Let $T:L^2(\mathbb{R}^n,\mathbb{R}^n)\rightarrow L^2(\mathbb{R}^n,\mathbb{R}^n)$ be a linear operator of the form: for all $x\in \mathbb{R}^n$ $$ T_{\kappa}(f)(x):=\int_{u\in\mathbb{R}^n} \, f(u)\kappa(u,x)du, $$ where $\kappa:\mathbb{R}^{2n}\rightarrow \mathbb{R}^n$ is a continuous function and multiplication is componentwise.
Is there a class of $\kappa$ for the operator norm of $T_{\kappa}$ is known and is expressed explicitly in terms of $\kappa$?
Edit The types of integral operators I am interested in are specified by the following type types of kernel functions:
- $\kappa(u,x)=(\tilde{\kappa}(u_1-x_1),\dots,\tilde{\kappa}(u_n-x_n))$ where $\tilde{\kappa}$ is a strictly increasing continuous map from the $\mathbb{R}$ to $[0,1]$, (Similar in spirit to what Yemon wrote)
- $\kappa(u,x)=K_x \,u$ where each $K_x$ is a symmetric positive-definite matrix-valued function satisfying the non-degeneracy condition $$ 0<m:=\inf_{x}\,|\sigma_{\min}(K_x)|\leq \sup_{x}\,|\sigma_{\max}(K_x)|=:M<\infty $$
- $\kappa(u,x)=K_x \,u$ where each $K_x$ is a square matrix satisfying $$ \sup_{x}\, \|K_x\|_{\ell^{\infty}}=:M^{\prime}<\infty $$
If anyone knows, and purely out of curiosity, what about examples from these classes:
- The components of $\kappa$ belongs to a Sobolev space $W^{k,\infty}$ for large $k$,
- The components of $\kappa$ are non-constant and compactly supported smooth (class-$C^{\infty}$) function.
Though, I'm asking about integral operators of types $1$, $2$, or $3$ above (only curious about the other types if anyone happens to know).
