Multilinear operator-valued Calderón-Zygmund theory
(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
- Di Plinio, Francesco ; Li, Kangwei ; Martikainen, Henri and Vuorinen, Emil LU
- organization
- publishing date
- 2020-11-01
- 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}}, }