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On a Muckenhoupt-type condition for Morrey spaces
Luleå University of Technology, Department of Engineering Sciences and Mathematics, Mathematical Science.ORCID iD: 0000-0002-8595-4326
2013 (English)In: Mediterranean Journal of Mathematics, ISSN 1660-5446, E-ISSN 1660-5454, Vol. 10, no 2, p. 941-951Article in journal (Refereed) Published
Abstract [en]

As is known, the class of weights for Morrey type spaces {Mathematical expression} for which the maximal and/or singular operators are bounded, is different from the known Muckenhoupt class A p of such weights for the Lebesgue spaces L p(Ω). For instance, in the case of power weights {Mathematical expression}, {Mathematical expression}, the singular operator (Hilbert transform) is bounded in {Mathematical expression}, if and only if -1 < ν < p - 1, while it is bounded in the Morrey space {Mathematical expression}, if and only if the exponent α runs the shifted interval λ - 1 < ν < λ + p - 1. A description of all the admissible weights similar to the Muckenhoupt class A p is an open problem. In this paper, for the one-dimensional case, we introduce the class A p,λ of weights, which turns into the Muckenhoupt class A p when λ = 0 and show that the belongness of a weight to A p,λ is necessary for the boundedness, in Morrey spaces, of the Hilbert transform in the one-dimensional case. In the case n > 1 we also provide some λ-dependent á priori assumptions on weights and give some estimates of weighted norms {Mathematical expression} of the characteristic functions of balls

Place, publisher, year, edition, pages
2013. Vol. 10, no 2, p. 941-951
National Category
Mathematical Analysis
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:ltu:diva-8843DOI: 10.1007/s00009-012-0208-2ISI: 000318250000019Scopus ID: 2-s2.0-84876686676Local ID: 764b4a1c-a9f3-4555-9871-e4dd4ac286e3OAI: oai:DiVA.org:ltu-8843DiVA, id: diva2:981781
Note
Validerad; 2013; 20120628 (andbra)Available from: 2016-09-29 Created: 2016-09-29 Last updated: 2018-07-10Bibliographically approved

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Samko, Natasha

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