On a problem for integral convolution operators

This article considers convolution operators $T_k\colon L^2(R^N)\to L^2(R^N)$ of the form $T_kf(x)=\int_{R^N}k(x-y)f(y)\,dy$ which are integral operators on the whole class $L^2(R^N)$, i.e., the kernel $k(x)$ is such that $\int_{R^N}|k(x-y)f(y)|\,dy<\infty$ for almost all $x\in R^N$. An answer is obtained to the following question of Korotkov: if $T_k\colon L^2(R^N)\to L^2(R^N)$ is a convolution operator which is an integral operator on the whole of $L^2(R^N)$, does it follow that $\operatorname{mes}\{\xi\in R^N:|k^\wedge(\xi)|>\lambda\}<\infty$ for any $\lambda>0$? Here $k^\wedge(\xi)$ is the Fourier transform of $k(x)$. An example answering the question in the negative is given by the operator $T_{\mathscr K}\colon L^2(R^1)\to L^2(R^1)$ with kernel $\mathscr K(x)$ such that $\mathscr K^\wedge(\xi)=\sum\limits_{n\ne0}\operatorname{sign}n\chi_{\bigl[-\frac1{2|n|},\frac1{2|n|}\bigr]}(\xi-n),$ where $\chi_{[a,b]}$ is the characteristic function of $[a,b]$.
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