Gradient estimates for the heat semigroup on \(\mathbb H\)-type groups (Q711719)

The Heisenberg type group \(\mathbb H(2n, m)\) is the set of points \(g=(x,t) \in\mathbb R^{2n} \times\mathbb R^m\) with the group law \[ (x,t)(x',t')= \big(x+x', t+t'+\tfrac{1}{2}\langle x, Ux'\rangle\big) \] and \[ \langle x, Ux' \rangle= \big(\langle x, U^{(1)}x' \rangle, \dots, \langle x, U^{(m)}x' \rangle\big) \] where the matrices \(U^{(1)}, \dots , U^{(m)}\) have the following properties: (1) \(U^{(j)}(1 \leq j \leq m)\) is an \((2n) \times (2n)\) skew-symmetric and orthogonal matrix, (2) \(U^{(i)}U^{(j)}+U^{(j)}U^{(i)}=0\) for all \(1 \leq i \neq j \leq m\). Let \(U^{(j)}=(U_{kl}^{(j)})_{1 \leq k,l \leq 2n}\) \((1\leq j \leq m)\). The canonical sub-Laplacian on \(\mathbb H(2n,m)\) is given by \(\Delta= \sum_{l=1}^{2n}X_{l}^2\), where \(X_{l}\) \((1 \leq l \leq 2n)\) are the left-invariant vector fields on \(\mathbb H(2n,m)\) defined by \[ X_{l}= \frac{\partial}{\partial x_{l}} + \frac{1}{2}\sum_{j=1}^{m}\bigg(\sum_{k=1}^{2n}x_{k}U_{kl}^{(j)}\bigg) \frac{\partial}{\partial t_{j}} \] and \(e^{t\triangle}\) \((t > 0)\) is the heat semigroup. Let \(\nabla=(X_{1}, \dots , X_{2n})\) be the gradient operator. In this paper, by utilizing the Poincaré inequality and representation formulae, it is shown that on the Heisenberg type group \(\mathbb H(2n,m)\), there exists a constant \(C > 0\) such that \[ |\nabla e^{t \Delta}f|(g) \leq Ce^{t\Delta}(|\nabla f|)(g) \quad \forall g\in\mathbb H(2n,m),\;t >0,\;f \in C_0^\infty (\mathbb H(2n,m)). \]