---
_id: '13318'
abstract:
- lang: eng
  text: Bohnenblust–Hille inequalities for Boolean cubes have been proven with dimension-free
    constants that grow subexponentially in the degree (Defant et al. in Math Ann
    374(1):653–680, 2019). Such inequalities have found great applications in learning
    low-degree Boolean functions (Eskenazis and Ivanisvili in Proceedings of the 54th
    annual ACM SIGACT symposium on theory of computing, pp 203–207, 2022). Motivated
    by learning quantum observables, a qubit analogue of Bohnenblust–Hille inequality
    for Boolean cubes was recently conjectured in Rouzé et al. (Quantum Talagrand,
    KKL and Friedgut’s theorems and the learnability of quantum Boolean functions,
    2022. arXiv preprint arXiv:2209.07279). The conjecture was resolved in Huang et
    al. (Learning to predict arbitrary quantum processes, 2022. arXiv preprint arXiv:2210.14894).
    In this paper, we give a new proof of these Bohnenblust–Hille inequalities for
    qubit system with constants that are dimension-free and of exponential growth
    in the degree. As a consequence, we obtain a junta theorem for low-degree polynomials.
    Using similar ideas, we also study learning problems of low degree quantum observables
    and Bohr’s radius phenomenon on quantum Boolean cubes.
acknowledgement: The research of A.V. is supported by NSF DMS-1900286, DMS-2154402
  and by Hausdorff Center for Mathematics. H.Z. is supported by the Lise Meitner fellowship,
  Austrian Science Fund (FWF) M3337. This work is partially supported by NSF DMS-1929284
  while both authors were in residence at the Institute for Computational and Experimental
  Research in Mathematics in Providence, RI, during the Harmonic Analysis and Convexity
  program.
article_processing_charge: No
article_type: original
arxiv: 1
author:
- first_name: Alexander
  full_name: Volberg, Alexander
  last_name: Volberg
- first_name: Haonan
  full_name: Zhang, Haonan
  id: D8F41E38-9E66-11E9-A9E2-65C2E5697425
  last_name: Zhang
citation:
  ama: Volberg A, Zhang H. Noncommutative Bohnenblust–Hille inequalities. <i>Mathematische
    Annalen</i>. 2023. doi:<a href="https://doi.org/10.1007/s00208-023-02680-0">10.1007/s00208-023-02680-0</a>
  apa: Volberg, A., &#38; Zhang, H. (2023). Noncommutative Bohnenblust–Hille inequalities.
    <i>Mathematische Annalen</i>. Springer Nature. <a href="https://doi.org/10.1007/s00208-023-02680-0">https://doi.org/10.1007/s00208-023-02680-0</a>
  chicago: Volberg, Alexander, and Haonan Zhang. “Noncommutative Bohnenblust–Hille
    Inequalities.” <i>Mathematische Annalen</i>. Springer Nature, 2023. <a href="https://doi.org/10.1007/s00208-023-02680-0">https://doi.org/10.1007/s00208-023-02680-0</a>.
  ieee: A. Volberg and H. Zhang, “Noncommutative Bohnenblust–Hille inequalities,”
    <i>Mathematische Annalen</i>. Springer Nature, 2023.
  ista: Volberg A, Zhang H. 2023. Noncommutative Bohnenblust–Hille inequalities. Mathematische
    Annalen.
  mla: Volberg, Alexander, and Haonan Zhang. “Noncommutative Bohnenblust–Hille Inequalities.”
    <i>Mathematische Annalen</i>, Springer Nature, 2023, doi:<a href="https://doi.org/10.1007/s00208-023-02680-0">10.1007/s00208-023-02680-0</a>.
  short: A. Volberg, H. Zhang, Mathematische Annalen (2023).
date_created: 2023-07-30T22:01:03Z
date_published: 2023-07-24T00:00:00Z
date_updated: 2023-12-13T11:36:20Z
day: '24'
department:
- _id: JaMa
doi: 10.1007/s00208-023-02680-0
external_id:
  arxiv:
  - '2210.14468'
  isi:
  - '001035665500001'
isi: 1
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://doi.org/10.1007/s00208-023-02680-0
month: '07'
oa: 1
oa_version: Published Version
project:
- _id: eb958bca-77a9-11ec-83b8-c565cb50d8d6
  grant_number: M03337
  name: Curvature-dimension in noncommutative analysis
publication: Mathematische Annalen
publication_identifier:
  eissn:
  - 1432-1807
  issn:
  - 0025-5831
publication_status: epub_ahead
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Noncommutative Bohnenblust–Hille inequalities
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2023'
...
---
_id: '10588'
abstract:
- lang: eng
  text: We prove the Sobolev-to-Lipschitz property for metric measure spaces satisfying
    the quasi curvature-dimension condition recently introduced in Milman (Commun
    Pure Appl Math, to appear). We provide several applications to properties of the
    corresponding heat semigroup. In particular, under the additional assumption of
    infinitesimal Hilbertianity, we show the Varadhan short-time asymptotics for the
    heat semigroup with respect to the distance, and prove the irreducibility of the
    heat semigroup. These results apply in particular to large classes of (ideal)
    sub-Riemannian manifolds.
acknowledgement: "The authors are grateful to Dr. Bang-Xian Han for helpful discussions
  on the Sobolev-to-Lipschitz property on metric measure spaces, and to Professor
  Kazuhiro Kuwae, Professor Emanuel Milman, Dr. Giorgio Stefani, and Dr. Gioacchino
  Antonelli for reading a preliminary version of this work and for their valuable
  comments and suggestions. Finally, they wish to express their gratitude to two anonymous
  Reviewers whose suggestions improved the presentation of this work.\r\n\r\nL.D.S.
  gratefully acknowledges funding of his position by the Austrian Science Fund (FWF)
  grant F65, and by the European Research Council (ERC, grant No. 716117, awarded
  to Prof. Dr. Jan Maas).\r\n\r\nK.S. gratefully acknowledges funding by: the JSPS
  Overseas Research Fellowships, Grant Nr. 290142; World Premier International Research
  Center Initiative (WPI), MEXT, Japan; JSPS Grant-in-Aid for Scientific Research
  on Innovative Areas “Discrete Geometric Analysis for Materials Design”, Grant Number
  17H06465; and the Alexander von Humboldt Stiftung, Humboldt-Forschungsstipendium."
article_processing_charge: Yes (via OA deal)
article_type: original
arxiv: 1
author:
- first_name: Lorenzo
  full_name: Dello Schiavo, Lorenzo
  id: ECEBF480-9E4F-11EA-B557-B0823DDC885E
  last_name: Dello Schiavo
  orcid: 0000-0002-9881-6870
- first_name: Kohei
  full_name: Suzuki, Kohei
  last_name: Suzuki
citation:
  ama: Dello Schiavo L, Suzuki K. Sobolev-to-Lipschitz property on QCD- spaces and
    applications. <i>Mathematische Annalen</i>. 2022;384:1815-1832. doi:<a href="https://doi.org/10.1007/s00208-021-02331-2">10.1007/s00208-021-02331-2</a>
  apa: Dello Schiavo, L., &#38; Suzuki, K. (2022). Sobolev-to-Lipschitz property on
    QCD- spaces and applications. <i>Mathematische Annalen</i>. Springer Nature. <a
    href="https://doi.org/10.1007/s00208-021-02331-2">https://doi.org/10.1007/s00208-021-02331-2</a>
  chicago: Dello Schiavo, Lorenzo, and Kohei Suzuki. “Sobolev-to-Lipschitz Property
    on QCD- Spaces and Applications.” <i>Mathematische Annalen</i>. Springer Nature,
    2022. <a href="https://doi.org/10.1007/s00208-021-02331-2">https://doi.org/10.1007/s00208-021-02331-2</a>.
  ieee: L. Dello Schiavo and K. Suzuki, “Sobolev-to-Lipschitz property on QCD- spaces
    and applications,” <i>Mathematische Annalen</i>, vol. 384. Springer Nature, pp.
    1815–1832, 2022.
  ista: Dello Schiavo L, Suzuki K. 2022. Sobolev-to-Lipschitz property on QCD- spaces
    and applications. Mathematische Annalen. 384, 1815–1832.
  mla: Dello Schiavo, Lorenzo, and Kohei Suzuki. “Sobolev-to-Lipschitz Property on
    QCD- Spaces and Applications.” <i>Mathematische Annalen</i>, vol. 384, Springer
    Nature, 2022, pp. 1815–32, doi:<a href="https://doi.org/10.1007/s00208-021-02331-2">10.1007/s00208-021-02331-2</a>.
  short: L. Dello Schiavo, K. Suzuki, Mathematische Annalen 384 (2022) 1815–1832.
date_created: 2022-01-02T23:01:35Z
date_published: 2022-12-01T00:00:00Z
date_updated: 2023-08-02T13:39:05Z
day: '01'
ddc:
- '510'
department:
- _id: JaMa
doi: 10.1007/s00208-021-02331-2
ec_funded: 1
external_id:
  arxiv:
  - '2110.05137'
  isi:
  - '000734150200001'
file:
- access_level: open_access
  checksum: 2593abbf195e38efa93b6006b1e90eb1
  content_type: application/pdf
  creator: alisjak
  date_created: 2022-01-03T11:08:31Z
  date_updated: 2022-01-03T11:08:31Z
  file_id: '10596'
  file_name: 2021_MathAnn_DelloSchiavo.pdf
  file_size: 410090
  relation: main_file
  success: 1
file_date_updated: 2022-01-03T11:08:31Z
has_accepted_license: '1'
intvolume: '       384'
isi: 1
keyword:
- quasi curvature-dimension condition
- sub-riemannian geometry
- Sobolev-to-Lipschitz property
- Varadhan short-time asymptotics
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '12'
oa: 1
oa_version: Published Version
page: 1815-1832
project:
- _id: 256E75B8-B435-11E9-9278-68D0E5697425
  call_identifier: H2020
  grant_number: '716117'
  name: Optimal Transport and Stochastic Dynamics
- _id: fc31cba2-9c52-11eb-aca3-ff467d239cd2
  grant_number: F6504
  name: Taming Complexity in Partial Differential Systems
- _id: B67AFEDC-15C9-11EA-A837-991A96BB2854
  name: IST Austria Open Access Fund
publication: Mathematische Annalen
publication_identifier:
  eissn:
  - 1432-1807
  issn:
  - 0025-5831
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: Sobolev-to-Lipschitz property on QCD- spaces and applications
tmp:
  image: /images/cc_by.png
  legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
  name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
  short: CC BY (4.0)
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 384
year: '2022'
...