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.

In this paper we make use of semigroup methods on the space of compactly supported measures to obtain a Bochner representation for α-bounded positive-definite functions on a commutative hypergroup.

We use the second derivative of intertwining operators to realize a unitary structure for the irreducible subrepresentations in the reducible spherical principal series of U(1, n). These representations can also be realized as the kernels of certain invariant first-order differential operators acting on sections of homogeneous bundles over the hyperboloid (U(1) × U(n))/U(1, n).

The distributions of nearest neighbour random walks on hypercubes in continuous time t 0 can be expressed in terms of binomial distributions; their limit behaviour for t, N → ∞ is well-known. We study here these random walks in discrete time and derive explicit bounds for the deviation of their distribution from their counterparts in continuous time with respect to the total variation norm. Our results lead to a recent asymptotic result of Diaconis, Graham and Morrison for the deviation from uniformity for N →∞. Our proofs use Krawtchouk polynomials and a version of the Diaconis–Shahshahani upper bound lemma. We also apply our methods to certain birth-and-death random walks associated with Krawtchouk polynomials.

The Plancherel formula for the horocycle space, and several generalizations, is derived within the framework of quasi-regular representations which have monomial spectrum. The proof uses only machinery from the Penney-Fujiwara distribution-theoretic technique; no special semisimple harmonic analysis is needed. The Plancherel formulas obtained include the spectral distributions and the intertwining operators that effect the direct integral decomposition of the quasi-regular representation.

A weighted generalization of a p-Sidon set, called an (a, p)-Sidon set, is introduced and studied for infinite, non-abelian, connected, compact groups G. The entire dual object Ĝ is shown never to be central (p − 1, p)-Sidon for 1 ≦ p < 2, nor central (1 + ε, 2)-Sidon for ε > 0. Local (p, p)-Sidon sets are shown to be identical to local Sidon sets. Examples are constructed of infinite non-Sidon sets which are central (2p − 1, p)-Sidon, or (p − 1, p)-Sidon, for 1 < p < 2. Full m-fold FTR sets are proved not to be central (a, 2m/(m + 1))-Sidon for any a > 1.

Lipschitz spaces are important function spaces with relations to Hp spaces and Campanato spaces, the other two important function spaces in harmonic analysis. In this paper we give some characterizations for Lipschitz spaces on compact Lie groups, which are analogues of results in Euclidean spaces.

Let (ℋ, G, U) be a continuous representation of the Lie group G by bounded operators g ↦ U(g) on the Banach space ℋ and let (ℋ, g, dU) denote the representation of the Lie algebra g obtained by differentiation. If a1,…, ad′ is a Lie algebra basis of g and Ai = dU(ai) then we examine elliptic regularity properties of the subelliptic operators where (cij) is a real-valued strictly positive-definite matrix and c0, c1,…, cd′ ∈ C. We first introduce a family of Lipschitz subspaces ℋγ, γ > 0, of ℋ which interpolate between the Cn-subspaces of the representation and for which the parameter γ is a continuous measure of differentiability. Secondly, we give a variety of characterizations of the spaces in terms of the semigroup generated by the closure of H and the group representation. Thirdly, for sufficiently large values of Re c0 the fractional powers of the closure of H are defined, and we prove that D()γ⊆γ′, for γ′ < 2γ/r where r is the rank of the basis. Further we establish that 2γ/r is the optimal regularity value and it is attained for unitary representations or for the representations obtained by restricting U to ℋγ. Many other regularity properties are obtained.

It is proved that on certain kinds of homogeneous spaces, the only Lp function, 1≤ p < ∞, satisfying the mean value property is the zero function.

Let Δ be a thick building of type Ã2, and let be its set of vertices. We study a commutative algebra of ‘averaging’ operators acting on the space of complex valued functions on . This algebra may be identified with a space of ‘biradial functions’ on , or with a convolution algebra of bi-K-invariant functions on G, if G is a sufficiently large group of ‘type-rotating’ automorphisms of Δ, and K is the subgroup of G fixing a given vertex. We describe the multiplicative functionals on and the corresponding spherical functions. We consider the C*-algebra induced by on l2, find its spectrum Σ, prove positive definiteness of a kernel kz for each z ∈ Σ, find explicity the spherical Plancherel formula for any group G of type rotating automorphisms, and discuss the irreducibility of the unitary representations appearing therein. For the class of buildings ΔJ arising from the groups ΓJ introduced in [2], this involves proving that the weak closure of is maximal abelian in the von Neumann algebra generated by the left regular representation of ΓJ.

In a recent paper In a recent paper the authors proved a multiplier theorem for Hardy spaces Hp (G), 0 < p ≤ 1, defined on a locally compact Vilenkin group G. The assumptions on the multiplier were expressed in terms of the “norms” of certain Herz spaces K (1/p − 1/?r, r, p) with r restricted to 1 ≤ r < ∞ and p < r. In the present paper we show how this restriction on r may be weakened to p ≤ r ∞. Furthermore, we present two modifications of our main theorem and compare these with certain results for multipliers on LP (Rn)-spaces, 1 < p < ∞, due to Seeger and to Cowling, Fendler and Foumier. We also discuss the sharpness of some of our results.