## The Poincaré inequality for vector fields satisfying Hörmander’s condition.(English)Zbl 0614.35066

From author’s summary: The purpose of this paper is to prove a Poincaré-type inequality of the form $\int_{B(r)}| f- f_{B(r)}|^ 2\leq Cr^ 2\int_{B(r)}\sum^{m}_{i=1}| X_ if|^ 2,\text{ for all } f\in C^{\infty}(\overline{B(r)}),$ where $$X_ 1,...,X_ m$$ denote vector fields on $$R^ d$$ satisfying Hörmander’s condition, B(r) denotes a ball of radius r with respect to a natural metric associated with $$X_ 1,...,X_ m$$ and $$f_{B(r)}$$ denotes the average value of f on the ball B(r). This inequality is the same as finding a lower bound $$1/Cr^ 2$$ of the least nonzero eigenvalue in the Neumann problem for $$L=\sum^{m}_{i=1}X^*_ iX_ i$$ of B(r).
Reviewer: A.H.Nasr

### MSC:

 35P15 Estimates of eigenvalues in context of PDEs 35J99 Elliptic equations and elliptic systems 35B99 Qualitative properties of solutions to partial differential equations
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### References:

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