26. CMB 2001 (vol 44 pp. 231)
|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.
27. CMB 2000 (vol 43 pp. 330)
|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 set
28. CMB 1999 (vol 42 pp. 169)
|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, eigenvalues
Categories:35K05, 43A85, 35K15, 80A20
29. CMB 1998 (vol 41 pp. 392)
|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.
30. CMB 1997 (vol 40 pp. 316)
|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.
31. CMB 1997 (vol 40 pp. 296)
|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
32. CMB 1997 (vol 40 pp. 183)
|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
33. CMB 1997 (vol 40 pp. 133)
|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