The sub-Laplacian on the Heisenberg group is first decomposed into twisted Laplacians parametrized by Planck's constant. Using Fourier-Wigner transforms so parametrized, we prove that the twisted Laplacians are globally hypoelliptic in the setting of tempered distributions. This result on global hypoellipticity is then used to obtain Liouville's theorems for harmonic functions for the sub-Laplacian on the Heisenberg group.

Let $E$ be a subset of a discrete abelian group whose compact dual is $G$. $E$ is exactly $p$-Sidon (respectively, exactly non-$p$-Sidon) when$(*)C_E(G{)}^{\wedge }\subset {\ell }^r$ holds if and only if $r\in [p,\infty ]$ (respectively, $r\in (p,\infty )$). $E$ is said to be exactly ${\Lambda }^{\beta }$ (respectively, exactly non-${\Lambda }^{\beta }$) if $E$ has the property${\displaystyle (**)\text{every}f\in L_E^2\left(G\right)\text{satisfies}{\int }_G\exp \left(\lambda \right|f{|}^{2/\alpha }\<\infty ,\text{for}\text{all}\lambda \>0,}$ if and only if $\alpha \in [\beta ,\infty )$ (respectively, $\alpha \in (\beta ,\infty )$).In this paper, for every $p\in [1,2)$ and $\beta \in [1,\infty )$, we display sets which are exactly $p$-Sidon, exactly non-$p$-Sidon, exactly ${\Lambda }^{\beta }$ and exactly non-${\Lambda }^{\beta }$.

In some recent papers ([1],[2],[3],[4]) we have investigated some general spectral properties of a multiplier defined on a commutative semi-simple Banach algebra. In this paper we expose some aspects concerning the Fredholm theory of multipliers.

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 compactly generated, locally compact group with polynomial growth and let ω be a weight on G. We look for general conditions on the weight which allow us to develop a functional calculus on a total part of L1(G,ω). This functional calculus is then used to study harmonic analysis properties of L1(G,ω), such as the Wiener property and Domar's theorem.

Let A be a complex, commutative Banach algebra and let $M_A$ be the structure space of A. Assume that there exists a continuous homomorphism h:L¹(G) → A with dense range, where L¹(G) is a group algebra of the locally compact abelian group G. The main results of this note can be summarized as follows: (a) If every weakly almost periodic functional on A with compact spectra is almost periodic, then the space $M_A$ is scattered (i.e., $M_A$ has no nonempty perfect subset). (b) Weakly almost periodic functionals...