We derive various integral representation formulas for a function minus a polynomial in terms of vector field gradients of the function of appropriately high order. Our results hold in the general setting of metric spaces, including those associated with Carnot-Carathéodory vector fields, under the assumption that a suitable $L^1$ to $L^1$ Poincaré inequality holds. Of particular interest are the representation formulas in Euclidean space and stratified groups, where polynomials exist and $L^1$ to $L^1$ Poincaré...

The aim of this paper is to extend the study of Riesz transforms associated to Dunkl Ornstein-Uhlenbeck operator considered by A. Nowak, L. Roncal and K. Stempak to higher order.

Continuous actions of topological groups on compact Hausdorff spaces X are investigated which induce almost periodic functions in the corresponding commutative C*-algebra. The unique invariant mean on the group resulting from averaging allows one to derive a C*-valued inner product and a Hilbert C*-module which serve as an environment to describe characteristics of the group action. For Lyapunov stable actions the derived invariant mean $M\left({\varphi }_x\right)$ is continuous on X for any ϕ ∈ C(X), and the induced C*-valued...

The aim of this paper is to show that the integral and derivative operators defined by local regularities are homeomorphisms for generalized Besov and Triebel-Lizorkin spaces with local regularities. The underlying geometry is that of homogeneous type spaces and the functions defining local regularities belong to a larger class of growth functions than the potentials tα, related to classical fractional integral and derivative operators and Besov and Triebel-Lizorkin spaces.

This paper considers the Lipschitz subalgebras $\Lambda (\alpha ,p,𝒜)$ of a homogeneous algebra on the circle. Interpolation space theory is used to derive estimates for the multiplier norm on closed primary ideals in $\Lambda (\alpha ,p;𝒜)$, $\alpha \ne \left[\alpha \right]$. From these estimates the Ditkin and Analytic Ditkin conditions for $\Lambda (\alpha ,p;𝒜)$ follow easily. Thus the well-known theory of (regular) Banach algebras satisfying the Ditkin condition applies to $\Lambda (\alpha ;,p;𝒜)$ as does the theory developed in part I of this series which requires the Analytic Ditkin condition.Examples are discussed...

Let S be a Rees semigroup, and let ℓ¹(S) be its convolution semigroup algebra. Using Morita equivalence we show that bounded Hochschild homology and cohomology of ℓ¹(S) are isomorphic to those of the underlying discrete group algebra.

Let A and B be semisimple commutative Banach algebras with bounded approximate identities. We investigate the problem of extending a homomorphism φ: A → B to a homomorphism of the multiplier algebras M(A) and M(B) of A and B, respectively. Various sufficient conditions in terms of B (or B and φ) are given that allow the construction of such extensions. We exhibit a number of classes of Banach algebras to which these criteria apply. In addition, we prove a polar decomposition for homomorphisms from...

We prove that, for a distinguished laplacian on an Iwasawa AN group corresponding to a complex semisimple Lie group, a Hörmander type multiplier theorem holds. Our argument is based on Littlewood-Paley theory.

Let X be a Polish space, and let ${\left(A_p\right)}_{p\in \omega }$ be a sequence of $G_{\delta }$ hereditary subsets of K(X) (the space of compact subsets of X). We give a general criterion which allows one to decide whether ${\cup }_{p\in \omega }{A}_p$ is a true ${\sum }_3^0$ subset of K(X). We apply this criterion to show that several natural families of thin sets from harmonic analysis are true ${\sum }_3^0$.

For every closed subset C in the dual space ${Ĥ}_n$ of the Heisenberg group $H_n$ we describe via the Fourier transform the elements of the hull-minimal ideal j(C) of the Schwartz algebra $S\left(H_n\right)$ and we show that in general for two closed subsets $C_1,C_2$ of ${Ĥ}_n$ the product of $j\left(C_1\right)$ and $j\left(C_2\right)$ is different from $j\left(C_1\cap C_2\right)$.