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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.
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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