Geodesic
Negative Sobolev Spaces in the Cauchy Problem for the Cauchy--Riemann Operator
Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 2 (2009) no. 1, pp. 17-30

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Let $D$ be a bounded domain in $\mathbb C^n$ ($n\ge1$) with a smooth boundary $\partial D$. We indicate appropriate Sobolev spaces of negative smoothness to study the non-homogeneous Cauchy problem for the Cauchy–Riemann operator $\overline\partial$ in $D$. In particular, we describe traces of the corresponding Sobolev functions on $\partial D$ and give an adequate formulation of the problem. Then we prove the uniqueness theorem for the problem, describe its necessary and sufficient solvability conditions and produce a formula for its exact solution.
Keywords: negative Sobolev spaces, ill-posed Cauchy problem. 
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Ivan V. Shestakov; Alexander A. Shlapunov. Negative Sobolev Spaces in the Cauchy Problem for the Cauchy--Riemann Operator. Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 2 (2009) no. 1, pp. 17-30. http://geodesic.mathdoc.fr/item/JSFU_2009_2_1_a1/