Fundamental solutions for powers of the Heisenberg sub-Laplacian
Let \(\Delta_{H_ n}\) denote the sub-Laplacian operator on the Heisenberg group \(H_ n\) of dimension \(2n+1\). In this paper we compute explicit fundamental solutions for powers \(\Delta^ p_{H_ n}\) of \(\Delta_{H_ n}\) with \(p \leq n\), generalizing a result of G. Folland for the case \(p=1\). We show that \(\Delta^ p_{H_ n}\) has a tempered fundamental solution which is given by a smooth function away from the identity in \(H_ n\). The solution can be written in terms of iterated antiderivatives of elementary functions or obtained as the weak limit of an infinite series. We recover Folland's solution when \(p=1\) and obtain an explicit (finite) form when \(p=2\). The key idea in our derivation exploits the \(U(n)\)-invariance of the operator \(\Delta_{H_ n}\) to express fundamental solutions in terms of \(U(n)\)-spherical functions. Our methods should be applicable to other differential operators on certain nilpotent Lie groups that satisfy strong invariance conditions.
scientific article; zbMATH DE number 561223