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Hardy type operators in local vanishing Morrey spaces on fractal sets. (English) Zbl 1345.46023

Summary: We obtain two-weight estimates for the Hardy type operators from local generalized Morrey spaces \(\mathcal L^{p,\phi}_{\mathrm{loc}} (X,w_{1})\) defined on an arbitrary underlying quasi-metric measure space \((X, \mu, \varrho)\) with the growth condition, to \(\mathcal L^{q,\psi}_{\mathrm{loc}}(X,w_{2})\), where \(w_{1} = w_{1}[\varrho(x, x_{0})]\), \(x_{0} \in X\), is a weight of radial type, while \(w_{2} = w_{2}(x)\) may be an arbitrary weight. The proof allows to simultaneously treat a similar boundedness \(V \mathcal L^{p,\phi}_{\mathrm{loc}} (X,w_{1}) \to V \mathcal L^{q,\psi}_{\mathrm{loc}}(X,w_{2})\) for vanishing Morrey spaces. We obtain sufficient conditions for such estimates in terms of some integral inequalities imposed on \(\phi\), \(\psi\) and \(w_{1}.w_{2}\). We also specially treat the one weight case where \(w_{2}(x)\) is also of radial type. We do not impose doubling condition on the measure \(\mu\), but base our result on the growth condition. { }The obtained results show the explicit dependence of the mapping properties of the Hardy type operators on the fractional dimension of the set \((X, \mu, \varrho)\). An application to spherical Hardy type operators is also given.

MSC:

46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
42B35 Function spaces arising in harmonic analysis
42B25 Maximal functions, Littlewood-Paley theory
47B38 Linear operators on function spaces (general)
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