Hardy operators of monotone functions and sequences in Orlicz spaces. (English) Zbl 0853.42012

Let \(P\) be an \(N\)-function and \(L_{P(v)}\) the weighted Orlicz space, with weight \(v\) and norm \[ |f|_{P(v)}= \inf\Biggl\{ \lambda> 0: \int^\infty_0 P\Biggl[ {|f(x)|\over \lambda} \Biggr] v(x) dx\leq 1\Biggr\}. \] If \(G(x)= \int^x_0 g\), \(g\geq 0\), \(V(x)= \int^x_0 v\) and \(\widetilde P\) the complementary function of \(P\), then it is shown that for \(P\) and \(\widetilde P\) in \(\Delta_2\), \[ \sup_{0\leq f\downarrow} {\int^\infty_0 fg\over |f|_{P(v)}}\approx \Biggl|{G\over V}\Biggr|_{\widetilde P(v)}+ {\int^\infty_0 g\over |1|_{P(v)}}\tag{\(*\)} \] is satisfied. A result similar to \((*)\) for non-decreasing functions is also given. This extends to Orlicz spaces corresponding weighted \(L^p\)-results of E. T. Sawyer [Stud. Math. 96, No. 2, 145-158 (1990; Zbl 0705.42014)] and V. D. Stepanov [J. Lond. Math. Soc., II. Ser. 48, No. 3, 465-487 (1993; Zbl 0837.26011)].
In addition, discrete converse Hölder inequalities for weighted Orlicz sequence spaces on the cone of monotone functions are also obtained. The proofs depend on weighted Orlicz modular inequalities for the Hardy operator both in the continuous and discrete case.


42B25 Maximal functions, Littlewood-Paley theory
26D10 Inequalities involving derivatives and differential and integral operators
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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