Eric A. Carlen

Sharp uniform convexity and smoothness inequalities for trace norms

SummaryWe prove several sharp inequalities specifying the uniform convexity and uniform smoothness properties of the Schatten trace idealsCp, which are the analogs of the Lebesgue spacesLp in non-commutative integration. The inequalities are all precise analogs of results which had been known inLp, but were only known inCp for special values ofp. In the course of our treatment of uniform convexity and smoothness inequalities forCp we obtain new and simple proofs of the known inequalities forLp.

Superadditivity of Fisher's information and logarithmic Sobolev inequalities

Abstract We prove a theorem characterizing Gaussian functions and we prove a strict superaddivity property of the Fisher information. We use these results to determine the cases of equality in the logarithmic Sobolev inequality on R n equipped with Lebesgue measure and with Gauss measure. We also prove a strengthened form of Grosss logarithmic Sobolev inequality with a “remainder term” added to the left side. Finally we show that the strict form of Grosss inequality is a direct consequence of an inequality due to Blachman and Stam, and that this in turn is a direct consequence of strict superadditivity of the Fisher information.

Competing symmetries, the logarithmic HLS inequality and Onofri's inequality ons n

The sharp version of the logarithmic Hardy-Littlewood-Sobolev inequality including the cases of equality is established. We then show that this implies Beckners generalization of Onofris inequality to arbitrary dimensions and determines the cases of equality.

Extremals of functionals with competing symmetries

Abstract We present a new method of producing optimizing sequences for highly symmetric functionals. The sequences have good convergence properties built in. We apply the method in different settings to give elementary proofs of some classical inequalities—such as the Hardy-Littlewood-Sobolev and the logarithmic Sobolev inequality—in their sharp form.

Differential Calculus and Integration by Parts on Poisson Space

We define a gradient operator on random variables defined on the “standard Poisson space” (the sample space of paths which have unit jumps and are constant between their jumps). An “integration by parts” formula shows that the adjoint of that operator extends the usual Poisson stochastic integral. We prove a “Malliavin calculus” type of result, which is closely related to the co-area formula of geometric measure theory.

Entropy production by block variable summation and central limit theorems

We prove a strict lower bound on the entropy produced when independent random variables are summed and rescaled. Using this, we develop an approach to central limit theorems from a dynamical point of view in which the entropy is a Lyapunov functional governing approach to the Gaussian limit. This dynamical approach naturally extends to cover dependent variables, and leads to new results in pure probability theory as well as in statistical mechanics. It also provides a unified framework within which many previous results are easily derived.

A sharp analog of Young’s inequality on SN and related entropy inequalities

We prove a sharp analog of Young’s inequality on SN, and deduce from it certain sharp entropy inequalities. The proof turns on constructing a nonlinear heat flow that drives trial functions to optimizers in a monotonic manner. This strategy also works for the generalization of Young’s inequality on RN to more than three functions, and leads to significant new information about the optimizers and the constants.

Strict entropy production bounds and stability of the rate of convergence to equilibrium for the Boltzmann equation

We first consider the Boltzmann equation with a collision kernel such that all kinematically possible collisions are run at equal rates. This is the simplest Boltzmann equation having the compressible Euler equations as a scaling limit. For it we prove a stability result for theH-theorem which says that when the entropy production is small, the solution of the spatially homogeneous Boltzmann equation is necessarily close to equilibrium in the entropie sense, and therefore strongL1 sense. We use this to prove that solutions to the spatially homogeneous Boltzmann equation converge to equilibrium in the entropie sense with a rate of convergence which is uniform in the initial condition for all initial conditions belonging to certain natural regularity classes. Every initial condition with finite entropy andpth velocity moment for some p>2 belongs to such a class. We then extend these results by a simple monotonicity argument to the case where the collision rate is uniformly bounded below, which covers a wide class of slightly modified physical collision kernels. These results are the basis of a study of the relation between scaling limits of solutions of the Boltzmann equation and hydrodynamics which will be developed in subsequent papers; the program is described here.

Determination of the spectral gap for Kac's master equation and related stochastic evolution

We present a method for bounding, and in some cases computing, the spectral gap for systems of many particles evolving under the influence of a random collision mechanism. In particular, the method yields the exact spectral gap in a model due to Mark Kac of energy conserving collisions with one dimensional velocities. It is also sufficiently robust to provide qualitatively sharp bounds also in the case of more physically realistic momentum and energy conserving collisions in three dimensions, as well as a range of related models.

Optimal Hypercontractivity for Fermi Fields and Related Non-Commutative Integration Inequalities

Optimal hypercontractivity bounds for the fermion oscillator semigroup are obtained. These are the fermion analogs of the optimal hypercontractivity bounds for the boson oscillator semigroup obtained by Nelson. In the process, several results of independent interest in the theory of non-commutative integration are established.