Extremals for a class of convolution operators (Q2705832)

Suppose that \(q>1\) and that \(g\) on \({\mathbb R}^n\) is symmetrically decreasing. If there is an \(\varepsilon>0\) such that \(g\in L^{q-\varepsilon}\cap L^{q+\varepsilon}\), the author then proves that the convolution operator \(T_g:\;L^p\to L^r\), \(1/r=1/p+1/q-1\), attains its norm on the unit ball of \(L^p\) for any \(p>1\). Applications to and extensions of the well-known results for fractional integrals and Sobolev inequalities are also discussed.