# Translations of functions iv vector Hardy classes on the unit disk

- Publisher: Instytut Matematyczny Polskiej Akademi Nauk(Warszawa), 1996

## Access Full Book

top## Abstract

top## How to cite

topMichalak Artur. Translations of functions iv vector Hardy classes on the unit disk. Warszawa: Instytut Matematyczny Polskiej Akademi Nauk, 1996. <http://eudml.org/doc/270065>.

@book{MichalakArtur1996,

abstract = {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...................................................................................................771991 Mathematics Subject Classification: Primary 46E40, 46E27, 46B20, 46B22, 46G10; Secondary 46B03, 28C10, 46J15, 32A35},

author = {Michalak Artur},

keywords = {Hardy spaces of holomorphic vector functions; Hardy spaces of harmonic vector functions; spaces of vector measures; Radon-Nikodym property; analytic Radon-Nikodym property; Pettis integral; uncomplemented subspace; translations of vector measures on compact abelian groups; translation mappings; Radon-Nikodým property; analytic Radon-Nikodým property; topological and geometric structure; Hardy classes; rotation},

language = {eng},

location = {Warszawa},

publisher = {Instytut Matematyczny Polskiej Akademi Nauk},

title = {Translations of functions iv vector Hardy classes on the unit disk},

url = {http://eudml.org/doc/270065},

year = {1996},

}

TY - BOOK

AU - Michalak Artur

TI - Translations of functions iv vector Hardy classes on the unit disk

PY - 1996

CY - Warszawa

PB - Instytut Matematyczny Polskiej Akademi Nauk

AB - 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...................................................................................................771991 Mathematics Subject Classification: Primary 46E40, 46E27, 46B20, 46B22, 46G10; Secondary 46B03, 28C10, 46J15, 32A35

LA - eng

KW - Hardy spaces of holomorphic vector functions; Hardy spaces of harmonic vector functions; spaces of vector measures; Radon-Nikodym property; analytic Radon-Nikodym property; Pettis integral; uncomplemented subspace; translations of vector measures on compact abelian groups; translation mappings; Radon-Nikodým property; analytic Radon-Nikodým property; topological and geometric structure; Hardy classes; rotation

UR - http://eudml.org/doc/270065

ER -

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.