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 on the Phillips functional calculus also known as spectral multiplier estimates. In this paper we consider specific group generators which are abstraction of first order differential operators and prove similar spectral multiplier estimates assuming only that the group is bounded on L² rather than L ^p. We also prove an R-bounded Hörmander calculus result by assuming an abstract Sobolev embedding property and show that the square of a perturbed Hodge–Dirac operator has such calculus.