  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.

Classical Analysis and ODEs;Functional Analysis