Coercive inequalities on metric measure spaces

Abstract

In this paper we study coercive inequalities on finite dimensional metric spaces with probability measures which do not have the volume doubling property. Crown

Logarithmic Sobolev Inequalities for Infinite Dimensional Hörmander Type Generators on the Heisenberg Group

“…We will often drop the ω in the notation for convenience. The work is strongly motivated by the methods of Hebisch and Zegarlinski described in [15]. …”

“…Independently, and by very different methods, in [15] the authors were able to show that a related class of measures on H satisfy (LS q ) inequalities (see Theorem 2.8 below). To describe these we first need to introduce the natural distance function on H, which is the so-called Carnot-Carathéodory distance.…”

“…Proof. For the sake of completeness, we recall part of the proof given in [15]. It suffices to show that ∆d ≤ K on {d(x) = 1}.…”

Bounded harmonic functions on Riemannian manifolds of nonpositive curvature

“…The new Laplacian comparison theorem has applications in eigenvalue estimates [7] and rigidity of harmonic maps from bounded symmetric domains [25]. Some other applications in geometry and probability theory are presented in [6,9,14,18].…”

Linear and Nonlinear Dissipative Dynamics

“…For symmetric semigroups, after a recent progress in proving the log-Sobolev inequality for infinite dimensional Hörmander type generators L symmetric in L 2 (µ) defined with a suitable nonproduct measure µ ( [32], [25], [28], [26], [27], [43]), one can expect an extension of the established strategy ( [51]) for proving strong pointwise ergodicity for the corresponding Markov semigroups P t ≡ e tL , (respectively in the uniform norm in case of the compact spaces as in [24] and refs therein). One could obtain more results in this direction, including configuration spaces given by infinite products of general noncompact nilpotent Lie groups other than Heisenberg type groups, by conquering a (finite dimensional) problem of subLaplacian bounds (of the corresponding control distance) which for a moment remains still very hard.…”