Lai, Qinsheng Weighted modular inequalities for Hardy type operators. (English) Zbl 1030.46030 Proc. Lond. Math. Soc., III. Ser. 79, No. 3, 649-672 (1999). Let \(T\) be a Hardy type operator \(Tf(x)= \int^x_0 K(x,y) f(y)dy\), \(x\geq 0\). Let \(P\) be a Young function and let \(Q:\mathbb{R}^+ \to\mathbb{R}^+\) be strictly increasing with \(Q(0)=0\), \(Q(\infty) =\infty\), satisfying the condition: there is a constant \(M\) such that \(2Q(x)\leq Q(Mx)\) for all \(x\geq 0\). Let \(\Theta,w, \rho,v\) be weight functions. Necessary and sufficient conditions are obtained in order that the following weighted modular inequality \[ Q^{-1}\left( \int^\infty_0 Q\bigl(\theta(x) Tf(x)\bigr) w(x)dx\right)\leq P^{-1}\left (\int^\infty_0 P\bigl(C\rho (x)f(x)\bigr) v(x)dx\right), \] where \(f\geq 0\), be true with some constant \(C\). Reviewer: J.Musielak (Poznań) Cited in 24 Documents MSC: 46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 42B25 Maximal functions, Littlewood-Paley theory 26D15 Inequalities for sums, series and integrals Keywords:Orlicz space; Hardy type operator; weighted modular inequality PDFBibTeX XMLCite \textit{Q. Lai}, Proc. Lond. Math. Soc. (3) 79, No. 3, 649--672 (1999; Zbl 1030.46030) Full Text: DOI