Inhomogenous Cauchy Riemann equation
I am stock with the following problem, I hope it is well known, therefore I ask you...
Using a Cauchy formula it is possible to find a function u : D --> C such that
d-bar u = v
where d-bar is 1/2 (d/dx + id/dy) and v is smooth and compactly supported. My question is whether it is possible to find u such that u is compactly supported.
One easy condition for this is that v is orthogonal to the holomorphic functions :
\int_D f v = 0 for all f holomorphic.
Is this sufficient ? And in this case, is it possible to find a solution u with an estimate on its gradients of the form
|| Du ||_infty \leq C || v ||_infty
where C is a constant ?
Thanks for your reading !