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

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.
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