Let N be a nilpotent simply connected Lie group, and A a commutative connected d-dimensional Lie group of automorphisms of N which correspond to semisimple endomorphisms of the Lie algebra of N with positive eigenvalues. Form the split extension S = N × A ≅ N × a, a being the Lie algebra of A. We consider a family of “rectangles” Br in S, parameterized by r > 0, such that the measure of Br behaves asymptotically as a fixed power of r. One can construct the Hardy-Littlewood maximal function operator f → Mf relative to left translates of the family {Br}. We prove that M is of weak type (1, 1). This complements a result of J.-O. Strömberg concerning maximal functions defined relative to hyperbolic balls in a symmetric space.

Let X be either the d-dimensional sphere or a compact, simply connected, simple, connected Lie group. We define a mean-value operator analogous to the spherical mean-value operator acting on integrable functions on Euclidean space. The value of this operator will be written as ℳ f (x, a), where x ∈ X and a varies over a torus A in the group of isometries of X. For each of these cases there is an interval pO < p ≦ 2, where the p0 depends on the geometry of X, such that if f is in Lp (X) then there is a set full measure in X and if x lies in this set, the function a ↦ℳ f(x, a) has some Hölder continuity on compact subsets of the regular elements of A.

The Lebesgue measure, λ (E + F), of the algebraic sum of two Borel sets, E, F of the classical “middle-thirds’ Cantor set on the circle can be estimated by evaluating the Cantor meaure, μ of the summands. For example log λ (E + F) exceeds a fixed scalar multiple of log μ (E)+ log μ (F). Several numerical inequalities which are required to prove this and related results look tantalizingly simple and basic. Here we isolate them from the measure theory and present a common format and proof.

According to an extension of a classical theorem of Bernstein, due to C. Herz, a function on Rn belonging to a Besov space of appropriate order has an absolutely convergent Fourier transform. We establish extensions of this result to Cartan motion groups, for Besov spaces defined with respect to both isotropic and non-isotropic differences.

We consider the distribution μ of numbers whose binary digits are generated from infinitely many tosses of a biased coin. It is shown that, if E has positive μ measure, then some n-fold sum of E with itself must contain an interval. This contrasts with the known result that all convolution powers of μ are singular.

In this paper it is proved that the principal series of representations of Γ = Z2*…*Z2 may be analytically continued to give uniformly bounded representations on the same Hilbert space, and that these representations are irreducible. Further, the reducibility of the restrictions to Γ ⊂ SL(2, Qp) of the irreducible unitary representations of SL(2, Qp) is examined.

Let G/K be a compact symmetric space, and let G = KAK be a Cartan decomposition of G. For f in L1(G) we define the spherical means f(g, t) = ∫k∫k ∫(gktk′) dk dk′, g ∈ G, t ∈ A. We prove that if f is in Lp(G), 1 ≤ p ≤ 2, then for almost every g ∈ G the functions t → f(g, t) belong to certain Soblev spaces on A. From these regularity results for the spherical means we deduce, if G/K is a compact rank one symmetric space, a theorem on the almost everywhere localization of spherical harmonic expansions of functions in L2 (G/K).

Let G be an exponential Lie group. We study primitive ideals (i.e. kernels of irreducible *-representations of L1(G)), with bounded approximate units (b.a.u.). We prove a result relating the existence of b.a.u. in certain primitive ideals with the geometry of the corresponding Kirillov orbits. This yields for a solvable group of class 2, a characterization of the primitive ideals with b.a.u.

It is shown that if G is a non-amenable group, then there are no non-zero translation invariant functionals on Lp(G) for 1 < p < ∞. Furthermore, if G contains a closed, non-abelian free subgroup, then there are no non-zero translation invariant functionals on C0(G). The latter is proved by showing that a certain non-invertible convolution operator on C0(G) is surjective.

In 1947 I. E. Segal proved that to each non-degenerate ~ -representation R of L1 (= L1 (G) for a compact group G) with representation space , there corresponds a continuous unitary representation W of G, also with representation space , which satisfies

for each fL1 and hk . This was extended to Lp,1 p < , in 1970 by E. Hewitt and K. A. Ross. We now generalize this result to any symmetric homogeneous convolution Banach alebra of pseudomeasures on G. Further we prove that the correspondence preserves irreduibility.

Let A be a commutative Banach algebra with identity of norm 1, X a Banach A-module and G a locally compact abeian group with Haar measure. Then the multipliers from an A -valued function algebra into an X-valued function space is studied. We characterize the multiplier spaces as the following isometrically isomorphic relations under some appropriate conditions:

Let G be a locally compact abeian group, (μρ) a net of bounded Radon measures on G. In this paper we consider conditions under which (μρ) is saturated in Lp (G) and apply these results to the Fejér and Picard approximation processes.

We describe on the Heisenberg group Hn a family of spaces M(h, X) of functions which play a role analogous to the trigonometric polynomials in Tn or the functions of exponential type in Rn. In particular we prove that for the space M(h, X), Jackson's theorem holds in the classical form while Bernstein's inequality hold in a modified form. We end of the paper with a characterization of the functions of the Lipschitz space by the behavior of their best approximations by functions in the space M(h, X).

The special Denjoy-Bochner integral (the D*B-integral) which are generalisations of Lebesgue-Bochner integral are discussed in [7, 6, 5]. Just as the concept of numerical almost periodicity was extended by Burkill [3] to numerically valued D*- or D-integrable function, we extend the concept of almost periodicity for Banach valued function to Banach valued D*B-integrable function. For this purpose we introduce as in [3] a distance in the space of all D*B-integrable functions with respect to which the D*B-almost periodicity is defined. It is shown that the D*B-almost periodicity shares many of the known properties of the almost periodic Banach valued function [1, 4].

Random Fourier series are studied for a class of compact abelian hypergroups. The randomizing factors are assumed to be independent and uniformly subgaussian. In the presence of a natural teachnical hypothesis, an entropy condition of Dudley is shown to be sufficient for almost sure continuity. The classical results on almost sure membership in Lp, where p < ∞, are generalized to this setting. As an application, it is shown that a simple condition on the dual object implies that the de Leeuw-Kahane-Katznelson phenomenon occurs. Another application is the analogue of a classical sufficient condition for almost sure continuity. Examples illustrating the general theory are given for the hypergroup of conjugacy classes of SU(2) and for a class of compact countable hypergroups.

Let G be a compact abelian group with dual Ĝ and let K be a Banach L1 (G)-module. We introduce the notion of character convolution transformation of K which reduces to ordinary Fourier or Fourier-Stieltjes transformation when K is one of the spaces Lp(G), M(G). We show that the question of what maps Ĝ → K extend to multipliers of K is a question of asking for descriptions of the character convolution transforms. In this setting some results of Helson-Edward and Schoenberg-Eberlein find generalizations, as do some classical results, including the inversion formula and the Parseval relation. We then apply these results to transformation groups, obtaining a variant of a theorem of Bochner and an extension of a theorem of Ryan.

Let G denote any locally compact abelian group with the dual group Γ. We construct a new kind of subalgebra L1(G) ⊗ΓS of L1(G) from given Banach ideal S of L1(G). We show that L1(G) ⊗гS is the larger amoung all strongly character invariant homogeneous Banach algebras in S. when S contains a strongly character invariant Segal algebra on G, it is show that L1(G) ⊗гS is also the largest among all strongly character invariant Segal algebras in S. We give applications to characterizations of two kinds of subalgebras of L1(G)-strongly character invariant Segal algebras on G and Banach ideal in L1(G) which contain a strongly character invariant Segal algebra on G.

This paper shows that the idempotent generalized characters associated with a Raikov System generated by a K2 set in is contained in the closure of the characters D2 in ΔM(D2).

Let G be a compact, simple, simply connected Lie group. The Lp-norm of a central trigonometric polynomial reduces naturally to a weighted Lp-norm of a trigonometric polynomial on a maximal torus T. The weight is | Δ |2-p, where Δ is the usual Weyl function. If p ≥ 2, we prove that | Δ |2-p satisfies Muckenhoupt's Ap condition if and only if the Lp-norms of the irreducible characters of G are uniformly bounded.

In this paper we study commutative topological semigroups S admitting an absolutely continuous measure. When S is cancellative we show that S admits a weaker topology J with respect to which (S, J) is embeddable as a subsemigroup with non-empty interior in some locally compact topological group. As a consequence, we deduce certain results related to the existence of invariant measures on S and for a large class of locally compact topological semigroups S, we associate S with some useful topological subsemigroup of a locally compact group.