Consider the integral operator $$ I[u](x) = \int K(x,y)u(y)dy. $$ The intuition is that, $I[u]$ will be more regular than $u$ in terms of the Sobolev space exponents, and we should get some estimate of the form $$ \|I[u]\|_{H^s} \lesssim \|u\|_{L^2} $$
And we know the particular example of Laplacians, and many others.
The question is: where can I find more about general results that works for a generic function $K?$ Is there a name for this area of study (more specific than "harmonic analysis")? It would be very helpful if anyone can share some references.