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Fractional Integration on Mixed Norm Spaces. I

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Abstract

In this paper we characterize completely the septuple

$$\begin{aligned} (p_1, p_2, q_1, q_2; \alpha _1, \alpha _2; t) \in (0, \infty ]^4 \times (0, \infty )^2 \times {\mathbb {C}} \end{aligned}$$

such that the fractional integration operator \({\mathfrak {I}}_t\), of order \(t \in {\mathbb {C}}\), is bounded between two mixed norm spaces:

$$\begin{aligned} {\mathfrak {I}}_t: H(p_1, q_1, \alpha _1) \rightarrow H(p_2, q_2, \alpha _2). \end{aligned}$$

We treat three types of definitions for \({\mathfrak {I}}_t\): 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 \(H^p({\mathbb {D}})\), of type Riemann-Liouville, is due to Hardy and Littlewood in 1932.

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Acknowledgements

X. Fang is supported by Ministry of Science and Technology (Taiwan) (MOST) (108-2628-M-008-003-MY4). S. Hou is supported by National Natural Science Foundation (NNSF) of China (Grant No. 11971340).

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Authors and Affiliations

  1. School of Mathematics Science, Soochow University, Suzhou, 215006, People’s Republic of China

    Feng Guo, Shengzhao Hou & Xiaolin Zhu

  2. Department of Mathematics, National Central University, Chungli, Taiwan R.O.C.

    Xiang Fang

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Communicated by H. Turgay Kaptanoglu.

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Guo, F., Fang, X., Hou, S. et al. Fractional Integration on Mixed Norm Spaces. I. Complex Anal. Oper. Theory 18, 45 (2024). https://doi.org/10.1007/s11785-024-01488-3

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