We develop methods for studying transition operators on metric spaces that are invariant under a co-compact group which acts properly. A basic requirement is a decomposition of such operators with respect to the group orbits. We then introduce "reduced" transition operators on the compact factor space whose norms and spectral radii are upper bounds for the $L^p$-norms and spectral radii of the original operator. If the group is amenable then the spectral radii of the original and reduced operators coincide, and under additional hypotheses, this is also sufficient for amenability. Further bounds involve the modular function of the group. In this framework, we prove among other things that the bottom of the spectrum of the Laplacian on a co-compact Riemannian manifold is 0 if and only if the group is amenable and unimodular. The same result holds for Euclidean simplicial complexes. On a geodesic, proper metric space with co-compact isometry group action, the averaging operator over balls with a fixed radius has norm equal to 1 if and only if the group is amenable and unimodular. The technique also allows explicit computation of spectral radii when the group is amenable.

We study the appropriate versions of parabolicity stochastic completeness and related Liouville properties for a general class of operators which include the $p$-Laplace operator, and the non linear singular operators in non-diagonal form considered by J. Serrin and collaborators.

In this note we continue a theme taken up in part I, see [Gzyl and Recht: The geometry on the class of probabilities I: The finite dimensional case. Rev. Mat. Iberoamericana 22 (2006), 545-558.], namely to provide a geometric interpretation of exponential families as end points of geodesics of a non-metric connection in a function space. For that we characterize the space of probability densities as a projective space in the class of strictly positive functions, and these will be regarded as a homogeneous reductive space in the class of all bounded complex valued functions. We shall develop everything in a generic $\mathcal{C}^*$-algebra setting, but shall have the function space model in mind.

Toeplitz operators on strongly pseudoconvex domains in $\mathbb{C}^n$, constructed from the Bergman projection and with symbol equal to a positive power of the distance to the boundary, are considered. The mapping properties of these operators on $L^p$, as the power of the distance varies, are established.

$m$-Berezin transforms are introduced for bounded operators on the Bergman space of the unit ball. The norm of the $m$-Berezin transform as a linear operator from the space of bounded operators to $L^{\infty}$ is found. We show that the $m$-Berezin transforms are commuting with each other and Lipschitz with respect to the pseudo-hyperbolic distance on the unit ball. Using the $m$-Berezin transforms we show that a radial operator in the Toeplitz algebra is compact iff its Berezin transform vanishes on the boundary of the unit ball.

Let $X$ be a set with a symmetric kernel $d$ (not necessarily a distance). The space $(X,d)$ is said to have the weak (resp. strong) covering property of degree $\leq m$ [briefly \textbf{prf}$(m)$ (resp. \textbf{prF}$(m)$)], if, for each family $\mathcal{B}$ of closed balls of $(X,d)$ with radii in a decreasing sequence (resp. with bounded radii), there is a subfamily, covering the center of each element of $\mathcal{B}$, and of order $\leq m$ (resp. spliting into $m$ disjoint families). Since Besicovitch, covering properties are known to be the main tool for proving derivation theorems for any pair of measures on $(X,d)$. Assuming that any ball for $d$ belongs to the Baire $\sigma$-algebra for $d$, we show that the \textbf{prf} implies an almost sure derivation theorem. This implication was stated by D. Preiss when $(X,d)$ is a complete separable metric space. With stronger measurability hypothesis (to be stated later in this paper), we show that the \textbf{prf} restricted to balls with constant radius implies a derivation theorem with convergence in measure. We show easily that an equivalent to the \textbf{prf}$(m+1)$ (resp. to the \textbf{prf}$(m+1)$ restricted to balls with constant radius) is that the Nagata-dimension (resp. the De Groot-dimension) of $(X,d)$ is $\leq m$. These two dimensions (see J.I. Nagata) are not lesser than the topological dimension; for $\mathbb{R}^n$ with any given norm ($n > 1$), they are $> n$. For spaces with nonnegative curvature $\geq 0$ (for example for $\mathbb{R}^n$ with any given norm), we express these dimensions as the cardinality of a net; in these spaces, we give a similar upper bound for the degree of the \textbf{prF} (generalizing a result of Furedi and Loeb for $\mathbb{R}^n$) and try to obtain the exact degree in $\mathbb{R}$ and $\mathbb{R}^2$.

Let $R$ be a commutative Noetherian ring, and let $\mathbf{x}= x_1, \ldots, x_d$ be a regular $R$-sequence contained in the Jacobson radical of $R$. An ideal $I$ of $R$ is said to be a monomial ideal with respect to $\mathbf{x}$ if it is generated by a set of monomials $x_1^{e_1}\ldots x_d^{e_d}$. The monomial closure of $I$, denoted by $\widetilde{I}$, is defined to be the ideal generated by the set of all monomials $m$ such that $m^n\in I^n$ for some $n\in \mathbb{N}$. It is shown that the sequences $\mathrm{Ass}_RR/\widetilde{I^n}$ and $\mathrm{Ass}_R\widetilde{I^n}/I^n$, $n=1,2, \ldots,$ of associated prime ideals are increasing and ultimately constant for large $n$. In addition, some results about the monomial ideals and their integral closures are included.

We prove that the classical Coifman-Meyer theorem holds on any polydisc $\mathbf{T}^d$ of arbitrary dimension $d\geq 1$.

We study $L^p (\mathbb{R}^n)\rightarrow L^{\alpha ,\infty}_{d\mu (\sigma)}(L^{\infty}_{dt})$ estimates for the Radon transform in certain cases where the dimension of the measure $\mu$ on $\Sigma^{(n-1)}$ is less than $n-1$.

We introduce and study a notion of Orlicz hypercontractive semigroups. We analyze their relations with general $F$-Sobolev inequalities, thus extending Gross hypercontractivity theory. We provide criteria for these Sobolev type inequalities and for related properties. In particular, we implement in the context of probability measures the ideas of Maz'ja's capacity theory, and present equivalent forms relating the capacity of sets to their measure. Orlicz hypercontractivity efficiently describes the integrability improving properties of the Heat semigroup associated to the Boltzmann measures $\mu_{\alpha}(dx) = (Z_{\alpha})^{-1} e^{-2|x|^{\alpha}} dx$, when $\alpha\in (1,2)$. As an application we derive accurate isoperimetric inequalities for their products. This completes earlier works by Bobkov-Houdré and Talagrand, and provides a scale of dimension free isoperimetric inequalities as well as comparison theorems.

We continue the investigation initiated in [Grafakos and Li: Uniform bounds for the bilinear Hilbert transforms, I. Ann. of Math. (2) 159 (2004), 889-933] of uniform $L^{p}$ bounds for the family of bilinear Hilbert transforms $$ H_{\alpha,\beta} (f,g)(x) = \text{p.v.} \displaystyle\int_{\mathbb{R}} f(x-\alpha t) g(x-\beta t) \frac{dt}{t} \,. $$ In this work we show that $H_{\alpha,\beta}$ map $L^{p_1}(\mathbb R)\times L^{p_2}(\mathbb R)$ into $L^p(\mathbb R)$ uniformly in the real parameters $\alpha$, $\beta$ satisfying $|\frac{\alpha}{\beta}-1|\ge c > 0$ when $1 < p_1, p_2 < 2$ and $\frac{2}{3} < p= \frac{p_1p_2}{p_1+p_2} < \infty$. As a corollary we obtain $L^p \times L^\infty \to L^p$ uniform bounds in the range $4/3 < p < 4 $ for the $H_{1,\alpha}$'s when $\alpha\in [0,1)$.