Second Order Differential Operators Associated to the Space of Generalized Analytic Functions

Let \(\mathcal {F}\) be a given differential operator, then a function space \(\mathcal {E}\) is called an associated space to \(\mathcal {F}\) if \(\mathcal {F}\) transforms \(\mathcal {E}\) into itself. The interest in the theory of associated spaces is its application to initial value problems. A problem like \(\partial _t w = \mathcal {F}(t, z, w, \partial _z w)\) ,     \(w(0, z) = \varphi (z)\) , where t means time, can be solved in the case that the initial function \(\varphi (z)\) belongs to an associated space whose elements satisfy an interior estimate. In this work we show sufficient conditions for all second order operators with complex coefficients to be associated with the space of generalized analytic functions.