Subelliptic operators on Lie groups: Variable coefficients
Let \(G\) be a Lie group with Lie algebra \(\mathfrak g\), let \(a_1, \dots, a_n\) be an algebraic basis of \(\mathfrak g\) and \(dg\) a right Haar measure. The authors consider the second-order subelliptic differential operator \(H = \sum_{\alpha : |\alpha|\leq 2} c_\alpha A^\alpha\) with real bounded coefficients \(c_\alpha : G \to \mathbb{R}\), where \(A_i = dL(a_i)\) are the corresponding generators of the left translation by \(G\) on one of the Banach spaces \(L_p(G)\), \(p \in [1,\infty)\). The authors prove that if the principal coefficients \(\{C_\alpha : |\alpha |= 2\}\) are left differentiable in the directions \(a_1,\dots, a_n\) with bounded derivates, then the operator \(H\) has a family of semigroup generator extensions on the \(L_p\)-spaces, and the corresponding holomorphic semigroups \(\{S(t)\), \(t \geq 0\}\) are given by the positive integral kernel \[ (S(t) \varphi) (g) = \int_G K_t (g;h) \varphi(h) dh. \]
scientific article; zbMATH DE number 841766
scientific article