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Weighted multidimensional inequalities for monotone functions. (English) Zbl 0942.26028
Summary: We discuss the characterization of the inequality $\Big (\int _{{\mathbb R}^N_+} f^q u\Big)^{1/q} \leq C \Big (\int _{{\mathbb R}^N_+} f^p v \Big)^{1/p},\quad 0<q, p <\infty ,$ for monotone functions $$f\geq 0$$ and nonnegative weights $$u$$ and $$v$$ and $$N\geq 1$$. We prove a new multidimensional integral modular inequality for monotone functions. This inequality generalizes and unifies some recent results in one and several dimensions.
##### MSC:
 26D15 Inequalities for sums, series and integrals 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 26A48 Monotonic functions, generalizations 26B25 Convexity of real functions of several variables, generalizations
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