Translations of functions iv vector Hardy classes on the unit disk

Michalak Artur

GDML_Books, (1996), p.

Résumé

AbstractThe paper contains studies of relationships between properties of the “translation” mappings $T_F$ and the topological and geometric structure of spaces X and Hardy classes $h^p(,X)$ of X-valued harmonic functions on the open unit disk in ℂ (X is a Banach space). The mapping $T_F$ transforming the unit circle of ℂ into $h^p(,X)$ is associated with a function $F ∈ h^p(,X)$ by the formula $T_F(t) = F ∘ ϕₜ$, where ϕₜ is the rotation of through t.AcknowledgmentsThis work is based in part on the author’s doctoral thesis written at the Institute of Mathematics of the Polish Academy of Sciences under the supervision of Professor Lech Drewnowski. I wish to thank Professor Z. Lipecki for bringing the paper [I-M] to my attention, and Professor P. Wojtaszczyk for his remarks about my doctoral thesis. I would particularly like to thank Professor L. Drewnowski whose remarks allowed me to improve the paper. This research was supported in part by Komitet Badań Naukowych (State Committee for Scientific Research), Poland, grant no. 2 P301 003 07.

CONTENTS   Introduction...................................................................................................50. Preliminaries................................................................................................71. Fundamental properties of harmonic vector functions...............................132. Hardy spaces of vector functions...............................................................15   Relations between scalar and vector Hardy classes...................................15   The factorization theorem for $H^p(,X)$...................................................19   Nontangential limits of functions in $h^p(,X)$...........................................22   Properties of functions in $h^p(,X)$..........................................................273. Spaces $h^p(,X)$ and $M_p(,X)$..........................................................294. The sets of translates of harmonic functions..............................................335. Translations of functions from Hardy classes..............................................376. Translations of functions from Smirnov classes...........................................417. Translations of measures from $M_p(G,X)$................................................438. A criterion of uncomplementability of $L^p(λ_G,X)$ in $M_p(G,X)$.............539. Pettis integrability of the translation function for vector measures...............64   References...................................................................................................77

1991 Mathematics Subject Classification: Primary 46E40, 46E27, 46B20, 46B22, 46G10; Secondary 46B03, 28C10, 46J15, 32A35

Détail

Zbl 0872.46023

EUDML-ID : urn:eudml:doc:270065