Abstract We consider the ill-posed Cauchy problem in a bounded domain 𝒟 {\mathcal{D}} of ℝ n {\mathbb{R}^{n}} for an elliptic differential operator 𝒜 ( x , ∂ ) {\mathcal{A}(x,\partial)} with data on a relatively open subset S of the boundary ∂ 𝒟 {\partial\mathcal{D}} . We do it in weighted Sobolev spaces H s , γ ( 𝒟 ) {H^{s,\gamma}(\mathcal{D})} containing the elements with prescribed smoothness s ∈ ℕ {s\in\mathbb{N}} and growth near ∂ S {\partial S} in 𝒟 {\mathcal{D}} , controlled by a real number γ. More precisely, using proper (left) fundamental solutions of 𝒜 ( x , ∂ ) {\mathcal{A}(x,\partial)} , we obtain a Green-type integral formula for functions from H s , γ ( 𝒟 ) {H^{s,\gamma}(\mathcal{D})} . Then a Neumann-type series, constructed with the use of iterations of the (bounded) integral operators applied to the data, gives a solution to the Cauchy problem in H s , γ ( 𝒟 ) {H^{s,\gamma}(\mathcal{D})} whenever this solution exists.