,\mu)\mid g\leq f\leq h\}$ for $g,h\in L^p$. A subset $\mathfrak{F}$ of $L^q(Y,\nu)$ is called equi-measurable if for every $\epsilon>0$ there exists a measurable subset $Y_\epsilon$ of $Y$ such that $\nu(Y\setminus Y_\epsilon)<\epsilon$ and $\mathfrak{F}|_{Y_\epsilon}$ is relatively norm compact in $L^\infty(Y_\epsilon,\mu|_{Y_\epsilon})$.

Related results can be found in the articles Kernel operators and Compactness properties of an operator which imply that it is an integral operator by Schep, and The integral representation of linear operators by Buhvalov.