Let G be a locally compact abelian group, 1 < p < ∞, and A be a commutative Banach algebra. In this paper we study the space of multipliers on Lp (G, A) and characterize it as the space of multipliers of certain banach algebra. We also study the multipliers space on L1 (G, A) ∩ Lp (G, A).

Let G be a locally compact Hausdorif abelian group and X be a complex Banach space. Let C(G, X) denote the space of all continuous functions f: G → X, with the topology of uniform convergence on compact sets. Let X′ denote the dual of X with the weak* topology. Let Mc(G, X′) denote the space of all X′-valued compactly supported regular measures of finite variation on G. For a function f ∈ C(G, X) and μ ∈ Mc(G, X′), we define the notion of convolution f * μ. A function f ∈ C(G, X) is called mean-periodic if there exists a non-trivial measure μ ∈ Mc(G, X′) such that f * μ = 0. For μ ∈ Mc(G, X′), let M P(μ) = {f ∈ C(G, X): f * μ = 0} and let M P(G, X) = ∪μ M P(μ). In this paper we analyse the following questions: Is M P(G, X) ≠ 0? Is M P(G, X) ≠ C(G, X)? Is M P(G, X) dense in C(G, X)? Is M P(μ) generated by ‘exponential monomials’ in it? We answer these questions for the groups G = ℝ, the real line, and G = T, the circle group. Problems of spectral analysis and spectral synthesis for C(ℝ, X) and C(T, X) are also analysed.

We show that the left regular representation of a countably infinite (discrete) group admits no finite-dimensional invariant subspaces. We also discuss a consequence of this fact, and the reason for our interest in this statement.

We then formally state, as a ‘conjecture’, a possible generalisation of the above statement to the context of fusion algebras. We prove the validity of this conjecture in the case of the fusion algebra arising from the dual of a compact Lie group.

We finally show, by example, that our conjecture is false as stated, and raise the question of whether there is a ‘good’ class of fusion algebras, which contains (a) the two ‘good classes’ discussed above, namely, discrete groups and compact group duals, and (b) only contains fusion algebras for which the conjecture is valid.

A harmonic NA group is a suitable solvable extension of a two-step nilpotent Lie group N of Heisenberg type by R+, which acts on N by anisotropic dilations. A hypergroup is a locally compact space for which the space of Borel measures has a convolution structure preserving the probability measures and satisfying suitable conditions. We describe a class of hypergroups associated to NA groups.

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.

Pointwise bounds for characters of representations of the classical, compact, connected, simple Lie groups are obtained with which allow us to study the singularity of central measures. For example, we find the minimal integer k such that any continuous orbital measure convolved with itself k times belongs to L2. We also prove that if k = rank G then μ 2k ∈ L1 for all central, continuous measures μ. This improves upon the known classical result which required the exponent to be dimension of the group G.

Let be the classical system of the Walsh functions, the multiplicative semigroup of the functions represented by series of functions Wk(t)with non-negative coefficients which sum equals 1. We study the arithmetic of . The analogues of the well-known [ related to the arithmetic of the convolution semigroup of probability measures on the real line are valid in . The classes of idempotent elements, of infinitely divisible elements, of elements without indecomposable factors, and of elements without indecomposable and non-degenerate idempotent factors are completely described. We study also the class of indecomposable elements. Our method is based on the following fact: is isomorphic to the semigroup of probability measures on the groups of characters of the Cantor-Walsh group.

In this paper we consider pseudo differential operators on local Hardy spaces hp (0 < p ≤ 1) on Chébli-Trimèche hypergroups of exponential growth.

We describe a generalization of the Hardy theorem on the motion group. We prove that for some weight functions νω growing very rapidly and a measurable function f, the finiteness of the Lp-norm of vf and the Lq-norm of ωf implies f=0 (almost everywhere).

Let ψ be a positive function defined near the origin such lim1 →0+ ψ(t) = 0. We consider the operator Tzƒ, defined as the pricipal value of the convolution of function ƒ and a kernel K(t) = eiy(t)t−z /ψ(t)1−z, where z is a complex number, 0 ≤ Re(z) ≤ 1, 0 < t ≤ 1 and γ is a real function. Assuming certain regularity conditions on ψ and γ and certain relations between ψ and γ we show that Tθ is a bounded operator on Lp (R) for 1/p = (1+ θ) /2 and 0 ≤ θ < 1, and T1 is bounded from H1 (R) to L1 (R).

Let G be a locally compact Vilenkin group with dual group Γ. We prove Littlewood-Paley type inequalities corresponding to arbitrary coset decompositions of Γ. These inequalities are then applied to obtain new Lp(G) multiplier theorems. The sharpness of some of these results is also discussed.

Let ℳ be a von Neumann algebra with a faithful normal trace τ, and let H∞ be a finite, maximal. subdiagonal algebra of ℳ. We prove that the Hilbert transform associated with H∞ is a linear continuous map from L1 (ℳ, τ) into L1.∞ (ℳ, τ). This provides a non-commutative version of a classical theorem of Kolmogorov on weak type boundedness of the Hilbert transform. We also show that if a positive measurable operator b is such that b log+b ∈ L1 (ℳ, τ) then its conjugate b, relative to H∞ belongs to L1 (ℳ, τ). These results generalize classical facts from function algebra theory to a non-commutative setting.

We introduce a new homology theory for infinite graphs in order to generalize some results of Willis and Woodward on translation invariant functionals. We also extend some theorems of Gerl and Gromov.

Iseki [11] defined a general notion of ergodicity suitable for functions ϕ: J → X where J is an arbitrary abelian semigroup and X is a Banach space. In this paper we develop the theory of such functions, showing in particular that it fits the general framework established by Eberlein [9] for ergodicity of semigroups of operators acting on X. Moreover, let A be a translation invariant closed subspace of the space of all bounded functions from J to X. We prove that if A contains the constant functions and ϕ is an ergodic function whose differences lie in A then ϕ ∈ A. This result has applications to spaces of sequences facilitating new proofs of theorems of Gelfand and Katznelson-Tzafriri [12]. We also obtain a decomposition for the space of ergodic vectors of a representation T: J → L(X) generalizing results known for the case J = Z+. Finally, when J is a subsemigroup of a locally compact abelian group G, we compare the Iseki integrals with the better known Cesàro integrals.

Let be a homogeneous tree of degree at least three. In this paper we investigate for which values of p and r the (σθ)-Poisson semigroup is Lp – Lr,-bounded, and we sharp estimate for the corresponding operator norms.

We study the Cesàro operator on the classical group G and give a necessary and sufficient condition on the index α = α(G) for which the operator is convergent to f(U) for any continuous function f as N → ∞. The result in this paper solves a question posed by Gong in the book Harmonic analysis on classical groups.

Let F ⊂ G be closed and A(F) = A(G)/IF. If F is a Helson set then A(F)** is an amenable (semisimple) Banach algebra. Our main result implies the following theorem: Let G be a locally compact group, F ⊂ G closed, a ∈ G. Assume either (a) For some non-discrete closed subgroup H, the interior of F ∩ aH in aH is non-empty, or (b) R ⊂ G, S ⊂ R is a symmetric set and aS ⊂ F. Then A(F)** is a non-amenable non-semisimple Banach algebra. This raises the question: How ‘thin’ can F be for A(F)** to remain a non-amenable Banach algebra?

Let G1, G2 be locally compact real-compact spaces. A linear map T defined from C(G1) into C(G2) is said to be separating or disjointness preserving if f = g ≡ 0 implies Tf = Tg ≡ 0 f or all f, g ∈ C(G1). In this paper we prove that both a separating map which preserves non-vanishing functions and a separating bijection which satisfies condition (M) (see Definition 4) are automatically continuous and can be written as weighted composition maps. We also study the effect of separating surjections (respectively injections) on the underlying spaces G1 and G2.

Next we apply the above results to give an algebraic characterization of locally compact Abelian groups, similar to the one given in [7] for compact Abelian groups in the presence of ring isomorphisms.

Finally, locally compact (not necessarily Abelian) groups are considered. We provide a sharpening of a result of Edwards and study the effect of onto (respectively injective) weighted composition maps on the groups G1 and G2.

We restrict the metaplectic representation to subgroups G of the symplectic group associated to equivariant holomorphic maps into the Siegel disc. We describe the invariant subspaces of the decomposition, and reduce the problem to the decomposition of a space of ‘harmonic’ polynomials under the action of the maximal compact subgroup of G.

We characterize the Hardy spaces Hp(G) of a compact Lie groupG by means of S-functions in analogy with the theorem of Fefferman-Stein for Rn. We also characterize Hp(G) by means of the -functions.