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Sobolev, Besov and Triebel-Lizorkin Spaces on Quantum Tori

About this Title

Xiao Xiong, Quanhua Xu and Zhi Yin

Publication: Memoirs of the American Mathematical Society
Publication Year: 2018; Volume 252, Number 1203
ISBNs: 978-1-4704-2806-8 (print); 978-1-4704-4375-7 (online)
Published electronically: January 29, 2018
Keywords:Quantum tori, noncommutative $L_p$-spaces, Bessel and Riesz potentials, (potential) Sobolev spaces, Besov spaces, Triebel-Lizorkin spaces, Hardy spaces, characterizations, Poisson and heat semigroups, embedding inequalities, interpolation, (completely) bounded Fourier multipliers

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Table of Contents


  • Chapter 1. Introduction
  • Chapter 2. Preliminaries
  • Chapter 3. Sobolev spaces
  • Chapter 4. Besov spaces
  • Chapter 5. Triebel-Lizorkin spaces
  • Chapter 6. Interpolation
  • Chapter 7. Embedding
  • Chapter 8. Fourier multiplier


This paper gives a systematic study of Sobolev, Besov and Triebel-Lizorkin spaces on a noncommutative -torus (with a skew symmetric real -matrix). These spaces share many properties with their classical counterparts. We prove, among other basic properties, the lifting theorem for all these spaces and a Poincaré type inequality for Sobolev spaces. We also show that the Sobolev space coincides with the Lipschitz space of order , already studied by Weaver in the case . We establish the embedding inequalities of all these spaces, including the Besov and Sobolev embedding theorems. We obtain Littlewood-Paley type characterizations for Besov and Triebel-Lizorkin spaces in a general way, as well as the concrete ones in terms of the Poisson, heat semigroups and differences. Some of them are new even in the commutative case, for instance, our Poisson semigroup characterizations improve the classical ones. As a consequence of the characterization of the Besov spaces by differences, we extend to the quantum setting the recent results of Bourgain-Brézis -Mironescu and Maz'ya-Shaposhnikova on the limits of Besov norms. The same characterization implies that the Besov space for is the quantum analogue of the usual Zygmund class of order . We investigate the interpolation of all these spaces, in particular, determine explicitly the K-functional of the couple , which is the quantum analogue of a classical result due to Johnen and Scherer. Finally, we show that the completely bounded Fourier multipliers on all these spaces do not depend on the matrix , so coincide with those on the corresponding spaces on the usual -torus. We also give a quite simple description of (completely) bounded Fourier multipliers on the Besov spaces in terms of their behavior on the -components in the Littlewood-Paley decomposition.

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