We establish the spectral gap property for dense subgroups of SU$\left(d\right)$$(d\ge 2)$, generated by finitely many elements with algebraic entries; this result was announced...

We prove a spectral Paley-Wiener theorem for the Heisenberg group by means of a support theorem for the twisted spherical means on ${ℂ}^n.$ If $f\left(z\right)e^{\frac{1}{4}{\left|z\right|}^2}$ is a Schwartz class function we show that $f$ is supported in a ball of radius $B$ in ${ℂ}^n$ if and only if $f\times {\mu }_r\left(z\right)=0$ for $r\>B+\left|z\right|$ for all $z\in {ℂ}^n.$ This is an analogue of Helgason’s support theorem on Euclidean and hyperbolic spaces. When $n=1$ we show that the two conditions $f\times {\mu }_r\left(z\right)={\mu }_r\times f\left(z\right)=0$ for $r\>B+\left|z\right|$ imply a support theorem for a large class of functions with exponential growth. Surprisingly enough,this latter...

Let A be a commutative Banach algebra with Gelfand space ∆ (A). Denote by Aut (A) the group of all continuous automorphisms of A. Consider a σ(A,∆(A))-continuous group representation α:G → Aut(A) of a locally compact abelian group G by automorphisms of A. For each a ∈ A and φ ∈ ∆(A), the function ${\phi }_a\left(t\right):=\phi \left({\alpha }_{t}a\right)$ t ∈ G is in the space C(G) of all continuous and bounded functions on G. The weak-star spectrum ${\sigma }_w*\left({\phi }_a\right)$ is defined as a closed subset of the dual group Ĝ of G. For φ ∈ ∆(A) we define ${Ʌ}_{\phi }^a$ to be the union of all...

Let 𝓢(Hₙ) be the space of Schwartz functions on the Heisenberg group Hₙ. We define a spherical transform on 𝓢(Hₙ) associated to the action (by automorphisms) of U(p,q) on Hₙ, p + q = n. We determine its kernel and image and obtain an inversion formula analogous to the Godement-Plancherel formula.

We prove an approximation lemma on (stratified) homogeneous groups that allows one to approximate a function in the non-isotropic Sobolev space ${\dot{NL}}^{1,Q}$ by $L^{\infty }$ functions, generalizing a result of Bourgain–Brezis. We then use this to obtain a Gagliardo–Nirenberg inequality for $$ on the Heisenberg group ${ℍ}^n$.

Let $D$ be a Hermitian symmetric space of tube type, $S$ its Silov boundary and $G$ the neutral component of the group of bi-holomorphic diffeomorphisms of $D$. Our main interest is in studying the action of $G$ on $S^3=S\times S\times S$. Sections 1 and 2 are part of a joint work with B. Ørsted (see [4]). In Section 1, as a pedagogical introduction, we study the case where $D$ is the unit disc and $S$ is the circle. This is a fairly elementary and explicit case, where one can easily get a flavour of the more general results. In Section...

We give a simple proof of a result of R. Rochberg and M. H. Taibleson that various maximal operators on a homogeneous tree, including the Hardy-Littlewood and spherical maximal operators, are of weak type (1,1). This result extends to corresponding maximal operators on a transitive group of isometries of the tree, and in particular for (nonabelian finitely generated) free groups.