Wave equations in a complex domain (Q2814327)

In this paper the author considers the Cauchy problem for 2-dimensional wave operator in \(C^3\) as follows NEWLINE\[NEWLINE(\partial^2_0- \partial^2_1- \partial^2_2) u(x)+ P_1(x_0) u(x)+ P_0(x_0) u(x)= 0,\tag{1}NEWLINE\]NEWLINE NEWLINE\[NEWLINEu(0, x_1,x_2)= u_0(x_1,x_2),\;\partial_0 u(0,x_1,x_2)= u_1(x_1,x_2)\tag{2}NEWLINE\]NEWLINE Then he proves the following theorem: Let \(Y= Y_0\cup Y_{1,+}\cup Y_{1,-}\cup Y_{2,+}\cup Y_{2,-}\), where NEWLINE\[NEWLINEY_0= \{x\in C^3; -x^2_0+ x^2_1+ x^2_2= 0\}NEWLINE\]NEWLINE is the light cone issuing from the origin, and NEWLINE\[NEWLINEY_{j,\pm}= \{x\in C^3; x_j\pm x_0= 0\},\;j= 1,2NEWLINE\]NEWLINE is a characteristic hypersurface issuing from \(\{(x_1,x_2)\in C^2; x_j\neq 0\}\) and propagating backwards or forwards. Assume that \(x^{n_0}_1 x^{n_0}_2 u_j(x_1,x_2)\), \(j= 0,1\) are holomorphic near the origin for some positive integer \(n_0\). Then the solution of (1) and (2) is holomorphic on the universal covering space \({\mathcal R}(\omega\setminus Y)\) of \(\omega\setminus Y\), where \(\omega\) is a small neighborhood of the origin.