Sobolev Inequalities, Riesz Transforms, and the Ricci Flow

Abstract

A number of results about deriving further Sobolev inequalities from a given Sobolev inequality are presented. Various techniques are employed, including Bessel potentials and Riesz transforms. Combining these results with the $W^{1,2}$ Sobolev inequality along the Ricci flow established by the author in earlier papers then yields various new Sobolev inequalities along the Ricci flow.

Keywords

Ricci flow / Sobolev inequalities / Riesz transforms

References

[1] Bakry, D.: Étude des transfromations de Riesz dans les variété riemanniennes à courbure de Ricci minorée, Séminaire de Probabilités, vol. XXI, pp. 137–172. Lecture notes in mathematics, vol. 1247. Springer, Berlin (1987)

[2] Gallot S. Inégalités isopérimétriques et analytiques sur les variétés riemanniennes. Astérisque. 1988, 163–164 31-91

[3] Gallot S. Isoperimetric inequalities based on integral norms of Ricci curvature. Astérisque. 1988, 157–158 191-216

[4] Gilbarg D, Trudinger NS. Elliptic Partial Differential Equations of Second Order. 1983 2 New York: Springer

[5] Li XD. Riesz transforms for symmetric diffusion operators on complete Riemannian manifolds. Rev. Mat. Iberoamericana. 2006, 22 2 591-648

[6] Perelman, G.: Ricci flow with surgery on three-manifolds. arXiv.math/0303109v1 (2003). Accessed March 2003

[7] Seeley RT. Complex powers of an elliptic operator. Proc. Symp. Pure Math.. 1967, 10 288-307

[8] Strichartz R. Analysis of the Laplacian on the complete Riemannian manifold. J. Funct. Anal.. 1983, 52 48-79

[9] Ye, R.: The logarithmic Sobolev inequality along the Ricci flow. arXiv:math/0707.2424

[10] Ye, R.: The logarithmic Sobolev inequality along the Ricci flow in dimension 2. arXiv:0708.2003

[11] Ye, R.: The logarithmic Sobolev inequality along the Ricci flow: the case $\lambda _0(g_0)=0$. arXiv:0708.2005

[12] Ye, R.: The log entropy functional along the Ricci flow. arXiv:0708.2008