## Treatise on analysis, Volume 8 |

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apply assume Bd(X belongs BOUNDARY VALUE PROBLEMS Cauchy problem chart coefficients compact support consider converges Dchm(u deduce defined denote derivatives differential operator Dirichlet problem distribution eigenvalues ELLIPTIC DIFFERENTIAL OPERATORS equal essential spectrum exists a constant expression finite follows Fourier Frechet space function g function of class Green's formula hence Hilbert space holomorphic Hs'r(Rn Hs'r(U+ hyperplane hypotheses implies inequality integral intersection Ji x Xo kernel Lax-Maslov operators Lebesgue measure linear map matrix Moreover neighborhood notations oo[xXo open interval open set operator of order positively homogeneous principal symbol prove pseudodifferential operator rational type real numbers replace resp restriction right hand side Rn_1 Rn+1 satisfying the conditions scalar Section self-adjoint extension self-adjoint operator set in Rn Show solution square-integrable strictly hyperbolic subspace suffices support contained symbol of order tends to oo theorem upper bound vector weak topology weakly