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Reduction theorems for operators on the cones of monotone functions. (English) Zbl 1326.47063

Let \(\mathfrak M^+\) be the set of all non-negative measurable functions on the semiaxis \([0, \infty)\), and \(\mathfrak M^{\downarrow} \subset \mathfrak M^+\) be the subset of all non-increasing functions. The authors consider a wide class of positive quasi-linear operators on \(\mathfrak M^+\), including operators of the form \(f \mapsto \left( \int |k(\cdot, y) (Tf)(y)|^r\right)^{1/r}\), where \(T\) is a positive linear operator and \(k(x,y) \geq 0\). The paper deals with reduction of weighted \(L_p \to L_q\) type inequalities for positive quasi-linear operators on \(\mathfrak M^{\downarrow}\) to related inequalities on \(\mathfrak M^+\).

MSC:

47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
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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