Extrapolation of Functional Calculus of Dirac Operators and Applications

Several rather general sufficient conditions for the extrapolation of
the calculus of generalized Dirac operators from L2 to Lp are presented.
Using the resolvent approach and showing the irrelevance of the semi group
one, we extrapolate (with natural generalisations) the model considered by
Axelsson, Keith and McIntosh in L2 in order to generalise the setting of
the Kato problem.
As applications, one obtains some embedding theorems, quadratic
estimates and Littlewood-Paley-type theorems in terms of this calculus
in Lebesgue spaces.
Among the auxiliary results are high order counterparts of the Hilbert
identity, new forms of “off-diagonal” estimates, the study of the structure
of the model in reflexive Banach spaces (especially, Lebesgue ones) and
its interpolation properties, and up-to-date analogy of the Calderón-Zygmund
theory. We do not use any stability. In particular, some coercively conditions
for bilinear forms in Banach spaces are used as substitutions for the
ellipticity ones.
We also discuss the definitions of functional calculus and make
an attempt to show how the algebraic and geometric structures come into
and how the localisation problem is fought with.