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Vectorial Hankel operators, Carleson embeddings, and notions of BMOA

Rydhe, Eskil LU (2017) In Geometric and Functional Analysis 27(2). p.427-451
Abstract

Let (Formula presented.) denote the space of (Formula presented.)-valued analytic functions (Formula presented.) for which the Hankel operator (Formula presented.) is (Formula presented.)-bounded. Obtaining concrete characterizations of (Formula presented.) has proven to be notoriously hard. Let (Formula presented.) denote fractional differentiation. Motivated originally by control theory, we characterize (Formula presented.)-boundedness of (Formula presented.), where (Formula presented.), in terms of a natural anti-analytic Carleson embedding condition. We obtain three notable corollaries: The first is that (Formula presented.) is not characterized by said embedding condition. The second is that when we add an adjoint embedding... (More)

Let (Formula presented.) denote the space of (Formula presented.)-valued analytic functions (Formula presented.) for which the Hankel operator (Formula presented.) is (Formula presented.)-bounded. Obtaining concrete characterizations of (Formula presented.) has proven to be notoriously hard. Let (Formula presented.) denote fractional differentiation. Motivated originally by control theory, we characterize (Formula presented.)-boundedness of (Formula presented.), where (Formula presented.), in terms of a natural anti-analytic Carleson embedding condition. We obtain three notable corollaries: The first is that (Formula presented.) is not characterized by said embedding condition. The second is that when we add an adjoint embedding condition, we obtain a sufficient but not necessary condition for boundedness of (Formula presented.). The third is that there exists a bounded analytic function for which the associated anti-analytic Carleson embedding is unbounded. As a consequence, boundedness of an analytic Carleson embedding does not imply that the anti-analytic ditto is bounded. This answers a question by Nazarov, Pisier, Treil, and Volberg.

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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Geometric and Functional Analysis
volume
27
issue
2
pages
427 - 451
publisher
Birkhauser Verlag Basel
external identifiers
  • scopus:85014529498
  • wos:000398727300005
ISSN
1016-443X
DOI
10.1007/s00039-017-0400-4
language
English
LU publication?
yes
id
d2012cd0-f443-4981-af5c-3b2570c282bf
date added to LUP
2017-03-14 12:01:03
date last changed
2018-01-07 11:55:14
@article{d2012cd0-f443-4981-af5c-3b2570c282bf,
  abstract     = {<p>Let (Formula presented.) denote the space of (Formula presented.)-valued analytic functions (Formula presented.) for which the Hankel operator (Formula presented.) is (Formula presented.)-bounded. Obtaining concrete characterizations of (Formula presented.) has proven to be notoriously hard. Let (Formula presented.) denote fractional differentiation. Motivated originally by control theory, we characterize (Formula presented.)-boundedness of (Formula presented.), where (Formula presented.), in terms of a natural anti-analytic Carleson embedding condition. We obtain three notable corollaries: The first is that (Formula presented.) is not characterized by said embedding condition. The second is that when we add an adjoint embedding condition, we obtain a sufficient but not necessary condition for boundedness of (Formula presented.). The third is that there exists a bounded analytic function for which the associated anti-analytic Carleson embedding is unbounded. As a consequence, boundedness of an analytic Carleson embedding does not imply that the anti-analytic ditto is bounded. This answers a question by Nazarov, Pisier, Treil, and Volberg.</p>},
  author       = {Rydhe, Eskil},
  issn         = {1016-443X},
  language     = {eng},
  number       = {2},
  pages        = {427--451},
  publisher    = {Birkhauser Verlag Basel},
  series       = {Geometric and Functional Analysis},
  title        = {Vectorial Hankel operators, Carleson embeddings, and notions of BMOA},
  url          = {http://dx.doi.org/10.1007/s00039-017-0400-4},
  volume       = {27},
  year         = {2017},
}