Weighted multidimensional inequalities for monotone functions. (English) Zbl 0942.26028

Barza, Sorina; Persson, Lars-Erik

Math. Bohem. 124, No. 2-3, 329-335 (1999).

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

Keywords:

integral inequalities; monotone functions; several variables; weighted \(L^p\) spaces; modular functions; convex functions; weakly convex functions