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On a general weighted Hardy type inequality in the variable exponent Lebesgue spaces. (English) Zbl 1284.42053

Authors’ abstract: “We study the Hardy type, two-weight inequality for the multidimensional Hardy operator in the variable exponent Lebesgue space \(L^{p(\cdot)}(\mathbb R^n)\). We prove equivalent conditions for \(L^{p(\cdot)}\to L^{q(\cdot)}\) boundness of the Hardy operator in the case of so called “mixed” exponents: \(q(0)\geq p(0)\), \(q(\infty)<p(\infty)\) or \(q(0)<p(0)\), \(q(\infty)\geq p(\infty)\). We show that a necessary and sufficient condition for such an inequality to hold coincides with conditions for the validity of two weight Hardy inequalities with constant exponents, provided that the exponents are regular at zero and at infinity.”

MSC:

42B25 Maximal functions, Littlewood-Paley theory
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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