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On the Characterization of Triebel–Lizorkin Type Spaces of Analytic Functions

Rydhe, Eskil LU (2017) In Journal of Fourier Analysis and Applications
Abstract

We consider different characterizations of Triebel–Lizorkin type spaces of analytic functions on the unit disc. Even though our results appear in the folklore, detailed descriptions are hard to find, and in fact we are unable to discuss the full range of parameters. Without additional effort we work with vector-valued analytic functions, and also consider a generalized scale of function spaces, including for example so-called Q-spaces. The primary aim of this note is to generalize, and clarify, a remarkable result by Cohn and Verbitsky, on factorization of Triebel–Lizorkin spaces. Their result remains valid for functions taking values in an arbitrary Banach space, provided that the vector-valuedness “sits in the right factor”. On the... (More)

We consider different characterizations of Triebel–Lizorkin type spaces of analytic functions on the unit disc. Even though our results appear in the folklore, detailed descriptions are hard to find, and in fact we are unable to discuss the full range of parameters. Without additional effort we work with vector-valued analytic functions, and also consider a generalized scale of function spaces, including for example so-called Q-spaces. The primary aim of this note is to generalize, and clarify, a remarkable result by Cohn and Verbitsky, on factorization of Triebel–Lizorkin spaces. Their result remains valid for functions taking values in an arbitrary Banach space, provided that the vector-valuedness “sits in the right factor”. On the other hand, if we impose vector-valuedness on the “wrong” factor, then the factorization theorem fails even for functions taking values in a separable Hilbert space.

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author
organization
publishing date
type
Contribution to journal
publication status
epub
subject
keywords
Factorization, Q-spaces, Triebel–Lizorkin spaces, Vector-valued analytic functions
in
Journal of Fourier Analysis and Applications
pages
27 pages
publisher
Springer
external identifiers
  • scopus:85036498420
ISSN
1069-5869
DOI
10.1007/s00041-017-9584-0
language
English
LU publication?
yes
id
bd3a4680-7263-4b01-b17b-033df6a7b7d9
date added to LUP
2017-12-18 08:42:14
date last changed
2018-01-07 12:29:04
@article{bd3a4680-7263-4b01-b17b-033df6a7b7d9,
  abstract     = {<p>We consider different characterizations of Triebel–Lizorkin type spaces of analytic functions on the unit disc. Even though our results appear in the folklore, detailed descriptions are hard to find, and in fact we are unable to discuss the full range of parameters. Without additional effort we work with vector-valued analytic functions, and also consider a generalized scale of function spaces, including for example so-called Q-spaces. The primary aim of this note is to generalize, and clarify, a remarkable result by Cohn and Verbitsky, on factorization of Triebel–Lizorkin spaces. Their result remains valid for functions taking values in an arbitrary Banach space, provided that the vector-valuedness “sits in the right factor”. On the other hand, if we impose vector-valuedness on the “wrong” factor, then the factorization theorem fails even for functions taking values in a separable Hilbert space.</p>},
  author       = {Rydhe, Eskil},
  issn         = {1069-5869},
  keyword      = {Factorization,Q-spaces,Triebel–Lizorkin spaces,Vector-valued analytic functions},
  language     = {eng},
  month        = {12},
  pages        = {27},
  publisher    = {Springer},
  series       = {Journal of Fourier Analysis and Applications},
  title        = {On the Characterization of Triebel–Lizorkin Type Spaces of Analytic Functions},
  url          = {http://dx.doi.org/10.1007/s00041-017-9584-0},
  year         = {2017},
}