Bounds on the Phillips Calculus of Abstract First Order Differential Operators

For an operator generating a group on $$L^p$$ spaces transference results give bounds the Phillips functional calculus also known as spectral multiplier estimates. In this paper we consider specific generators which are abstraction of first order differential operators and prove similar estimates assuming only that is bounded $$L^2$$ rather than . We R-bounded Hörmander result by abstract Sobolev embedding property show square perturbed Hodge–Dirac has such calculus.