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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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