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26. CMB 2010 (vol 54 pp. 126)

Jin, Yongyang; Zhang, Genkai
 Fundamental Solutions of Kohn Sub-Laplacians on Anisotropic Heisenberg Groups and H-type Groups We prove that the fundamental solutions of Kohn sub-Laplacians $\Delta + i\alpha \partial_t$ on the anisotropic Heisenberg groups are tempered distributions and have meromorphic continuation in $\alpha$ with simple poles. We compute the residues and find the partial fundamental solutions at the poles. We also find formulas for the fundamental solutions for some matrix-valued Kohn type sub-Laplacians on H-type groups. Categories:22E30, 35R03, 43A80

27. CMB 2010 (vol 54 pp. 3)

Bakonyi, M.; Timotin, D.
 Extensions of Positive Definite Functions on Amenable Groups Let $S$ be a subset of an amenable group $G$ such that $e\in S$ and $S^{-1}=S$. The main result of this paper states that if the Cayley graph of $G$ with respect to $S$ has a certain combinatorial property, then every positive definite operator-valued function on $S$ can be extended to a positive definite function on $G$. Several known extension results are obtained as corollaries. New applications are also presented. Categories:43A35, 47A57, 20E05

28. CMB 2010 (vol 53 pp. 491)

Jizheng, Huang; Liu, Heping
 The Weak Type (1,1) Estimates of Maximal Functions on the Laguerre Hypergroup In this paper, we discuss various maximal functions on the Laguerre hypergroup $\mathbf{K}$ including the heat maximal function, the Poisson maximal function, and the Hardy--Littlewood maximal function which is consistent with the structure of hypergroup of $\mathbf{K}$. We shall establish the weak type $(1,1)$ estimates for these maximal functions. The $L^p$ estimates for $p>1$ follow from the interpolation. Some applications are included. Keywords:Laguerre hypergroup, maximal function, heat kernel, Poisson kernelCategories:42B25, 43A62

29. CMB 2010 (vol 53 pp. 447)

Choi, Yemon
 Injective Convolution Operators on l∞(Γ) are Surjective Let $\Gamma$ be a discrete group and let $f \in \ell^{1}(\Gamma)$. We observe that if the natural convolution operator $\rho_f: \ell^{\infty}(\Gamma)\to \ell^{\infty}(\Gamma)$ is injective, then $f$ is invertible in $\ell^{1}(\Gamma)$. Our proof simplifies and generalizes calculations in a preprint of Deninger and Schmidt by appealing to the direct finiteness of the algebra $\ell^{1}(\Gamma)$. We give simple examples to show that in general one cannot replace $\ell^{\infty}$ with $\ell^{p}$, $1\leq p< \infty$, nor with $L^{\infty}(G)$ for nondiscrete $G$. Finally, we consider the problem of extending the main result to the case of weighted convolution operators on $\Gamma$, and give some partial results. Categories:43A20, 46L05, 43A22

30. CMB 2009 (vol 40 pp. 133)

Blackmore, T. D.
 Derivations from totally ordered semigroup algebras into their duals For a well-behaved measure $\mu$, on a locally compact totally ordered set $X$, with continuous part $\mu_c$, we make $L^p(X,\mu_c)$ into a commutative Banach bimodule over the totally ordered semigroup algebra $L^p(X,\mu)$, in such a way that the natural surjection from the algebra to the module is a bounded derivation. This gives rise to bounded derivations from $L^p(X,\mu)$ into its dual module and in particular shows that if $\mu_c$ is not identically zero then $L^p(X,\mu)$ is not weakly amenable. We show that all bounded derivations from $L^1(X,\mu)$ into its dual module arise in this way and also describe all bounded derivations from $L^p(X,\mu)$ into its dual for $1 Categories:43A20, 46M20 31. CMB 2009 (vol 40 pp. 316) Hudzik, H.; Kamińska, A.; Mastyło, M.  On geometric properties of Orlicz-Lorentz spaces Criteria for local uniform rotundity and midpoint local uniform rotundity in Orlicz-Lorentz spaces with the Luxemburg norm are given. Strict$K$-monotonicity and Kadec-Klee property are also discussed. Category:43B20 32. CMB 2009 (vol 40 pp. 296) Hare, Kathryn E.  A general approach to Littlewood-Paley theorems for orthogonal families A general lacunary Littlewood-Paley type theorem is proved, which applies in a variety of settings including Jacobi polynomials in$[0, 1]$,$\su$, and the usual classical trigonometric series in$[0, 2 \pi)$. The theorem is used to derive new results for$\LP$multipliers on$\su$and Jacobi$\LP$multipliers. Categories:42B25, 42C10, 43A80 33. CMB 2009 (vol 40 pp. 183) Kepert, Andrew G.  The range of group algebra homomorphisms A characterisation of the range of a homomorphism between two commutative group algebras is presented which implies, among other things, that this range is closed. The work relies mainly on the characterisation of such homomorphisms achieved by P.~J.~Cohen. Categories:43A22, 22B10, 46J99 34. CMB 2008 (vol 51 pp. 60) Janzen, David  F{\o}lner Nets for Semidirect Products of Amenable Groups For unimodular semidirect products of locally compact amenable groups$N$and$H$, we show that one can always construct a F{\o}lner net of the form$(A_\alpha \times B_\beta)$for$G$, where$(A_\alpha)$is a strong form of F{\o}lner net for$N$and$(B_\beta)$is any F{\o}lner net for$H$. Applications to the Heisenberg and Euclidean motion groups are provided. Categories:22D05, 43A07, 22D15, 43A20 35. CMB 2007 (vol 50 pp. 291) Sarkar, Rudra P.; Sengupta, Jyoti  Beurling's Theorem and Characterization of Heat Kernel for Riemannian Symmetric Spaces of Noncompact Type We prove Beurling's theorem for rank$1$Riemannian symmetric spaces and relate its consequences with the characterization of the heat kernel of the symmetric space. Keywords:Beurling's Theorem, Riemannian symmetric spaces, uncertainty principleCategories:22E30, 43A85 36. CMB 2007 (vol 50 pp. 56) Gourdeau, F.; Pourabbas, A.; White, M. C.  Simplicial Cohomology of Some Semigroup Algebras In this paper, we investigate the higher simplicial cohomology groups of the convolution algebra$\ell^1(S)$for various semigroups$S$. The classes of semigroups considered are semilattices, Clifford semigroups, regular Rees semigroups and the additive semigroups of integers greater than$a$for some integer$a$. Our results are of two types: in some cases, we show that some cohomology groups are$0$, while in some other cases, we show that some cohomology groups are Banach spaces. Keywords:simplicial cohomology, semigroup algebraCategory:43A20 37. CMB 2006 (vol 49 pp. 549) Führ, Hartmut  Hausdorff--Young Inequalities for Group Extensions This paper studies Hausdorff--Young inequalities for certain group extensions, by use of Mackey's theory. We consider the case in which the dual action of the quotient group is free almost everywhere. This result applies in particular to yield a Hausdorff--Young inequality for nonunimodular groups. Categories:43A30, 43A15 38. CMB 2004 (vol 47 pp. 445) Pirkovskii, A. Yu.  Biprojectivity and Biflatness for Convolution Algebras of Nuclear Operators For a locally compact group$G$, the convolution product on the space$\nN(L^p(G))$of nuclear operators was defined by Neufang \cite{Neuf_PhD}. We study homological properties of the convolution algebra$\nN(L^p(G))$and relate them to some properties of the group$G$, such as compactness, finiteness, discreteness, and amenability. Categories:46M10, 46H25, 43A20, 16E65 39. CMB 2004 (vol 47 pp. 389) He, Jianxun  An Inversion Formula of the Radon Transform Transform on the Heisenberg Group In this paper we give an inversion formula of the Radon transform on the Heisenberg group by using the wavelets defined in [3]. In addition, we characterize a space such that the inversion formula of the Radon transform holds in the weak sense. Keywords:wavelet transform, Radon transform, Heisenberg groupCategories:43A85, 44A15 40. CMB 2004 (vol 47 pp. 475) Wade, W. R.  Uniqueness of Almost Everywhere Convergent Vilenkin Series D. J. Grubb [3] has shown that uniqueness holds, under a mild growth condition, for Vilenkin series which converge almost everywhere to zero. We show that, under even less restrictive growth conditions, one can replace the limit function 0 by an arbitrary$f\in L^q$, when$q>1$. Categories:43A75, 42C10 41. CMB 2004 (vol 47 pp. 215) Jaworski, Wojciech  Countable Amenable Identity Excluding Groups A discrete group$G$is called \emph{identity excluding\/} if the only irreducible unitary representation of$G$which weakly contains the$1$-dimensional identity representation is the$1$-dimensional identity representation itself. Given a unitary representation$\pi$of$G$and a probability measure$\mu$on$G$, let$P_\mu$denote the$\mu$-average$\int\pi(g) \mu(dg)$. The goal of this article is twofold: (1)~to study the asymptotic behaviour of the powers$P_\mu^n$, and (2)~to provide a characterization of countable amenable identity excluding groups. We prove that for every adapted probability measure$\mu$on an identity excluding group and every unitary representation$\pi$there exists and orthogonal projection$E_\mu$onto a$\pi$-invariant subspace such that$s$-$\lim_{n\to\infty}\bigl(P_\mu^n- \pi(a)^nE_\mu\bigr)=0$for every$a\in\supp\mu$. This also remains true for suitably defined identity excluding locally compact groups. We show that the class of countable amenable identity excluding groups coincides with the class of$\FC$-hypercentral groups; in the finitely generated case this is precisely the class of groups of polynomial growth. We also establish that every adapted random walk on a countable amenable identity excluding group is ergodic. Categories:22D10, 22D40, 43A05, 47A35, 60B15, 60J50 42. CMB 2004 (vol 47 pp. 168) Baake, Michael; Sing, Bernd  Kolakoski-$(3,1)$Is a (Deformed) Model Set Unlike the (classical) Kolakoski sequence on the alphabet$\{1,2\}$, its analogue on$\{1,3\}$can be related to a primitive substitution rule. Using this connection, we prove that the corresponding bi-infinite fixed point is a regular generic model set and thus has a pure point diffraction spectrum. The Kolakoski-$(3,1)$sequence is then obtained as a deformation, without losing the pure point diffraction property. Categories:52C23, 37B10, 28A80, 43A25 43. CMB 2003 (vol 46 pp. 268) Puls, Michael J.  Group Cohomology and$L^p$-Cohomology of Finitely Generated Groups Let$G$be a finitely generated, infinite group, let$p>1$, and let$L^p(G)$denote the Banach space$\{ \sum_{x\in G} a_xx \mid \sum_{x\in G} |a_x |^p < \infty \}$. In this paper we will study the first cohomology group of$G$with coefficients in$L^p(G)$, and the first reduced$L^p$-cohomology space of$G$. Most of our results will be for a class of groups that contains all finitely generated, infinite nilpotent groups. Keywords:group cohomology,$L^p$-cohomology, central element of infinite order, harmonic function, continuous linear functionalCategories:43A15, 20F65, 20F18 44. CMB 2002 (vol 45 pp. 483) Baake, Michael  Diffraction of Weighted Lattice Subsets A Dirac comb of point measures in Euclidean space with bounded complex weights that is supported on a lattice$\varGamma$inherits certain general properties from the lattice structure. In particular, its autocorrelation admits a factorization into a continuous function and the uniform lattice Dirac comb, and its diffraction measure is periodic, with the dual lattice$\varGamma^*$as lattice of periods. This statement remains true in the setting of a locally compact Abelian group whose topology has a countable base. Keywords:diffraction, Dirac combs, lattice subsets, homometric setsCategories:52C07, 43A25, 52C23, 43A05 45. CMB 2002 (vol 45 pp. 436) Sawyer, P.  The Spherical Functions Related to the Root System$B_2$In this paper, we give an integral formula for the eigenfunctions of the ring of differential operators related to the root system$B_2$. Categories:43A90, 22E30, 33C80 46. CMB 2001 (vol 44 pp. 231) Rosenblatt, Joseph M.; Willis, George A.  Weak Convergence Is Not Strong Convergence For Amenable Groups Let$G$be an infinite discrete amenable group or a non-discrete amenable group. It is shown how to construct a net$(f_\alpha)$of positive, normalized functions in$L_1(G)$such that the net converges weak* to invariance but does not converge strongly to invariance. The solution of certain linear equations determined by colorings of the Cayley graphs of the group are central to this construction. Category:43A07 47. CMB 2000 (vol 43 pp. 330) Hare, Kathryn E.  Maximal Operators and Cantor Sets We consider maximal operators in the plane, defined by Cantor sets of directions, and show such operators are not bounded on$L^2$if the Cantor set has positive Hausdorff dimension. Keywords:maximal functions, Cantor set, lacunary setCategories:42B25, 43A46 48. CMB 1999 (vol 42 pp. 169) Ding, Hongming  Heat Kernels of Lorentz Cones We obtain an explicit formula for heat kernels of Lorentz cones, a family of classical symmetric cones. By this formula, the heat kernel of a Lorentz cone is expressed by a function of time$t$and two eigenvalues of an element in the cone. We obtain also upper and lower bounds for the heat kernels of Lorentz cones. Keywords:Lorentz cone, symmetric cone, Jordan algebra, heat kernel, heat equation, Laplace-Beltrami operator, eigenvaluesCategories:35K05, 43A85, 35K15, 80A20 49. CMB 1998 (vol 41 pp. 392) Daly, James E.; Phillips, Keith L.  A note on$H^1$multipliers for locally compact Vilenkin groups Kitada and then Onneweer and Quek have investigated multiplier operators on Hardy spaces over locally compact Vilenkin groups. In this note, we provide an improvement to their results for the Hardy space$H^1\$ and provide examples showing that our result applies to a significantly larger group of multipliers. Categories:43A70, 44A35
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