On weighted multidimensional embeddings for monotone functions

Abstract

We characterize the inequality 7817 \left(\int_{\mathbf{R}^N_+} f^q u\right)^{1/q} \leq C \left(\int_{\mathbf{R}^N_+} f^p v \right)^{1/p},\,\,0<q ,p <\infty, 7817 for monotone functions $f\geq 0$ and nonnegative weights $u$ and $v$. The case $q < p$ is new and the case $0<p\leq q <\infty$ is extended to a modular inequality with N- functions. A remarkable fact concerning the calculation of $C$ is pointed out.