Form Inequalities for Symmetric Contraction Semigroups
Markus Haase
备注：
29 pages; submitted to: Proceedings of the IWOTA, Amsterdam, July 2014. For this updated version, the term "complete contraction" has been exchanged for "absolute contraction" in order to avoid confusion with terminology used in operator space theory. Some small misprints and errors have been corrected, and a reference has been added. The proof of Theorem 4.11 was incomplete and has been amended
考虑-对于在某个测度空间$\mathrm{X}$，$1\le p < \infty$，$q$上的对称压缩半群的生成元\({-}A\)，对偶指数和给定的可测函数$F_j,\: G_j : \mathbb{C}^d \to \mathbb{C}$ -陈述：$$ \mathrm{Re}\, \sum_{j=1}^m \int_{\mathrm{X}} A F_j(\mathbf{f}) \cdot G_j(\mathbf{f}) \,\,\ge \,\,0 $$ { \em for all $\mathbb{C}^d$ -valued measurable functions $\mathbf{f}$ on $\mathrm{X}$ such that $F_j(\mathbf{f}) \in \mathrm{dom}(A_p)$ and $G_j(\mathbf{f}) \in \mathrm{L}^q(\mathrm{X})$ for all $j$ .}证明了这一命题在一般情况下成立，如果$\mathrm{X}$是两点Bernoulli $(\frac{1}{2}, \frac{1}{2})$ -空间，$A$是一种特殊形式.由此得到了这类半群的$\mathrm{L}^{p}$ -解析性的最优角的一个新的证明，它与著名的次马尔可夫情形基本相同.主要定理的证明是结合了著名的约化技巧和$\mathrm{C}(K)$ -空间上算子的一些表示结果。本文的重点之一在于详细介绍这些辅助技术和结果。
英文摘要：
Consider --- for the generator \({-}A\) of a symmetric contraction semigroup over some measure space $\mathrm{X}$, $1\le p < \infty$, $q$ the dual exponent and given measurable functions $F_j,\: G_j : \mathbb{C}^d \to \mathbb{C}$ --- the statement: $$ \mathrm{Re}\, \sum_{j=1}^m \int_{\mathrm{X}} A F_j(\mathbf{f}) \cdot G_j(\mathbf{f}) \,\,\ge \,\,0 $$ {\em for all $\mathbb{C}^d$-valued measurable functions $\mathbf{f}$ on $\mathrm{X}$ such that $F_j(\mathbf{f}) \in \mathrm{dom}(A_p)$ and $G_j(\mathbf{f}) \in \mathrm{L}^q(\mathrm{X})$ for all $j$.}
It is shown that this statement is valid in general if it is valid for $\mathrm{X}$ being a two-point Bernoulli $(\frac{1}{2}, \frac{1}{2})$-space and $A$ being of a special form. As a consequence we obtain a new proof for the optimal angle of $\mathrm{L}^{p}$-analyticity for such semigroups, which is essentially the same as in the well-known sub-Markovian case.
The proof of the main theorem is a combination of well-known reduction techniques and some representation results about operators on $\mathrm{C}(K)$-spaces. One focus of the paper lies on presenting these auxiliary techniques and results in great detail.
We establish convolution inequalities for Besov spaces $B_{p,q}^s$ and Triebel--Lizorkin spaces $F_{p,q}^s$. As an application, we study the mapping properties of convolution semigroups, considered as operators on the function spaces $A_{p,q}^s$, $A \in \{B,F\}$. Our results apply to a wide class of convolution semigroups including the Gauß--Weierstraß semigroup, stable semigroups and heat kernels for higher-order powers of the Laplacian $(-Δ)^m$, and we can derive various caloric smoothing estimates.
For generators of Markov semigroups which lack a spectral gap, it is shown how bounds on the density of states near zero lead to a so-called "weak Poincaré inequality" (WPI), originally introduced by Liggett [Ann. Probab., 1991]. Applications to general classes of constant coefficient pseudodifferential operators are studied. Particular examples are the heat semigroup and the semigroup generated by the fractional Laplacian in the whole space, where the optimal decay rates are recovered. Moreover, the classical Nash inequality appears as a special case of the WPI for the heat semigroup.
Using the tools of stochastic analysis, we prove various gradient estimates and Harnack inequalities for Feynman-Kac semigroups with possibly unbounded potentials. One of the main results is a derivative formula which can be used to characterize a lower bound on Ricci curvature using a potential.
我们应用一种方法开发的作者之一，见\cite{Arl1}，本地化的数值域的\textit{quasi-sectorial}压缩半群。我们的主要定理建立了拟扇形压缩半群$\{\exp(- t S)\}_{t\ge 0}$的数值域与极大{扇形}生成元$S$之间的关系。我们还对这类半群给出了算子模Euler公式逼近的速率$O(1/n)$：$\exp(- t S)=\lim\limits_{n\to \infty}(I+tS/n)^{-n}$，$t\ge 0$的一个新的证明.
We study Poincar{é} inequalities and long-time behavior for diffusion processes on R^n under a variable curvature lower bound, in the sense of Bakry-Emery. We derive various estimates on the rate of convergence to equilibrium in L^1 optimal transport distance, as well as bounds on the constant in the Poincar{é} inequality in several situations of interest, including some where curvature may be negative. In particular, we prove a self-improvement of the Bakry-Emery estimate for Poincar{é} inequalities when curvature is positive but not constant.
In this paper we establish $L^p$-boundedness properties for variation, oscillation and jump operators associated with Riesz transforms and Poisson semigroups related to Laguerre polynomial expansions.
In this paper we describe the Euler semigroup $\{e^{-t\mathbb{E}^{*}\mathbb{E}}\}_{t>0}$ on homogeneous Lie groups, which allows us to obtain various types of the Hardy-Sobolev and Gagliardo-Nirenberg type inequalities for the Euler operator $\mathbb{E}$. Moreover, the sharp remainder terms of the Sobolev type inequality, maximal Hardy inequality and $|\cdot|$-radial weighted Hardy-Sobolev type inequality are established.
We study probability inequalities leading to tail estimates in a general semigroup $\mathscr{G}$ with a translation-invariant metric $d_{\mathscr{G}}$. (An important and central example of this in the functional analysis literature is that of $\mathscr{G}$ a Banach space.) Using our prior work [Ann. Prob. 2017] that extends the Hoffmann-Jorgensen inequality to all metric semigroups, we obtain tail estimates and approximate bounds for sums of independent semigroup-valued random variables, their moments, and decreasing rearrangements. In particular, we obtain the "correct" universal constants in…
We introduce heat semigroup-based Besov classes in the general framework of Dirichlet spaces. General properties of those classes are studied and quantitative regularization estimates for the heat semigroup in this scale of spaces are obtained. As a highlight of the paper, we obtain a far reaching $L^p$-analogue, $p \ge 1$, of the Sobolev inequality that was proved for $p=2$ by N. Varopoulos under the assumption of ultracontractivity for the heat semigroup. The case $p=1$ is of special interest since it yields isoperimetric type inequalities.
We extend the classical Bernstein inequality to a general setting including Schr{ö}dinger operators and divergence form elliptic operators on Riemannian manifolds or domains. Moreover , we prove a new reverse inequality that can be seen as the dual of the Bernstein inequality. The heat kernel will be the backbone of our approach but we also develop new techniques such as semi-classical Bernstein inequalities, weak factorization of smooth functions {à} la Dixmier-Malliavin and BM O -- L $\infty$ multiplier results (in contrast to the usual L $\infty$ -- BM O ones). Also, our approach reveals a …
The optimal transport map between the standard Gaussian measure and an $α$-strongly log-concave probability measure is $α^{-1/2}$-Lipschitz, as first observed in a celebrated theorem of Caffarelli. In this paper, we apply two classical covariance inequalities (the Brascamp-Lieb and Cramér-Rao inequalities) to prove a sharp bound on the Lipschitz constant of the map that arises from entropically regularized optimal transport. In the limit as the regularization tends to zero, we obtain an elegant and short proof of Caffarelli's original result. We also extend Caffarelli's theorem to the setting …
It is well known that Nash-type inequalities for symmetric Dirichlet forms are equivalent to on-diagonal heat kernel upper bounds for the associated symmetric Markov semigroups. In this paper, we show that both imply (and hence are equivalent to) off-diagonal heat kernel upper bounds under some mild assumptions. Our approach is based on a new generalized Davies' method. Our results extend that of \cite{CKS} for Nash-type inequalities with power order considerably and also extend that of \cite{Gri} for second order differential operators on a complete non-compact manifold.
We prove that every reversible Markov semigroup which satisfies a Poincaré inequality satisfies a matrix-valued Poincaré inequality for Hermitian $d\times d$ matrix valued functions, with the same Poincaré constant. This generalizes recent results [Aoun et al. 2019, Kathuria 2019] establishing such inequalities for specific semigroups and consequently yields new matrix concentration inequalities. The short proof follows from the spectral theory of Markov semigroup generators.
The purpose of the present notes is to examine the following issues related to the the Chernoff estimate: (1) For contractions on a Banach space we modify the $\sqrt{n}$-estimate and apply it in the proof of the Chernoff product formula for $C_0$-semigroups in the \textit{strong} operator topology. (2) We use the idea of a {probabilistic} approach, proving the Chernoff estimate in the strong operator topology, to uplift it to the \textit{operator-norm} estimate for \textit{quasi-sectorial} contraction semigroups. (3) The operator-norm Chernoff estimate is applied to {quasi-sectorial} contracti…
We prove that, in a large class of Banach lattices, the fixed space of each commuting family of positive linear contractions is a lattice subspace. As consequences, new cyclicity results for the peripheral point spectra of positive operators and semigroups are derived; we also pose an open problem that naturally occurs in this context. Finally, a variety of counterexamples is presented to point out some limits of our results.
A contraction semigroup T on a Hilbert space H and its cogenerator S define an algebra, the limit algebra - which determines the structure of the subspace of weakly Poisson recurrent vectors and gives a necessary and sufficient condition for T and S to be weakly stable equivalent.
This paper considers universal Hilbert space operators in the sense of Rota, and gives criteria for universality of semigroups in the context of uniformly continuous semigroups and contraction semigroups. Specific examples are given. Universal semigroups provide models for these classes of semigroups: following a line of research initiated by Shimorin, models for concave semigroups are developed, in terms of shifts on reproducing kernel Hilbert spaces.
