In this talk, I will present my work on the solvability of the inhomogenuous Cauchy-Riemann equation in spaces of smooth functions that decay fast at infinity on strip-like domains in the complex plane. My first main result is a complete characterization of the solvability of this equation in terms of the weights that model the decay at infinity. Secondly, I will discuss parameter dependence problems related to this equation: If the data of the equation decay fast on a strip-like domain in the complex plane and depend on a couple of parameters in some "nice" (real analytic, smooth,...) way, can we then always find a solution that decays as fast as the data and that depends on these parameters in the same "nice" way? I will present a characterization of this phenomenon in terms of the linear topological invariant \((A)\). As an application, we will see that real-analytic parameter dependence of the solution cannot always be guaranteed. Throughout this talk, I intend to give a taste of the main functional analytic methods we used: the theory of the first derived projective limit functor and the splitting theory for Fréchet spaces.
This talk will be based on the following articles:
On the surjectivity of the Caucy-Riemann and Laplace operators on weighted spaces of smooth functions, published in Proceedings of the American Mathematical Society, collaborative work with Andreas Debrouwere and Jasson Vindas.
A linear topological invariant for weighted spaces of holomorphic functions, Revista Matemática Complutense, collaborative work with Andreas Debrouwere.