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

Hulanicki, Andrzej

Stud. Math. 78, 253-266 (1984).

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

Keywords:

Schwartz function; Rockland operator; hypoelliptic differential; operator; homogeneous Lie group; spectral resolution