Many of the operators one meets in several complex variables, such as
the famous Lewy operator, are not locally solvable. Nevertheless, such
an operator L can be thoroughly studied if one can find a suitable
relative parametrix--an operator K such that LK is essentially the
orthogonal projection onto the range of L. The analysis is by far most
decisive if one is able to work in the real analytic, as opposed to the
smooth, setting. With this motivation, the author develops an analytic
calculus for the Heisenberg group. Features include: simple, explicit
formulae for products and adjoints; simple representation-theoretic
conditions, analogous to ellipticity, for finding parametrices in the
calculus; invariance under analytic contact transformations; regularity
with respect to non-isotropic Sobolev and Lipschitz spaces; and
preservation of local analyticity. The calculus is suitable for doing
analysis on real analytic strictly pseudoconvex CR manifolds. In this
context, the main new application is a proof that the Szego projection
preserves local analyticity, even in the three-dimensional setting.
Relative analytic parametrices are also constructed for the adjoint of
the tangential Cauchy-Riemann operator.
Originally published in 1990.
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