Let $\lambda>1$, $x,y\in [0,1]$, and $K_\lambda(x,y)=(1+\lambda|x-y|)^{-2}e^{i\lambda|x-y|}$, $a\in C_0^\infty([0,1])$. Consider the operator $$T_\lambda f(x)=\int_0^1 K_\lambda(x,y)f(y)dy.$$ We want to precisely estimate the norm $\|T_\lambda\|_{L^\infty([0,1])\to L^1([0,1])}$.

First, it is clear that $\|T_\lambda\|_{L^\infty([0,1])\to L^1([0,1])}\le \int_0^1\int_0^1|K_\lambda(x,y)|dxdy\approx \lambda^{-1}$.

Is this bound optimal? It looks doubtful, as it does not exploit the rapid oscillations in the kernel. Any comments are welcome!