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integral inequalities; monotone functions; several variables; modular functions; convex functions; weakly convex functions; weighted $L^p$ spaces
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.
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