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Weighted modular inequalities for Hardy type operators. (English) Zbl 1030.46030

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 \(C\).

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
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