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Your operator $K$ is a Hilbert-Schmidt operator since its kernel belongs to $L^2$. As a result this is a compact operator whose spectrum contains a sequence of eigenvalues $\\{\lambda_k\not=0\\}$ with …

Let us give only an expansion in $a,b$. Calling $I(a,b)$ the integral, we get easily $$ I(a,b)=\sum_{k\ge 0}\frac{b^{2k}}{(2k)!}\underbrace{e^{a}\int_0^{+\infty} e^{-a (x^2+1)}(1+x^2)^k dx}_{J_k(a)}. …

Following your assumptions, it seems that the mapping $L^{-1}$ sends linearly and continuously the smooth compactly supported functions into distributions and thus, from the Schwartz (Laurent) kernel …

Too long for a comment. Your requirement is too stringent and it is quite likely that to get continuity from $L^\infty$ into itself, it is indeed necessary to have $$ \text{esssup}_x\int\vert k(x,y)\v …

Assuming the function $f$ of class $C^1$, you find that, using Taylor's formula with integral remainder, $$ f(t)-f(s)=(t-s) f_1(t,s),\quad \text{with $f_1$ continuous},$$ so that $ (T_\alpha f)(t)= …