Abstract
By estimating the intrinsic distance and using known heat kernel upper bounds, the global Nash inequality with exact dimension is established for a class of square fields with algebraic growth induced by vector fields satisfying the Hormander condition. As an application, a sufficient condition is presented for the log-Sobolev inequality to hold. Typical examples for Gruschin type operators and generalized Kohn-Lapacians on Heisenberg groups are provided.
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