On the vanishing of the symbol of a convolution integral operator

The class $\operatorname{Int}(p,p)$ of kernels of convolution integral operators is defined, and a criterion for a measurable function to belong to $\operatorname{Int}(p,p)$ is given. The question of the behavior of the Fourier transform of a kernel (the symbol) in $\operatorname{Int}(2,2)$ is considered, and it is shown that in the sense of order the symbol can vanish at infinity in an arbitrarily slow manner, and more slowly than any power in the mean.
Bibliography: 6 titles.