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∗-Algebras in Several Complex Variables
Published in Kehe Zhu, Handbook of Analytic Operator Theory, 2019
The parameter shift (5.37) is compatible with the discrete Wallach points (5.29) and the special continuous Wallach points (5.31). In particular, the Hardy space for Ω is mapped to the Hardy space for Ωu. Thus Theorem 6.1 can be viewed as a special case of the parameter dependent Theorem 6.4. On the other hand, the standard Bergman space parameter v = 2 + a(r − 1) + b is not preserved, since the shifted parameter v′ = 2 + a(r − 1 − ℓ/2) + b exceeds the standard Bergman parameter 2 + a(r − ℓ − 1) + b for Ωu. This means that the image representation becomes ‘more classical’.
Hilbert function spaces
Published in Orr Moshe Shalit, A First Course in Functional Analysis, 2017
Example 6.1.3 (The Hardy space). Let D denote the open unit disc in the complex plane ℂ. Recall that every function f analytic in D has a power series representation f(z)=∑n=0∞anzn. We define the Hardy spaceH2(D) to be the space of analytic functions on the disc with square summable Taylor coefficients, that is H2(D)={f(z)=∑n=0∞anzn:∑|an|2<∞}.
Fourier transform of Hardy spaces associated with ball quasi-Banach function spaces*
Published in Applicable Analysis, 2022
Long Huang, Der-Chen Chang, Dachun Yang
On the other hand, recall that, in 2017, Sawano et al. [8] first introduced the ball quasi-Banach function space X and the associated Hardy space . In their article, by assuming the boundedness of the (powered) Hardy–Littlewood maximal function on the associated space of X and the Fefferman–Stein vector-valued maximal inequality on X, Sawano et al. established various maximal function characterizations and several other characterizations of , respectively, in terms of atoms, molecules, and Lusin-area functions. Indeed, the real-variable theory of the Hardy space , associated with the ball quasi-Banach function space X, provides an appropriate framework that unifies the theory of various types of Hardy spaces, which have been investigated before, like classical Hardy spaces, mixed-norm Hardy spaces, variable Hardy spaces, and Orlicz Hardy spaces. For more recent developments on this topic, we refer the reader to [8–17]. Based on the recent rapid developments of the theory of the Hardy space and the aforementioned works on the characterization of the Fourier transform of the classical Hardy space or their generalization, it is natural to ask whether or not (1) and (2) also hold true for the Hardy space . Giving a positive answer to this question is the main goal of this article.