Abstract
A monogenic function of two vector variables is a function annihilated by the
operator consisting of two Dirac operators, which are associated to two
variables, respectively. We give the explicit form of differential operators in
the Dirac complex resolving this operator and prove its ellipticity directly.
This open the door to apply the method of several complex variables to
investigate this kind of monogenic functions. We prove the Poincar\'e lemma for
this complex, i.e. the non-homogeneous equations are solvable under the
compatibility condition by solving the associated Hodge Laplacian equations of
fourth order. As corollaries, we establish the Bochner--Martinelli integral
representation formula for this differential operator and the Hartogs'
extension phenomenon for monogenic functions. We also apply abstract duality
theorem to the Dirac complex to obtain the generalization of Malgrange's
vanishing theorem and establish the Hartogs--Bochner extension phenomenon for
monogenic functions under the moment condition.