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Multilinear singular integrals on non-commutative Lp spaces

Di Plinio, Francesco ; Li, Kangwei ; Martikainen, Henri and Vuorinen, Emil LU (2020) In Mathematische Annalen
Abstract

We prove Lp bounds for the extensions of standard multilinear Calderón–Zygmund operators to tuples of UMD spaces tied by a natural product structure. The product can, for instance, mean the pointwise product in UMD function lattices, or the composition of operators in the Schatten-von Neumann subclass of the algebra of bounded operators on a Hilbert space. We do not require additional assumptions beyond UMD on each space—in contrast to previous results, we e.g. show that the Rademacher maximal function property is not necessary. The obtained generality allows for novel applications. For instance, we prove new versions of fractional Leibniz rules via our results concerning the boundedness of multilinear singular integrals in... (More)

We prove Lp bounds for the extensions of standard multilinear Calderón–Zygmund operators to tuples of UMD spaces tied by a natural product structure. The product can, for instance, mean the pointwise product in UMD function lattices, or the composition of operators in the Schatten-von Neumann subclass of the algebra of bounded operators on a Hilbert space. We do not require additional assumptions beyond UMD on each space—in contrast to previous results, we e.g. show that the Rademacher maximal function property is not necessary. The obtained generality allows for novel applications. For instance, we prove new versions of fractional Leibniz rules via our results concerning the boundedness of multilinear singular integrals in non-commutative Lp spaces. Our proof techniques combine a novel scheme of induction on the multilinearity index with dyadic-probabilistic techniques in the UMD space setting.

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organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Calderón–Zygmund operators, Multilinear analysis, Non-commutative spaces, Representation theorems, Singular integrals, UMD spaces
in
Mathematische Annalen
publisher
Springer
external identifiers
  • scopus:85090243283
ISSN
0025-5831
DOI
10.1007/s00208-020-02068-4
language
English
LU publication?
yes
id
7ceb6896-7b83-4280-afa1-ddc01e6eb712
date added to LUP
2020-09-24 18:46:44
date last changed
2020-09-25 01:59:26
@article{7ceb6896-7b83-4280-afa1-ddc01e6eb712,
  abstract     = {<p>We prove L<sup>p</sup> bounds for the extensions of standard multilinear Calderón–Zygmund operators to tuples of UMD spaces tied by a natural product structure. The product can, for instance, mean the pointwise product in UMD function lattices, or the composition of operators in the Schatten-von Neumann subclass of the algebra of bounded operators on a Hilbert space. We do not require additional assumptions beyond UMD on each space—in contrast to previous results, we e.g. show that the Rademacher maximal function property is not necessary. The obtained generality allows for novel applications. For instance, we prove new versions of fractional Leibniz rules via our results concerning the boundedness of multilinear singular integrals in non-commutative L<sup>p</sup> spaces. Our proof techniques combine a novel scheme of induction on the multilinearity index with dyadic-probabilistic techniques in the UMD space setting.</p>},
  author       = {Di Plinio, Francesco and Li, Kangwei and Martikainen, Henri and Vuorinen, Emil},
  issn         = {0025-5831},
  language     = {eng},
  month        = {09},
  publisher    = {Springer},
  series       = {Mathematische Annalen},
  title        = {Multilinear singular integrals on non-commutative L<sup>p</sup> spaces},
  url          = {http://dx.doi.org/10.1007/s00208-020-02068-4},
  doi          = {10.1007/s00208-020-02068-4},
  year         = {2020},
}