Date | 2015-10-23 |
---|---|

Speaker | Xiao Xiong |

Dept. | 서울대 |

Room | 선택 |

Time | 10:30-12:00 |

* 장소: 25동 110호

We give a systematic study of Sobolev, Besov and Triebel-Lizorkin spaces on a noncommutative torus. These spaces share many properties with their classical counterparts. We prove the lifting theorem for all these spaces and a Poincare type inequality for Sobolev spaces. We also show that the Sobolev space coincides with the Lipschitz space studied by Weaver. 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. As a consequence of the characterization of the Besov spaces by differences, we extend to the quantum setting the recent results of Bourgain-Brezis -Mironescu and Maz'ya-Shaposhnikova on the limits of Besov norms. We investigate the interpolation of all these spaces. Finally, we show that the completely bounded Fourier multipliers on all these spaces coincide with those on the corresponding spaces on the usual torus.

We give a systematic study of Sobolev, Besov and Triebel-Lizorkin spaces on a noncommutative torus. These spaces share many properties with their classical counterparts. We prove the lifting theorem for all these spaces and a Poincare type inequality for Sobolev spaces. We also show that the Sobolev space coincides with the Lipschitz space studied by Weaver. 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. As a consequence of the characterization of the Besov spaces by differences, we extend to the quantum setting the recent results of Bourgain-Brezis -Mironescu and Maz'ya-Shaposhnikova on the limits of Besov norms. We investigate the interpolation of all these spaces. Finally, we show that the completely bounded Fourier multipliers on all these spaces coincide with those on the corresponding spaces on the usual torus.

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