Functional calculi for convolution operators on a discrete, periodic, solvable group

Suppose T is a bounded self-adjoint operator on the Hilbert space L2(X,μ) and let T = SpL2T λdE(λ) be its spectral resolution. Let F be a Borel bounded function on [−a, a], SpL2T ⊂ [−a, a]. We say that F is a spectral Lp-multiplier for T , if F(T ) = SpL2T F(λ)dE(λ)is a bounded operator on Lp(X,μ). The paper deals with l1-multipliers, where X = G is a discrete (countable) solvable group with ∀ x∈G, x4 = 1, μ is the counting measure and TΦ : l2(G) ξ → ξ ∗ Φ ∈ l2(G), where Φ = Φ ∗ is a l1(G) function, suppΦ generates G. The main result of the paper states that there exists a Ψ on G such that all l1-multipliers for TΨ are real analytic at every interior point of Spl2(G)TΨ. We also exhibit self-adjoint Φ s in l1(G) such that suppΦ generates G and F ∈ C2 c are l1-multipliers for TΦ.