### A cancellative amenable ascending union of nonamenable semigroups

We construct an example of a cancellative amenable semigroup which is the ascending union of semigroups, none of which are amenable.

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We construct an example of a cancellative amenable semigroup which is the ascending union of semigroups, none of which are amenable.

We prove three theorems on linear operators ${T}_{\tau ,p}:{H}_{p}\left(\mathcal{B}\right)\to {H}_{p}$ induced by rearrangement of a subsequence of a Haar system. We find a sufficient and necessary condition for ${T}_{\tau ,p}$ to be continuous for 0 < p < ∞.

We consider an arbitrary locally compact abelian group G, with an ordered dual group Γ, acting on a space of measures. Under suitable conditions, we define the notion of analytic measures using the representation of G and the order on Γ. Our goal is to study analytic measures by applying a new transference principle for subspaces of measures, along with results from probability and Littlewood-Paley theory. As a consequence, we derive new properties of analytic measures as well as extensions of previous...

The Riemann zeta-function ζ(s) extends to an outer function in ergodic Hardy spaces on ${}^{\omega}$, the infinite-dimensional torus indexed by primes p. This enables us to investigate collectively certain properties of Dirichlet series of the form $({a}_{p},s)={\prod}_{p}{(1-{a}_{p}{p}^{-s})}^{-1}$ for ${a}_{p}$ in ${}^{\omega}$. Among other things, using the Haar measure on ${}^{\omega}$ for measuring the asymptotic behavior of ζ(s) in the critical strip, we shall prove, in a weak sense, the mean-value theorem for ζ(s), equivalent to the Lindelöf hypothesis.

For the scalar holomorphic discrete series representations of $\mathrm{SU}(2,2)$ and their analytic continuations, we study the spectrum of a non-compact real form of the maximal compact subgroup inside $\mathrm{SU}(2,2)$. We construct a Cayley transform between the Ol’shanskiĭ semigroup having $\mathrm{U}(1,1)$ as Šilov boundary and an open dense subdomain of the Hermitian symmetric space for $\mathrm{SU}(2,2)$. This allows calculating the composition series in terms of harmonic analysis on $\mathrm{U}(1,1)$. In particular we show that the Ol’shanskiĭ Hardy space for $\mathrm{U}(1,1)$ is different...

Let G be a locally compact abelian group whose dual group Γ contains a Haar measurable order P. Using the order P we define the conjugate function operator on ${L}^{p}\left(G\right)$, 1 ≤ p < ∞, as was done by Helson [7]. We will show how to use Hahn’s Embedding Theorem for orders and the ergodic Hilbert transform to study the conjugate function. Our approach enables us to define a filtration of the Borel σ-algebra on G, which in turn will allow us to introduce tools from martingale theory into the analysis on groups...

In their celebrated paper [3], Burkholder, Gundy, and Silverstein used Brownian motion to derive a maximal function characterization of ${H}^{p}$ spaces for 0 < p < ∞. In the present paper, we show that the methods in [3] extend to higher dimensions and yield a dimension-free weak type (1,1) estimate for a conjugate function on the N-dimensional torus.