$L^p-L^q$ theory for holomorphic functions of perturbed first order Dirac operators

The aim of the article is to prove $L^p-L^q$ off-diagonal estimates and $L^p-L^q$ boundedness for operators in the functional calculus of certain perturbed first order differential operators of Dirac type for with $p\le q$ in a certain range of exponents. We describe the $L^p-L^q$ off-diagonal estimates and the $L^p-L^q$ boundedness in terms of the decay properties of the related holomorphic functions and give a necessary condition for $L^p-L^q$ boundedness. Applications to Hardy-Littlewood-Sobolev estimates for fractional operators will be given.