## A functional calculus for Rockland operators on nilpotent Lie groups.(English)Zbl 0595.43007

Let $$L$$ be a hypoelliptic left-invariant differential operator on a homogeneous Lie group $$G$$ which satisfies the estimate $\| \partial u\|_{L^ 2(G)}\leq c \| (l+L)^{\sigma (\partial)}u\|_{L^ 2(G)}\text{ for all } u\in Dom(\bar L^{\sigma (\partial)}),$ where $$\partial$$ is any left-invariant differential operator and $$\sigma$$ ($$\partial)$$ an integer depending on $$\partial$$. The aim of the present paper is to prove that the operator $$T_ mf=\int^{\infty}_{0}m(\lambda) \,dE_{\lambda}f$$ where $$m$$ is a bounded function on $$\mathbb{R}^+$$ and $$E_{\lambda}$$ is the spectral resolution of $$L$$, is of the form $$T_mf=f*M$$ with $$M\in\mathcal S(G)$$.
Reviewer: A. H. Nasr (Cairo)

### MSC:

 43A80 Analysis on other specific Lie groups 65H10 Numerical computation of solutions to systems of equations 22E30 Analysis on real and complex Lie groups 43A22 Homomorphisms and multipliers of function spaces on groups, semigroups, etc. 22E25 Nilpotent and solvable Lie groups
Full Text: