Spaces of analytic functions with integral norm.

*(English)*Zbl 1061.46021
Johnson, W. B. (ed.) et al., Handbook of the geometry of Banach spaces. Volume 2. Amsterdam: North-Holland (ISBN 0-444-51305-1/hbk). 1671-1702 (2003).

The main species among those mentioned in the title are Bergman and Hardy spaces, and the survey is confined to the information about these classes of functions from the point of view of a Banach space analyst. After showing that the Bergman space \(B_p (D)\) embeds in \(l^p\) for an arbitrary domain \(D\) in \({\mathbb C}^n\), the author discusses the reproducing kernels and the continuity of the Bergman and related projections for the case of the ball and the polydisc (then \(B_p\) is isomorphic to \(l^p\)).

The remaining part of the paper is devoted almost entirely to various aspects of the theory of \(H^p\)-spaces, specifically: classical results of the one-dimensional theory (boundary values, Poisson integrals, canonical factorization, isomorphism to \(L^p\) for \(1<p<\infty\)); coefficient multipliers on \(H^1\) (the theorems by Fefferman, Paley, and Stein); composition operators; duality between \(H^1\) and \(BMO\), atomic decompositions; a wavelet unconditional basis in \(H^1\); dyadic \(H^1\) and Müller’s theorem about isomorphic types of \(H^1\) on a space of homogeneous type; isomorphism of \(H^1\) on a complex ball to the dyadic \(H^1\); nonisomorphism of the spaces \(H^1\) on polydiscs; linear topological and metric questions about \(H^1\).

Proofs are presented in some cases; otherwise, references and hints are given.

For the entire collection see [Zbl 1013.46001].

The remaining part of the paper is devoted almost entirely to various aspects of the theory of \(H^p\)-spaces, specifically: classical results of the one-dimensional theory (boundary values, Poisson integrals, canonical factorization, isomorphism to \(L^p\) for \(1<p<\infty\)); coefficient multipliers on \(H^1\) (the theorems by Fefferman, Paley, and Stein); composition operators; duality between \(H^1\) and \(BMO\), atomic decompositions; a wavelet unconditional basis in \(H^1\); dyadic \(H^1\) and Müller’s theorem about isomorphic types of \(H^1\) on a space of homogeneous type; isomorphism of \(H^1\) on a complex ball to the dyadic \(H^1\); nonisomorphism of the spaces \(H^1\) on polydiscs; linear topological and metric questions about \(H^1\).

Proofs are presented in some cases; otherwise, references and hints are given.

For the entire collection see [Zbl 1013.46001].

Reviewer: Sergej V. Kislyakov (St. Peterburg)

##### MSC:

46E15 | Banach spaces of continuous, differentiable or analytic functions |

30H05 | Spaces of bounded analytic functions of one complex variable |

46B15 | Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces |

46-00 | General reference works (handbooks, dictionaries, bibliographies, etc.) pertaining to functional analysis |