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FRACTIONAL FOCK–SOBOLEV SPACES

Part of: Harmonic analysis in several variables Spaces and algebras of analytic functions Real functions Holomorphic functions of several complex variables

Published online by Cambridge University Press:  06 March 2018

HONG RAE CHO  and
SOOHYUN PARK
HONG RAE CHO
Affiliation:
Department of Mathematics, Pusan National University, Pusan 609-735, Republic of Korea email chohr@pusan.ac.kr
SOOHYUN PARK
Affiliation:
Department of Mathematics, Pusan National University, Pusan 609-735, Republic of Korea email shpark7@pusan.ac.kr
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Abstract

Let $s\in \mathbb{R}$ and $0<p\leqslant \infty$. The fractional Fock–Sobolev spaces $F_{\mathscr{R}}^{s,p}$ are introduced through the fractional radial derivatives $\mathscr{R}^{s/2}$. We describe explicitly the reproducing kernels for the fractional Fock–Sobolev spaces $F_{\mathscr{R}}^{s,2}$ and then get the pointwise size estimate of the reproducing kernels. By using the estimate, we prove that the fractional Fock–Sobolev spaces $F_{\mathscr{R}}^{s,p}$ are identified with the weighted Fock spaces $F_{s}^{p}$ that do not involve derivatives. So, the study on the Fock–Sobolev spaces is reduced to that on the weighted Fock spaces.

MSC classification

Primary: 30H20: Bergman spaces, Fock spaces 32A25: Integral representations; canonical kernels (Szegő, Bergman, etc.)
Secondary: 26A33: Fractional derivatives and integrals 42B35: Function spaces arising in harmonic analysis
Type
Article
Information
Nagoya Mathematical Journal , Volume 237 , March 2020 , pp. 79 - 97
Copyright
© 2018 Foundation Nagoya Mathematical Journal  

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Footnotes

The author was supported by NRF of Korea (NRF-2016R1D1A1B03933740).

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