On weighted multidimensional embeddings for monotone functions - Barza, Sorina; Persson, Lars-Erik; Stepanov, Vladimir D. (MATHEMATICA SCANDINAVICA, 2001)
We characterize the inequality 7817 \left(\int_{R^N_+} f^q u\right)^{1/q} ≤ C \left(\int_{R^N_+} f^p v \right)^{1/p},\,\,0<q ,p <∞, 7817 for monotone functions $f≥ 0$ and nonnegative weights $u$ and $v$. The case $q < p$ is new and the case $0<p≤ q <∞$ is extended to a modular inequality with N- functions. A remarkable fact concerning the calculation of $C$ is pointed out.