@article{13318,
  abstract     = {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.},
  author       = {Volberg, Alexander and Zhang, Haonan},
  issn         = {1432-1807},
  journal      = {Mathematische Annalen},
  publisher    = {Springer Nature},
  title        = {{Noncommutative Bohnenblust–Hille inequalities}},
  doi          = {10.1007/s00208-023-02680-0},
  year         = {2023},
}

@article{10588,
  abstract     = {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.},
  author       = {Dello Schiavo, Lorenzo and Suzuki, Kohei},
  issn         = {1432-1807},
  journal      = {Mathematische Annalen},
  keywords     = {quasi curvature-dimension condition, sub-riemannian geometry, Sobolev-to-Lipschitz property, Varadhan short-time asymptotics},
  pages        = {1815--1832},
  publisher    = {Springer Nature},
  title        = {{Sobolev-to-Lipschitz property on QCD- spaces and applications}},
  doi          = {10.1007/s00208-021-02331-2},
  volume       = {384},
  year         = {2022},
}