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)L2(X,μ) and letT=∫SpL2TλdE(λ) be its spectral resolution. Let F   be a Borel bounded function on [−a,a][−a,a], SpL2T⊂[−a,a]SpL2T⊂[−a,a]. We say that F   is a spectral LpLp-multiplier for T, ifF(T)=∫SpL2TF(λ)dE(λ) is a bounded operator on Lp(X,μ)Lp(X,μ). The paper deals with l1l1-multipliers, where X=GX=G is a discrete (countable) solvable group with ∀x∈G∀x∈G, x4=1x4=1, μ is the counting measure andTΦ:l2(G)∋ξ↦ξ∗Φ∈l2(G), where Φ=Φ∗Φ=Φ∗ is a l1(G)l1(G) function, suppΦ generates G. The main result of the paper states that there exists a Ψ on G   such that all l1l1-multipliers for TΨTΨ are real analytic at every interior point of Spl2(G)TΨSpl2(G)TΨ. We also exhibit self-adjoint Φ′sΦ′s in l1(G)l1(G) such that suppΦ generates G   and F∈Cc2 are l1l1-multipliers for TΦTΦ.