Logarithmic Sobolev inequalities and the growth of $L^p$ norms

We show that many of the recent results on exponential integrability of Lip 1 functions, when a logarithmic Sobolev inequality holds, follow from more fundamental estimates of the growth of LP norms under the same hypotheses. There have been a number of recent papers [1], [2], [3], [4] showing that a probability measure m on a Riemannian manifold M satisfying a logarithmic Sobolev inequality (1) p IVfl2dm > IM f2 lnfl2dm If fl2dm ln If l2dm -T If l2 dm has rather strong decay properties near infinity. These results take the form of integrability for exponentials of the powers of Lip 1 functions. Typically one has (2) J ef2 dm 0, the so called defect) in (1) is zero; in any case these techniques give considerably weaker results when T > 0. We intend to show that the exponential integrability holds in the defective case to the same extent as in the nondefective, despite the fact that the defective inequality does not in general give a mass gap or a Poincare inequality. To achieve our goal, we use an approach quite different from the ones typically used by others, who generally establish a result such as (2) by proving a differential equality for one or the other of the Laplace transforms E(eAf) or E(eAf ). Rather, we use the defective log Sobolev inequality to relate E(fP+2) inductively to E(fP), which in turn yields bounds on L2n norms and good estimates of E(eAf2). Our method works as well in the case T> 0 as the case T = 0, though it does not give as sharp inequalities in the case T = 0 as those obtained from previous techniques in [1]-[4]. It is well known how to approximate a globally Lip 1 function on a Riemannian manifold by smooth functions without increasing the Lip 1 norm. For simplicity of statement and without loss of generality, our theorems will be stated for a smooth function of compact support. Even more specifically, if p is Lip 1, so also is 1,1, with Received by the editors January 10, 1997. 1991 Mathematics Subject Classification. Primary 46E35, 46E39. (?)1998 American Mathematical Society 2309 This content downloaded from 157.55.39.170 on Wed, 19 Oct 2016 03:56:56 UTC All use subject to http://about.jstor.org/terms