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On weighted multidimensional embeddings for monotone functions. (English) Zbl 1075.46507
In this long and very interesting paper, the authors consider the inequality \[ \left(\int_{\mathbb R_+^N} f^q u \right)^{1/q} \leq C \left(\int_{\mathbb R_+^N} f^p v \right)^{1/p}, \tag{*} \] where \(0<p,q<\infty\) and the funktion \(f\) and the weights \(u,v\) are nonnegative. The authors investigate particular cases of the inequality (*) for various \(N,p,q,u,v\). The main results of this paper provides necessary and sufficient conditions in order that the inequality \[ \left(\int_{\mathbb R_+^N} f^q (x,y)u(xy)\,dx\,dy \right)^{1/q} \leq \left(\int_{\mathbb R_+^N} f^p (x,y)v(xy)\,dx\,dy \right)^{1/p} \] for \(0<q\leq p<\infty\) and the inequality (*) for \(0<p,q<\infty\), are valid. The paper contains interesting proofs of all the theorems.

MSC:
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
26D15 Inequalities for sums, series and integrals
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
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