We prove that on Iwasawa AN groups coming from arbitrary semisimple Lie groups there is a Laplacian with a nonholomorphic functional calculus, not only for $L^1\left(AN\right),$ but also for $L^p\left(AN\right)$, where 1 < p < ∞. This yields a spectral multiplier theorem analogous to the ones known for sublaplacians on stratified groups.

We study spectral multipliers for a distinguished Laplacian on certain groups of exponential growth. We obtain a stronger optimal version of the results proved in [CGHM] and [A].

Let G be a Lie group, Xj right invariant vector fields on G, which generate (as a Lie algebra) the Lie algebra of G,L = -Σ Xj2.(...) In this paper we consider L1(G) boundedness of F(L) for (some) metabelian G and a distinguished L on G. Of the main interest is that the group is of exponential growth, and possibly higher rank. Previously positive results about higher rank groups were only about Iwasawa type groups. Also, our groups may be unimodular, so it is the second positive result (after [13])...

Let n ≥ 1, d = 2n, and let (x,y) ∈ ℝⁿ × ℝⁿ be a generic point in ℝ²ⁿ. The twisted Laplacian $L=-1/2{\sum }_{j=1}^n[({\partial }_{x_j}+i{y}_j)²+({\partial }_{y_j}-i{x}_j)²]$ has the spectrum n + 2k = λ²: k a nonnegative integer. Let $P_{\lambda }$ be the spectral projection onto the (infinite-dimensional) eigenspace. We find the optimal exponent ϱ(p) in the estimate $\left|\right|P_{\lambda }{u\left|\right|}_{L^p\left({ℝ}^d\right)}\lesssim {\lambda }^{\varrho \left(p\right)}{\left|\right|u\left|\right|}_{L²\left({ℝ}^d\right)}$ for all p ∈ [2,∞], improving previous partial results by Ratnakumar, Rawat and Thangavelu, and by Stempak and Zienkiewicz. The expression for ϱ(p) is ϱ(p) = 1/p -1/2 if 2 ≤ p ≤ 2(d+1)/(d-1), ϱ(p) = (d-2)/2 - d/p if 2(d+1)/(d-1)...

In this note we define and explore, à la Godement, spectral subspaces of Banach space representations of the Fourier-Eymard algebra of a (nonabelian) locally compact group.

Relations between spectral synthesis in the Fourier algebra A(G) of a compact group G and the concept of operator synthesis due to Arveson have been studied in the literature. For an A(G)-submodule X of VN(G), X-synthesis in A(G) has been introduced by E. Kaniuth and A. Lau and studied recently by the present authors. To any such X we associate a $V^{\infty }\left(G\right)$-submodule X̂ of ℬ(L²(G)) (where $V^{\infty }\left(G\right)$ is the weak-* Haagerup tensor product $L^{\infty }\left(G\right){\otimes }_{w*h}{L}^{\infty }\left(G\right)$), define the concept of X̂-operator synthesis and prove that a closed set E...

For locally compact, second countable, type I groups G, we characterize all closed (two-sided) translation invariant subspaces of L²(G). We establish a similar result for K-biinvariant L²-functions (K a fixed maximal compact subgroup) in the context of semisimple Lie groups.

Let A be a semisimple commutative regular tauberian Banach algebra with spectrum ${\Sigma }_A$. In this paper, we study the norm spectra of elements of $\overline{span}{\Sigma }_A$ and present some applications. In particular, we characterize the discreteness of ${\Sigma }_A$ in terms of norm spectra. The algebra A is said to have property (S) if, for all $\phi \in \overline{}{\Sigma }_{A}0$, φ has a nonempty norm spectrum. For a locally compact group G, let ${ℳ}_2^d\left(Ĝ\right)$ denote the C*-algebra generated by left translation operators on $L^2\left(G\right)$ and $G_d$ denote the discrete group G. We prove that the Fourier...

We give a construction of an analytic series of uniformly bounded representations of a free group G, through the action of G on its Poisson boundary. These representations are irreducible and give as their coefficients all the spherical functions on G which tend to zero at infinity. The principal and the complementary series of unitary representations are included. We also prove that this construction and the other known constructions lead to equivalent representations.