The existence of a global fundamental solution for homogeneous Hörmander operators via a global lifting method. (English) Zbl 1383.35005

Biagi, Stefano; Bonfiglioli, Andrea

Proc. Lond. Math. Soc. (3) 114, No. 5, 855-889 (2017).

The authors introduce a general lifting method, namely given a linear partial differential operator \(P(x,D)\) with smooth coefficients, after addition of variables \(\lambda\), an operator \(Q(x,\lambda,D_x,D_\lambda)\) is considered so that \(Q(f\cdot \pi)(x,\lambda)=Pf(x)\) with \(\pi(x,\lambda)=x\). Under further assumptions on \(Q\), a fundamental solution \(K(x,\lambda,y,\eta)\) for \(Q\) provides a fundamental solution \(H(x,y)\) for \(P\) by the formula \[ H(x,y)= \int K(x,0,y,\eta)\,d\eta. \] The proceeding is applied to the construction of global fundamental solutions \(H(x,y)\) for homogeneous sum-of-squares operators, by applying the formula to the fundamental solution \(K(x,\lambda,y,\eta)\) of a lifting \(Q\), which turns out to be a sub-Laplacian on some Carnot group, cf. G. B. Folland [Commun. Partial Differ. Equations 2, 165–191 (1977; Zbl 0365.58016)]. A relevant example is given by the Grushin operator \[ P= D^2_{x_1}+ x^2_1 D^2_{x_2}, \] whose lifting is represented by the Kohn-Laplacian on the Heisenberg group.

Reviewer: Luigi Rodino (Torino)

MSC:

35A08	Fundamental solutions to PDEs
35H20	Subelliptic equations
35C15	Integral representations of solutions to PDEs
35J70	Degenerate elliptic equations
35R03	PDEs on Heisenberg groups, Lie groups, Carnot groups, etc.

Keywords:

sum-of-squares operators; Carnot groups; general lifting method; Grushin operator