Sharp Log-Sobolev inequalities

We show existence of a wide variety of Log-Sobolev inequalities in which the constant is exactly that required by the Poincare inequality which may be inferred from the Log-Sobolev. We are given a smooth compact Riemannian manifold M, intrinsic gradient V, and volume element du we assume W.L.O.G. that ,a(M) = 1 and a smooth positive function m, giving rise to a probability measure mda = dm, for which we have a log-Sobolev inequality (LSI): (1) p IVf 2dm > IfM 2 ln f 2dm If2dm ln IM f2dm. It is well known that p > 2/A, where A is the first non-zero eigenvalue of the Schr6dinger operator f-*Af+Vf m M (A the usual Laplacian) attached to the Dirichlet form in (1). We call the LSI sharp if p = 2/A. Many examples of sharp inequalities are known, the most familiar arising from M the sphere with the usual metric scaled to give M unit volume, and m = 1. We will show here that for every compact homogeneous Riemannian manifold, there are a continuum of choices of m for which sharp LSI's exist. We follow the notation and conclusions of [1], which we now briefly review. For every M as described initially there is a least constant po(M), the hypercontractive constant for M, such that (2) p(M) J IVf 12dt > ? If J 2 ln If 12da -J If 2d, IlnJ If 12da. For any positive p If 12 In If 12dp If 12dt, In If 12 d a(p) XIf l dpi with the inequality an equality for a real non-trivial minimizing function f = Jp satisfying f J,2du = 1 and the non-linear P.D.E. (4) pAJp + Jp ln J2 _ a(p)Jp = 0. Received by the editors September 27, 1996 and, in revised form, February 24, 1997. 1991 Mathematics Subject Classification. Primary 46E35, 46E39. (g)1998 American Mathematical Society 2903 This content downloaded from 207.46.13.52 on Sat, 22 Oct 2016 05:31:13 UTC All use subject to http://about.jstor.org/terms 2904 OSCAR S. ROTHAUS (If po(M) > 2/A, where A is the first non-trivial eigenvalue of the Laplacian, then there is known [1] to be a non-trivial (g 1) minimizer for p = po(M) as well.) If one replaces f in the defective LSI (3) by fJp, it is thrown into the nondefective form: (5) ,J Vf 12J2d > |I f 12 In If 12J2 d/ I f 12J2 da In I f 12J2d, i.e., a version of our equation (1). We know, as noted earlier, that in equation (5), p > 2/r where r is the first non-trivial eigenvalue of the Schr6dinger operator attached to the Dirichlet form in (5); i.e., the operator (6) f -+Af + 2 VJpVf. Jp Our principal result is Theorem. If M is a compact homogeneous Riemannian manifold, then (5) above is a sharp LSI. (Thus we have distinct choices of probability measure with sharp LSI's for M for each p which is a version of (7) with 0 = 2/p and g = XJp. Now since Jp is not constant, and M is homogeneous, XJp cannot be zero for all Killing vectors. So for the first non-trivial eigenvalue of (6), we have r 2/T, we must have p = 2/T, and the LSI is sharp. This completes the proof of our theorem. [1 It would be quite interesting to get sharp inequalities on the line, other than the usual one. REFERENCES [1] 0. S. Rothaus, Diffusion on Compact Riemannian Manifolds and Logarithmic Sobolev In- equalities, Journal of Functional Analysis 42, #1 (June 1981), 102-109. MR 83f:58080a DEPARTMENT OF 'MATHEMATICS, CORNELL UNIVERSITY, ITHACA, NEW YORK 14853-7901 E-mail address: rothausImath. cornell. edu This content downloaded from 207.46.13.52 on Sat, 22 Oct 2016 05:31:13 UTC All use subject to http://about.jstor.org/terms