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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.

##### 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.
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