A note on Poincaré, Sobolev, and Harnack inequalities.(English)Zbl 0769.58054

The author gives very interesting Poincaré and Sobolev inequalities for locally subelliptic, second order differential operators on a $$C^ \infty$$-connected manifold. Further he gives a parabolic Harnack inequality for the operator $$\partial_ t + L$$. As an application he states a Hölder continuity result for solutions of $$(\partial_ t + L)u = 0$$ and estimates for the heat kernel. All results are rather interesting and therefore the reviewer hopes that detailed proofs will appear elsewhere.
Reviewer: N.Jacob (Erlangen)

MSC:

 58J05 Elliptic equations on manifolds, general theory 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 58J35 Heat and other parabolic equation methods for PDEs on manifolds
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