Derivative formulas and Poincare inequality for Kohn-Laplacian type semigroups

作者：Wang Feng-Yu^*

来源：Science China Mathematics, 2016, 59(2): 261-280.

DOI：10.1007/s11425-015-5084-3

• Kohn-Laplacian type operator
• derivative formula
• Poincare inequality
• reverse Poincare inequality

As a generalization to the heat semigroup on the Heisenberg group, the diffusion semigroup generated by the subelliptic operator L := 1/2 Sigma(m)(i=1) X-i(2) on Rm+d := R-m x R-d is investigated, where @@@ X-i(x, y) = Sigma(m)(k=1) sigma(ki) partial derivative(xk) + Sigma(d)(l=1)(A(l)x)(i)partial derivative(yl), (x, y) is an element of Rm+d, 1 <= i <= m @@@ for sigma an invertible m x m-matrix and {A(l)}(1 <= l <= d) some m x m-matrices such that the Hormander condition holds. We first establish Bismut-type and Driver-type derivative formulas with applications on gradient estimates and the coupling/Liouville properties, which are new even for the heat semigroup on the Heisenberg group; then extend some recent results derived for the heat semigroup on the Heisenberg group.

• 出版日期2016-2
• 单位北京师范大学