Starting from a uniquely ergodic action of a locally compact group G on a compact space $X_0$, we consider non-commutative skew-product extensions of the dynamics, on the crossed product $C(X_0)\rtimes _\alpha {\mathbb Z}$, through a $1$-cocycle of G in ${\mathbb T}$, with $\alpha $ commuting with the given dynamics. We first prove that any two such skew-product extensions are conjugate if and only if the corresponding cocycles are cohomologous. We then study unique ergodicity and unique ergodicity with respect to the fixed-point subalgebra by characterizing both in terms of the cocycle assigning the dynamics. The set of all invariant states is also determined: it is affinely homeomorphic with ${\mathcal P}({\mathbb T})$, the Borel probability measures on the one-dimensional torus ${\mathbb T}$, as long as the system is not uniquely ergodic. Finally, we show that unique ergodicity with respect to the fixed-point subalgebra of a skew-product extension amounts to the uniqueness of an invariant conditional expectation onto the fixed-point subalgebra.

Let $({\mathcal{X}},d,\unicode[STIX]{x1D707})$ be a metric measure space satisfying the geometrically doubling condition and the upper doubling condition. In this paper, the authors establish the John-Nirenberg inequality for the regularized BLO space $\widetilde{\operatorname{RBLO}}(\unicode[STIX]{x1D707})$.

Let $\mu $ be a nonnegative Radon measure on ${{\mathbb{R}}^{d}}$ that satisfies the growth condition that there exist constants ${{C}_{0}}\,>\,0$ and $n\,\in \,(0,\,d]$ such that for all $x\,\in \,{{\mathbb{R}}^{d}}$ and $r\,>\,0$ , $\mu \left( B\left( x,\,r \right) \right)\,\le \,{{C}_{0}}{{r}^{n}}$ , where $B(x,\,r)$ is the open ball centered at $x$ and having radius $r$ . In this paper, the authors prove that if $f$ belongs to the $\text{BMO}$ -type space $\text{RBMO(}\mu \text{)}$ of Tolsa, then the homogeneous maximal function ${{\dot{\mathcal{M}}}_{s}}\left( f \right)$ (when ${{\mathbb{R}}^{d}}$ is not an initial cube) and the inhomogeneous maximal function ${{\overset{{}}{\mathop{\mathcal{M}}}\,}_{s}}\left( f \right)$ (when ${{\mathbb{R}}^{d}}$ is an initial cube) associated with a given approximation of the identity $S $ of Tolsa are either infinite everywhere or finite almost everywhere, and in the latter case, ${{\dot{\mathcal{M}}}_{s}}$ and ${{\mathcal{M}}_{s}}$ are bounded from $\text{RBMO(}\mu \text{)}$ to the $\text{BLO}$ -type space $\text{RBMO(}\mu \text{)}$ . The authors also prove that the inhomogeneous maximal operator ${{\mathcal{M}}_{s}}$ is bounded from the local $\text{BMO}$ -type space $\text{rbmo(}\mu \text{)}$ to the local $\text{BLO}$ -type space $\text{rblo(}\mu \text{)}$ .

Let 𝒳 be a space of homogeneous type in the sense of Coifman and Weiss. In this paper, two weighted estimates related to weights are established for singular integral operators with nonsmooth kernels via a new sharp maximal operator associated with a generalized approximation to the identity. As applications, the weighted Lp(𝒳) and weighted endpoint estimates with general weights are obtained for singular integral operators with nonsmooth kernels, their commutators with BMO (𝒳) functions, and associated maximal operators. Some applications to holomorphic functional calculi of elliptic operators and Schrödinger operators are also presented.

The concept of semi-bounded generalized hypergroups (SBG hypergroups) is developed. These hypergroups are more special than generalized hypergroups introduced by Obata and Wildberger and more general than discrete hypergroups or even discrete signed hypergroups. The convolution of measures and functions is studied. In the case of commutativity we define the dual objects and prove some basic theorems of Fourier analysis. Furthermore, we investigate the relationship between orthogonal polynomials and generalized hypergroups. We discuss the Jacobi polynomials as an example.

This paper studies the Laplacian operator on a metrized graph, and its spectral theory.

Let (X, ρ, μ)d, θ be a space of homogeneous type with d < 0 and θ ∈ (0, 1], b be a para-accretive function, ε ∈ (0, θ], ∣s∣ > ∈ and a0 ∈ (0, 1) be some constant depending on d, ∈ and s. The authors introduce the Besov space bBspq (X) with a0 > p ≧ ∞, and the Triebel-Lizorkin space bFspq (X) with a0 > p > ∞ and a0 > q ≧∞ by first establishing a Plancherel-Pôlya-type inequality. Moreover, the authors establish the frame and the Littlewood-Paley function characterizations of these spaces. Furthermore, the authors introduce the new Besov space b−1 Bs (X) and the Triebel-Lizorkin space b−1 Fspq (X). The relations among these spaces and the known Hardy-type spaces are presented. As applications, the authors also establish some real interpolation theorems, embedding theorems, T b theorems, and the lifting property by introducing some new Riesz operators of these spaces.

Let $K$ be a function on a unimodular locally compact group $G$ , and denote by ${{K}_{n}}\,=\,K\,*\,K\,*\cdots *\,K$ the $n$ -th convolution power of $K$ . Assuming that $K$ satisfies certain operator estimates in ${{L}^{2}}\left( G \right)$ , we give estimates of the norms ${{\left\| {{K}_{n}} \right\|}_{2}}$ and ${{\left\| {{K}_{n}} \right\|}_{\infty }}$ for large $n$ . In contrast to previous methods for estimating ${{\left\| {{K}_{n}} \right\|}_{\infty }}$ , we do not need to assume that the function $K$ is a probability density or nonnegative. Our results also adapt for continuous time semigroups on $G$ . Various applications are given, for example, to estimates of the behaviour of heat kernels on Lie groups.

Let µ be Radon measure on Rd which may be non doubling. The only condition that µ must satisfy is the size condition µ(B(x, r)) ≤ Crn for some fixed n є (0, d). Recently, Tolsa introduced the spaces RBMO(µ) and Hatb1.∞ (µ), which, in some ways, play the role of the classical spaces BMO and H1 in case u is a doubling measure. In this paper, the author considers the local versions of the spaces RBMO(µ) and Hatb1.∞ (µ) in the sense of Goldberg and establishes the connections between the spaces RBMO(µ) and Hatb1.∞ (µ) with their local versions. An interpolation result of linear operators is also given.

Calderón type reproducing formulae with applications have been studied on one- and higher-dimensional Lipschitz graphs. In this note we study higher order Calderón reproducing formulae on star-shaped and non-star-shaped closed Lipschitz curves and surfaces.

Let G be a locally compact group and H an open subgroup of G. The embeddings of group C*-algebras associated with H into the group C*-algebras associated with G are studied. Three conditions for the embeddings given in terms of C*-norms of the group algebras, group representations and positive definite functions are shown to be equivalent. As corollary, we prove that the full C*-algebra of H can be embedded into the full C*-algebra of G in a natural way as well as the case for the reduced group C*-algebras. We also show that the embeddings hold for their duals and double duals.