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485Inhomogenous Cauchy Riemann equation

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  • lakhmau
    Mar 21, 2011

      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 !