Abstract: We discuss the characterization of the inequality $$ \biggl(\int_{{\Bbb R}^N_+} f^q u\biggr)^{1/q} \leq C \biggl(\int_{{\Bbb R}^N_+} f^p v \biggr)^{1/p},\quad0<q, p <\infty, $$ for monotone functions $f\geq0$ and nonnegative weights $u$ and $v$ and $N\geq1$. We prove a new multidimensional integral modular inequality for monotone functions. This inequality generalizes and unifies some recent results in one and several dimensions.
Keywords: integral inequalities, monotone functions, several variables, weighted $L^p$ spaces, modular functions, convex functions, weakly convex functions
Classification (MSC2000): 26D15, 26B99
Full text of the article: