On the Cauchy problem for the Cauchy-Riemann operator in Sobolev spaces

Let D be a bounded domain in C-n (n >= 1) with a smooth boundary partial derivative D. We describe necessary and sufficient solvability conditions for the ill-posed Cauchy problem for the multi-dimensional Cauchy-Riemann opera, tor partial derivative in Sobolev spaces in D. Moreover, using bases with double orthogonality property we construct Carleman's formulae for functions from the Sobolev space H-s(D), s epsilon N, from their values on Gamma and the values of partial derivative u in D, where Gamma is an open (in the topology of partial derivative D) connected part of partial derivative D.