On embeddings between classical Lorentz spaces (Q2745656)

For a nonnegative measurable function \(\nu\) on \((0, \infty)\) (a ``weight''), the Lorentz spaces \(\Lambda^p (\nu)\) and \(\Lambda^{p,\infty} (\nu)\) are determined by the norms \(\|f\|_{\Lambda^p (\nu)} =(\int^\infty _0 (f^* (t))^p \nu(t) dt)^{1/p}\) and \(\|f\|_{\Lambda^{p,\infty} (\nu)} =\sup_{0<t<\infty} f^*(t)V(t)\), where \(V(t)=\int_0 ^\infty \nu(s) ds\). Replacing \(f^*\) by \(f^{**}(t)=t^{-1}\int_0 ^\infty f^* (s) ds\), we obtain the definitions of the spaces \(\Gamma^p (\nu)\) and \(\Gamma^{p,\infty} (\nu)\). The authors discuss various embeddings among these spaces. The main emphasis is on the embedding of the form \(\Gamma^{p,\infty} (\nu)\to\Lambda^p (w)\) (this case has not been considered previously). As a consequence, the authors characterize the cases where the norm of \(\Lambda^1 (\nu)\) can be expressed in terms of \(f^{**}\).