We present a new and simple approach to some of the deviation inequalities for product measures deeply investigated by M.

We study some problems on exponential integrability, concentration of measure, and transportation cost related to logarithmic Sobolev inequalities. On the real line, we then give a characterization of those probablity measures which satisfy these inequalities.

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In this paper, we consider the fractional Laplacian −(−∆) α/2 on an open subset in R d with zero exterior condition. We establish sharp two-sided estimates for the heat kernel of such Dirichlet fractional Laplacian in C 1,1 open sets. This heat kernel is also the transition density of a rotationally symmetric stable process killed upon leaving a C 1,1 open set. Our results are the first sharp two-sided estimates for the Dirichlet heat kernel of a non-local operator on open sets.

In this paper we introduce a large class of subordinators called special subordinators and study their potential theory. Then we study the potential theory of processes obtained by subordinating a killed symmetric stable process in a bounded open set D with special subordinators. We establish a one-to-one correspondence between the nonnegative harmonic functions of the killed symmetric stable process and the nonnegative harmonic functions of the subordinate killed symmetric stable process. We show that nonnegative harmonic functions of the subordinate killed symmetric stable process are continuous and satisfy a Harnack inequality. We then show that, when D is a bounded κ-fat set, both the Martin boundary and the minimal Martin boundary of the subordinate killed symmetric stable process in D coincide with the Euclidean boundary ∂D.

We extend the concept of intrinsic ultracontractivity to non-symmetric semigroups and prove the intrinsic ultracontractivity of the Dirichlet semigroups of non-symmetric second order elliptic operators in bounded Lipschitz domains.

In this paper we study the precise behavior of the transition density functions of censored (resurrected) α-stable-like processes in C 1,1 open sets in R d, where d ≥ 1 and α ∈ (1, 2). We first show that the semigroup of the censored α-stable-like process in any bounded Lipschitz open set is intrinsically ultracontractive. We then establish sharp two-sided estimates for the transition density functions of a large class of censored α-stable-like processes in C 1,1 open sets. We further obtain sharp two-sided estimates for the Green functions of these censored α-stable-like processes in bounded C 1,1 open sets.

Sharp smoothing estimates are proven for magnetic Schrödinger semigroups in two dimensions under the assumption that the magnetic field is bounded below by some positive constant B0. As a consequence the L ∞ norm of the associated integral kernel is bounded by the L ∞ norm of the Mehler kernel of the Schrödinger semigroup with the constant magnetic field B0.

For a symmetric ff-stable process X on R n with 0 ! ff ! 2, n 2 and a domain D ae R n , let L D be the infinitesimal generator of the subprocess of X killed upon leaving D. For a Kato class function q, it is shown that L D + q is intrinsic ultracontractive on a Holder domain D of order 0. This is then used to establish the conditional gauge theorem for X on bounded Lipschitz domains in R n . It is also shown that the conditional lifetimes for symmetric stable process in a Holder domain of order 0 are uniformly bounded. Keywords and phrases: Symmetric stable processes, Feynman-Kac semigroup, conditional gauge theorem, logarithmic Sobolev inequality, intrinsic ultracontractivity. Running Title: Conditional Gauge Theorem The research of this author is supported in part by NSA Grant MDA904-98-1-0044 1 Introduction A symmetric ff-stable process X on R n is a L'evy process whose transition density p(t; x \Gamma y) relative to Lebesgue measure is uniquely determined by ...

Let Xt be the relativistic α-stable process in Rd, α ∈ (0, 2), d>α, with infinitesimal generator H (α) 0 = −((− ∆ +m2/α) α/2 − m). We study intrinsic ultracontractivity (IU) for the Feynman-Kac semigroup Tt for this process with generator H (α) 0 − V, V ≥ 0, V locally bounded. We prove that if lim |x|→ ∞ V (x) =∞, then for every t>0 the operator Tt is compact. We consider the class V of potentials V such that V ≥ 0, lim |x|→ ∞ V (x) = ∞ and V is comparable to the function which is radial, radially nondecreasing and comparable on unit balls. For V in the class V we show that the semigroup Tt is IU if and only if lim |x|→ ∞ V (x)/|x | = ∞. If this condition is satisfied we also obtain sharp estimates of the first eigenfunction φ1 for Tt. Inparticular, when V (x) =|x | β, β>0, then the semigroup Tt is IU if and only if β>1. For β>1 the first eigenfunction φ1(x) is comparable to exp(−m 1/α |x|)(|x | +1) (−d−α−2β−1)/2.