The limitations of the Poincaré inequality for Gru?in type operators
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- 作者：Derek W. Robinson ; Adam Sikora
- 关键词：35J70 ; 35H20 ; 58J35
- 刊名：Journal of Evolution Equations
- 出版年：2014
- 出版时间：September 2014
- 年：2014
- 卷：14
- 期：3
- 页码：535-563
- 全文大小：362 KB
- 参考文献：1. Benjamini I., Chavel I., Feldman E. A.: kernel lower bounds on Riemannian manifolds using the old ideas of Nash. Proc. London Math. Soc. 72, 215-40 (1996) CrossRef
2. Chavel, I., and Feldman, E. A., Isoperimetric constants, the geometry of ends, and large time heat diffusion in Riemannian manifolds. / Proc. London Math. Soc.62 (1991), 427-48.
3. Davies E.B.: Non-Gaussian aspects of heat kernel behaviour. J. London Math. Soc. 55, 105-25 (1997) CrossRef
4. Franchi, B., Gutiérrez, C. E., and Wheeden, R. L., Weighted Sobolev–Poincaré inequalities for Grushin type operators. / Comm. Part. Diff. Eq. 19 (1994), 523-04.
5. Fabes, E. B., Kenig, C. E., and Serapioni, R. P., The local regularity of solutions of degenerate elliptic equations. / Comm. Part. Diff. Eq. 7 (1982), 77-16.
6. Grigor’yan, A., The heat equation on noncompact Riemannian manifolds. (in Russian). / Mat. Sb.182, No. 1, (1991), 55-7.
7. Grigor’yan, A., The heat equation on noncompact Riemannian manifolds. (English translation). / Math. USSR-Sb. 72, No. 1 (1992), 47-7.
8. Grigor’yan A., Saloff-Coste L.: Stability results for Harnack inequalities Ann. Inst. Fourier, Grenoble 55, 820-25 (2005)
9. Grigor’yan A., Saloff-Coste L.: Heat kernel on manifolds with ends. Ann. Inst. Fourier, Grenoble 59, 1917-974 (2009) CrossRef
10. Gru?in, V. V., A certain class of hypoelliptic operators. / Mat. Sb. (N.S.) 83 (125) (1970).
11. Gyrya, P., and Saloff-Coste, L., Neumann and Dirichlet heat kernels in inner uniform domains, vol. 336. Société Mathématique de France, Paris, 2011.
12. H?rmander L.: Hypoelliptic second order differential equations. Acta Math. 119, 147-71 (1967) CrossRef
13. Hassell A., Sikora A.: Riesz transforms in one-dimension. Ind. Univ. Math. J. 58, 823-52 (2009) CrossRef
14. Jerison D.: The Poincaré inequality for vector fields satisfying H?rmander’s condition. Duke Math. J. 53, 503-23 (1986) CrossRef
15. Jerison, D. S., and Sánchez-Calle, A. Estimates for the heat kernel for a sum of squares of vector fields. / Ind. Univ. Math. J. 35 (1986), 835-54.
16. Lu, G., The sharp Poincaré inequality for free vector fields: an endpoint result. / Revista Matemática Iberoamericana 10 (1994), 453-66.
17. Ma, Z. M., and R?ckner, M., / Introduction to the theory of (non symmetric) Dirichlet Forms. Universitext. Springer-Verlag, Berlin etc., 1992.
18. Moser, J. On Harnack’s theorem for elliptic differential equations. / Commun. Pure Appl. Math. 14 (1961), 577-91.
19. Moser, J. A Harnack inequality for parabolic differential equations. / Commun. Pure Appl. Math. 17 (1964), 101-34. Correction to: “A Harnack inequality for parabolic differential equations- Comm. Pure Appl. Math. 20 (1967), 231-36.
20. Moser, J. On pointwise estimate for parabolic differential equations. / Commun. Pure Appl. Math. 24 (1971), 727-40.
21. Robinson, D. W., and Sikora, A. Analysis of degenerate elliptic operators of Gru?in type. / Math. Z.260 (2008), 475-08.
22. Robinson, D. W., and Sikora, A. The limitations of the Poincaré inequality. Research report, Australian National University, 2013. arXiv:1305.6998.
23. Saloff-Coste, L., A note on Poincaré, Sobolev, and Harnack inequalities. / Internat. Math. Res. Notices 1992, No. 2 (1992), 27-8.
24. Saloff-Coste, L. Uniformly elliptic operators on Riemannian manifolds. / J. Diff. Geom. 36 (1992), 417-50.
25. Saloff-Coste L.: Parabolic Harnack inequality for divergence-form second-order differential operators. Potential Anal. 4, 429-67 (1995) CrossRef
26. Sturm K.-T.: Analysis on local Dirichlet spaces. II. Upper Gaussian estimates for the fundamental solutions of parabolic equations. Osaka J. Math. 32, 275-12 (1995)
27. Sturm K.-T.: Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality. J. Math. Pures Appl. 75, 273-97 (1996)
28. Trudinger N.S.: Linear elliptic operators with measurable coefficients. Ann. Scuola Norm. Sup. Pisa 27, 265-08 (1973) - 作者单位：Derek W. Robinson (1)
Adam Sikora (2)
1. Centre for Mathematics and its Applications, Mathematical Sciences Institute, Australian National University, Canberra, ACT 0200, Australia
2. Department of Mathematics, Macquarie University, Sydney, NSW 2109, Australia - ISSN：1424-3202
文摘
We examine the validity of the Poincaré inequality for degenerate, second-order, elliptic operators H in divergence form on \({L_2(\mathbf{R}^{n}\times \mathbf{R}^{m})}\) . We assume the coefficients are real symmetric and \({a_1H_\delta\geq H\geq a_2H_\delta}\) for some \({a_1,a_2>0}\) where H δ is a generalized Gru?in operator, $$H_\delta=-\nabla_{x_1}\,|x_1|^{\left(2\delta_1,2\delta_1'\right)} \,\nabla_{x_1}-|x_1|^{\left(2\delta_2,2\delta_2'\right)} \,\nabla_{x_2}^2.$$ Here \({x_1 \in \mathbf{R}^n,\; x_2 \in \mathbf{R}^m,\;\delta_1,\delta_1'\in[0,1\rangle,\;\delta_2,\delta_2'\geq0}\) and \({|x_1|^{\left(2\delta,2\delta'\right)}=|x_1|^{2\delta}}\) if \({|x_1|\leq 1}\) and \({|x_1|^{\left(2\delta,2\delta'\right)}=|x_1|^{2\delta'}}\) if \({|x_1|\geq 1}\) . We prove that the Poincaré inequality, formulated in terms of the geometry corresponding to the control distance of H, is valid if n??2, or if n?=?1 and \({\delta_1\vee\delta_1'\in[0,1/2\rangle}\) but it fails if n?=?1 and \({\delta_1\vee\delta_1'\in[1/2,1\rangle}\) . The failure is caused by the leading term. If \({\delta_1\in[1/2, 1\rangle}\) , it is an effect of the local degeneracy \({|x_1|^{2\delta_1}}\) , but if \({\delta_1\in[0, 1/2\rangle}\) and \({\delta_1'\in [1/2,1\rangle}\) , it is an effect of the growth at infinity of \({|x_1|^{2\delta_1'}}\) . If n?=?1 and \({\delta_1\in[1/2, 1\rangle}\) , then the semigroup S generated by the Friedrichs-extension of H is not ergodic. The subspaces \({x_1\geq 0}\) and \({x_1\leq 0}\) are S-invariant, and the Poincaré inequality is valid on each of these subspaces. If, however, \({n=1,\; \delta_1\in[0, 1/2\rangle}\) and \({\delta_1'\in [1/2,1\rangle}\) , then the semigroup S is ergodic, but the Poincaré inequality is only valid locally. Finally, we discuss the implication of these results for the Gaussian and non-Gaussian behaviour of the semigroup S.