On weighted multidimensional embeddings for monotone functions. (English) Zbl 1075.46507

Barza, Sorina; Persson, Lars-Erik; Stepanov, Vladimir D.

Math. Scand. 88, No. 2, 303-319 (2001).

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.

Reviewer: Aleksander Waszak (Poznań)

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