On maximizers of a convolution operator in $L_p$-spaces

The paper is concerned with convolution operators in $\mathbb R^d$, whose kernels are in $L_q$, which act from $L_p$ into $L_s$, where $1/p+1/q=1+1/s$. It is shown that for $1<q,p,s<\infty$ there exists a maximizer (a function with $L_p$-norm $1$) at which the supremum of the $s$-norm of the convolution is attained. A special analysis is carried out for the cases in which one of the exponents $q,p$, or $s$ is $1$ or $\infty$.
Bibliography: 12 titles.