Hardy-Littlewood-Sobolev theory on Lie groups with Lorentz norms (Q678662)

Let \(\Delta\) be the Euclidean Laplacian on \(\mathbb{R}^n\) and \(T_t\) the heat semigroup with infinitesimal generator \(\Delta\). It is well-known that the inequality \(|f |_{2n/(n-2)} \leq c(\Delta f,f)^{1/2}\) implies that \(|T_tf |_\infty\leq ct^{-n/2} |f |_1\) and vice versa. In order to generalize this Hardy-Littlewood-Sobolev type theorem for any semigroups \(T_t\) with infinitesimal generator \(A\) on Lie groups \(G\), it is necessary to treat the small time and large time behavior of functions on \(G\). The author uses the Lorentz spaces \(L^{rs}_{pq}\) on \(G\), which were introduced in his previous paper [J. Approximation Theory 84, 274-289 (1996; Zbl 0849.43002)]. Then he shows the equivalence relation between the following two inequalities: \[ |f|^\infty_{(2n/(n-2))}|f|^\infty_{(2m/(m-2))} \leq cRe (Af,f)^{1/2}, \] and \[ |T_t f|_\infty\leq ct^{-\delta(t)/2} |f|_1, \] where \(\delta(t) =n\) if \(t\leq 1\) and \(\delta (t)=m\) if \(t>1\).