Subcoercive and subelliptic operators on Lie groups: Variable coefficients
This paper extends the theory of second-order subelliptic differential operators with real coefficients acting on functions over a Lie group in two different directions: a generalization to higher-order operators characterized by a condition of coercivity and the consideration of complex-valued coefficients. Let \(a_ 1,\dots,a_ d\) be an algebraic basis of rank \(r\) in a Lie algebra \(\mathfrak g\) of a connected Lie group \(G\) and let \(A_ i\) be the left differential operator in the direction \(a_ i\) on the \(L_ p\)- spaces \(L_ p(G,d g)\) or \(L_ p(G,d \widehat{g})\) with respect to the left, or right, Haar measure, where \(p \in [1,\infty]\). An \(m\)-th order operator \(H = \sum C_ \alpha A^ \alpha\) with complex variable bounded coefficients \(C_ \alpha : G \to \mathbb{C}\) and multiindex products \(A^ \alpha\) of the operators \(A_ i\) is called subcoercive of step \(r\) if for all \(g \in G\) the form obtained by fixing the \(C_ \alpha\) at \(g \in G\) is subcoercive of step \(r\) and the ellipticity constant is bounded from below uniformly by a positive constant. The main result: the closure of \(H\) generates a continuous heat semigroup \(S\) which interpolates consistently between the spaces \(L_ p(G,d g)\) and \(L_ p(G,d \widehat{g})\), is determined by an integral kernel and which is holomorphic on a non-empty \(p\)-independent sector. In particular, if the matrix of principal coefficients of \(H\) is hermitian then \(S\) is holomorphic in the open right half-plane. With the help of regularization it is proved that these results are valid if only the principal coefficients are \(m\)-times differentiable in \(L_ \infty\). In this connection Gaussian bounds and smoothness properties of the corresponding integral kernel are studied.
second-order subelliptic differential operators
scientific article; zbMATH DE number 548928