Some gradient estimates on covering manifolds

• Autorzy: Dungey N.
• Języki publikacji: EN

Abstrakty

EN

Let M be a complete Riemannian manifold which is a Galois covering, that is, M is periodic under the action of a discrete group G of isometries. Assuming that G has polynomial volume growth, we provide a new proof of Gaussian upper bounds for the gradient of the heat kernel of the Laplace operator on M. Our method also yields a control on the gradient in case G does not have polynomial growth.

Słowa kluczowe

EN

heat kernel; Riemannian manifold; covering manifold; gradient estimates; periodic operator; polynomial growth

PL

rozmaitość Riemanna; operator okresowy

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2004
• Tom: Vol. 52, nr 4
• Strony: 437--443
• Opis fizyczny: Bibliogr. 9 poz.

Twórcy

autor

Dungey N.

• School of Mathematics, The University of New South Wales, Sydney 2052, Australia

Bibliografia

• [1] P. Auscher, T. Coulhon, X. T. Duong and S. Hofmann, Riesz transform on manifolds and heat kernel regularity, preprint.
• [2] T. Coulhon and X. T. Duong, Riesz transforms for 1 ≤ p ≤ 2, Trans. Amer. Math. Soc. 351 (1999), 1151-1169.
• [3] E. B. Davies, Non-Gaussian aspects of heat kernel behaviour, J. London Math. Soc. 55 (1997), 105-125.
• [4] N. Dungey, Heat kernel estimates and Riesz transforms on some Riemannian covering manifolds, Math. Z. 247 (2004), 765-794.
• [5] A. F. M. ter Elst, D. W. Robinson and A. Sikora, On second-order periodic elliptic operators in divergence form, Math. Z. 238 (2001), 569-637.
• [6] A. Grigor'yan, Upper bounds of derivatives of the heat kernel on an arbitrary complete manifold, J. Funct. Anal. 127 (1995), 363-389.
• [7] -, Estimates of heat kernels on Riemannian manifolds, in: Spectral Theory and Geometry, B. Davies and Y. Safarov (eds.), London Math. Soc. Lecture Note Ser. 273, Cambridge Univ. Press, Cambridge, 1999, 140-225.
• [8] L. Saloff-Coste, A note on Poincaré, Sobolev, and Harnack inequalities, Internat. Math. Res. Notices 1992, no. 2, 27-38.
• [9] N. T. Varopoulos, L. Saloff-Coste and T. Coulhon, Analysis and Geometry on Groups, Cambridge Tracts in Math. 100, Cambridge Univ. Press, Cambridge, 1992.