Boundedness of H ∞ Functional Calculus of Hodge-Dirac Operators

In this chapter, we introduce Hodge-Dirac operators associated with markovian semigroups of Fourier multipliers and we show that these operators admit a bounded H∞ functional calculus on a bisector. We also provide Hodge decompositions. We equally show a similar result for Hodge-Dirac operators associated with markovian semigroups of Schur multipliers. Particular attention is paid to the domain of the Hodge-Dirac operators and different choices are discussed in the course of the chapter. We also prove independence of the bounds of the H∞ functional calculus, on the group G or index set I, and on the markovian semigroup of multipliers.