Title: Logarithmic Sobolev, Hardy and Poincaré inequalities on the Heisenberg group Show Chinese title

Title: 对Heisenberg群上的对数Sobolev、Hardy和Poincaré不等式

Abstract: In this paper we first prove a number of important inequalities with explicit constants in the setting of the Heisenberg group. This includes the fractional and integer Sobolev, Gagliardo-Nirenberg, (weighted) Hardy-Sobolev, Nash inequalities, and their logarithmic versions. In the case of the first order Sobolev inequality, our constant recovers the sharp constant of Jerison and Lee. Remarkably, we also establish the analogue of the Gross inequality with a semi-probability measure on the Heisenberg group that allows -- as it happens in the Euclidean setting -- an extension to infinite dimensions, and particularly can be regarded as an inequality on the infinite dimensional $\mathbb{H}^{\infty}$. Finally, we prove the so-called generalised Poincar\'e inequality on the Heisenberg group both with respect to the aforementioned semi-probability measure and the Haar measure, also with explicit constants. Show Chinese abstract

Abstract: 在本文中，我们首先在海森堡群的背景下证明了一些带有显式常数的重要不等式。 这包括分数和整数次 Sobolev、Gagliardo-Nirenberg、(加权) Hardy-Sobolev、Nash 不等式及其对数版本。 在第一阶 Sobolev 不等式的情况下，我们的常数恢复了 Jerison 和 Lee 的精确常数。 值得注意的是，我们还在海森堡群上建立了 Gross 不等式的类似形式，该形式使用了一种半概率测度，正如在欧几里得情形中一样，可以扩展到无限维，并且特别可以视为无限维$\mathbb{H}^{\infty}$上的一个不等式。 最后，我们证明了所谓的广义 Poincaré 不等式，在海森堡群上，既相对于上述半概率测度，也相对于 Haar 测度，且都带有显式常数。