Operators interpolating between Riesz potentials and maximal operators
Let \(\lambda\) be normalized Lebesgue measure on either the unit ball or the unit sphere in \({\mathbb{R}}^ n\) and \(\lambda_ r\) denote the dilate of \(\lambda\). For a continuous function f with compact support, define in the unit ball case \[ S_{p,q,s}f(x)=[\int^{\infty}_{0}| r^{n/p-n/q}\lambda_ r*f(x)|^ s dr/r]^{1/s}, \] \[ S_{p,q,\infty}f(x)=\sup_{r>0}r^{n/p-n/q}| \lambda_ r*f(x)|. \] In the unit sphere case, define operators \(T_{p,q,s}\) and \(T_{p,q,\infty}\) analogously. For nonnegative f, both \(S_{p,q,1}\) and \(T_{p,q,1}\) are multiples of the Riesz potential \(I_{\alpha}(f)\) when \(\alpha =n/p-n/q\). On the other hand, \(S_{p,q,\infty}\) and \(T_{p,q,\infty}\) are maximal operators, especially \(S_{p,p,\infty}\) is the Hardy-Littlewood maximal operator and \(T_{p,p,\infty}\) is the spherical maximal operator. So, \(S_{p,q,s}'s\) \((T_{p,q,s}'s)\) are operators interpolating between Riesz potentials and maximal operators. The author gives a necessary and sufficient condition for \(S_{p,q,s}\) to be bounded from \(L^ p\) to \(L^ q\). As for the boundedness of \(T_{p,q,s}\), he gives several necessary conditions and sufficient ones.
scientific article; zbMATH DE number 4127852