Fractional Integration on Mixed Norm Spaces. I

Feng Guo; Xiang Fang; Shengzhao Hou; Xiaolin Zhu

doi:10.1007/s11785-024-01488-3

Fractional Integration on Mixed Norm Spaces. I

Feng Guo, Xiang Fang, Shengzhao Hou, Xiaolin Zhu

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

In this paper we characterize completely the septuple (Formula presented.) such that the fractional integration operator I_t, of order t∈C, is bounded between two mixed norm spaces: (Formula presented.) We treat three types of definitions for I_t: Hadamard, Flett, and Riemann-Liouville. Our main result (Theorem 2) extends that of Buckley-Koskela-Vukotić in 1999 on the Bergman spaces (Theorem B), and the case t=0 recovers the embedding theorem of Arévalo in 2015 (Corollary 3). The corresponding result for the Hardy spaces H^p(D), of type Riemann-Liouville, is due to Hardy and Littlewood in 1932.

Original language	English
Article number	45
Journal	Complex Analysis and Operator Theory
Volume	18
Issue number	3
DOIs	https://doi.org/10.1007/s11785-024-01488-3
State	Published - Apr 2024

Keywords

26A33
47B38
Fractional integration
Mixed norm space
Riemann-Liouville

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10.1007/s11785-024-01488-3

Cite this

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abstract = "In this paper we characterize completely the septuple (Formula presented.) such that the fractional integration operator It, of order t∈C, is bounded between two mixed norm spaces: (Formula presented.) We treat three types of definitions for It: Hadamard, Flett, and Riemann-Liouville. Our main result (Theorem 2) extends that of Buckley-Koskela-Vukoti{\'c} in 1999 on the Bergman spaces (Theorem B), and the case t=0 recovers the embedding theorem of Ar{\'e}valo in 2015 (Corollary 3). The corresponding result for the Hardy spaces Hp(D), of type Riemann-Liouville, is due to Hardy and Littlewood in 1932.",

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AU - Guo, Feng

AU - Fang, Xiang

AU - Hou, Shengzhao

AU - Zhu, Xiaolin

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PY - 2024/4

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N2 - In this paper we characterize completely the septuple (Formula presented.) such that the fractional integration operator It, of order t∈C, is bounded between two mixed norm spaces: (Formula presented.) We treat three types of definitions for It: Hadamard, Flett, and Riemann-Liouville. Our main result (Theorem 2) extends that of Buckley-Koskela-Vukotić in 1999 on the Bergman spaces (Theorem B), and the case t=0 recovers the embedding theorem of Arévalo in 2015 (Corollary 3). The corresponding result for the Hardy spaces Hp(D), of type Riemann-Liouville, is due to Hardy and Littlewood in 1932.

AB - In this paper we characterize completely the septuple (Formula presented.) such that the fractional integration operator It, of order t∈C, is bounded between two mixed norm spaces: (Formula presented.) We treat three types of definitions for It: Hadamard, Flett, and Riemann-Liouville. Our main result (Theorem 2) extends that of Buckley-Koskela-Vukotić in 1999 on the Bergman spaces (Theorem B), and the case t=0 recovers the embedding theorem of Arévalo in 2015 (Corollary 3). The corresponding result for the Hardy spaces Hp(D), of type Riemann-Liouville, is due to Hardy and Littlewood in 1932.

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VL - 18

JO - Complex Analysis and Operator Theory

JF - Complex Analysis and Operator Theory

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Fractional Integration on Mixed Norm Spaces. I

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Keywords

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