Norms of maximal functions between generalized and classical Lorentz spaces

Mustafayev, Rza; Bilgiçli, Nevin; Yılmaz, Merve

doi:10.48550/arxiv.2110.13698

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2023

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“…Note that various variants of the generalized fractional-maximal function were previously considered in [9][10][11][12][13][14][15]. For such variants of a generalized fractional-maximal function, the questions of boundedness in Lorentz spaces were considered in [8], [11].…”

Section: Introductionmentioning

confidence: 99%

On the Boundedness of a Generalized Fractional-Maximal Operator in Lorentz Spaces

Abek,

Turgumbayev,

Suleimenova

2023

JMMCS

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In this paper considers a generalized fractional-maximal operator, a special case of which is the classical fractional-maximal function. Conditions for the function Φ, which defines a generalized fractional-maximal function, and for the weight functions w and v, which determine the weighted Lorentz spaces Λ p (v) and Λ q (w) (1 < p ≤ q < ∞) under which the generalized maximal-fractional operator is bounded from one Lorentz space Λ p (v) to another Lorentz space Λ q (w) are obtained. For the classical fractional maximal operator and the classical maximal Hardy-Littlewood function such results were previously known. When proving the main result, we make essential use of an estimate for a nonincreasing rearrangement of a generalized fractional-maximal operator. In addition, we introduce a supremal operator for which conditions of boundedness in weighted Lebesgue spaces are obtained. This result is also essentially used in the proof of the main theorem.

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