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Multilinear operator-valued Calderón-Zygmund theory

Di Plinio, Francesco ; Li, Kangwei ; Martikainen, Henri and Vuorinen, Emil LU (2020) In Journal of Functional Analysis 279(8).
Abstract

We develop a general theory of multilinear singular integrals with operator-valued kernels, acting on tuples of UMD Banach spaces. This, in particular, involves investigating multilinear variants of the R-boundedness condition naturally arising in operator-valued theory. We proceed by establishing a suitable representation of multilinear, operator-valued singular integrals in terms of operator-valued dyadic shifts and paraproducts, and studying the boundedness of these model operators via dyadic-probabilistic Banach space-valued analysis. In the bilinear case, we obtain a T(1)-type theorem without any additional assumptions on the Banach spaces other than the necessary UMD. Higher degrees of multilinearity are tackled via a new... (More)

We develop a general theory of multilinear singular integrals with operator-valued kernels, acting on tuples of UMD Banach spaces. This, in particular, involves investigating multilinear variants of the R-boundedness condition naturally arising in operator-valued theory. We proceed by establishing a suitable representation of multilinear, operator-valued singular integrals in terms of operator-valued dyadic shifts and paraproducts, and studying the boundedness of these model operators via dyadic-probabilistic Banach space-valued analysis. In the bilinear case, we obtain a T(1)-type theorem without any additional assumptions on the Banach spaces other than the necessary UMD. Higher degrees of multilinearity are tackled via a new formulation of the Rademacher maximal function (RMF) condition. In addition to the natural UMD lattice cases, our RMF condition covers suitable tuples of non-commutative Lp-spaces. We employ our operator-valued theory to obtain new multilinear, multi-parameter, operator-valued theorems in the natural setting of UMD spaces with property α.

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author
; ; and
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Calderón–Zygmund operators, Multilinear analysis, Operator-valued analysis, UMD spaces
in
Journal of Functional Analysis
volume
279
issue
8
article number
108666
publisher
Elsevier
external identifiers
  • scopus:85085743741
ISSN
0022-1236
DOI
10.1016/j.jfa.2020.108666
language
English
LU publication?
yes
id
3bb54128-5208-4d89-83e1-15a5543d29c2
date added to LUP
2021-01-15 08:23:49
date last changed
2022-04-26 23:34:59
@article{3bb54128-5208-4d89-83e1-15a5543d29c2,
  abstract     = {{<p>We develop a general theory of multilinear singular integrals with operator-valued kernels, acting on tuples of UMD Banach spaces. This, in particular, involves investigating multilinear variants of the R-boundedness condition naturally arising in operator-valued theory. We proceed by establishing a suitable representation of multilinear, operator-valued singular integrals in terms of operator-valued dyadic shifts and paraproducts, and studying the boundedness of these model operators via dyadic-probabilistic Banach space-valued analysis. In the bilinear case, we obtain a T(1)-type theorem without any additional assumptions on the Banach spaces other than the necessary UMD. Higher degrees of multilinearity are tackled via a new formulation of the Rademacher maximal function (RMF) condition. In addition to the natural UMD lattice cases, our RMF condition covers suitable tuples of non-commutative L<sup>p</sup>-spaces. We employ our operator-valued theory to obtain new multilinear, multi-parameter, operator-valued theorems in the natural setting of UMD spaces with property α.</p>}},
  author       = {{Di Plinio, Francesco and Li, Kangwei and Martikainen, Henri and Vuorinen, Emil}},
  issn         = {{0022-1236}},
  keywords     = {{Calderón–Zygmund operators; Multilinear analysis; Operator-valued analysis; UMD spaces}},
  language     = {{eng}},
  month        = {{11}},
  number       = {{8}},
  publisher    = {{Elsevier}},
  series       = {{Journal of Functional Analysis}},
  title        = {{Multilinear operator-valued Calderón-Zygmund theory}},
  url          = {{http://dx.doi.org/10.1016/j.jfa.2020.108666}},
  doi          = {{10.1016/j.jfa.2020.108666}},
  volume       = {{279}},
  year         = {{2020}},
}