Solvability of a class of differential equations in the sheaf of microfunctions with holomorphic parameters (Q2780730)

Let \(M\subset{\mathbb R}^n\) be an open set, with coordinates \(x=(x'=(x_1,\ldots,x_{n-1}),x_n),\) let \(X\subset{\mathbb C}^n\) be a complex neighborhood of \(M\), with coordinates \(z=(z_1,\ldots,z_n),\) and let \(Y=\{z\in X\); \(z_n=0\}.\) Let \(T^*_MX\) be the conormal bundle to \(M\) in \(T^*X,\) and consider NEWLINE\[NEWLINE\begin{aligned} V&=\{(x,i\langle\xi,dx\rangle)\in T^*_MX\setminus M;\;\xi_1=\ldots=\xi_{n-1}=0\},\\ \Sigma&=\{(x,i\langle\xi,dx\rangle)\in T^*_MX\setminus M;\;\xi_1=\ldots=\xi_{n-1}=x_n=0\}. \end{aligned}NEWLINE\]NEWLINE Let \(\gamma_0=(x_0,i\langle\xi_0,dx\rangle)\in\Sigma,\) with \(x_0=(x'_0,0),\) \(\xi_0=(0,\xi_n).\) Consider the differential operator of order \(m\) with analytic coefficients NEWLINE\[NEWLINEP=P(x,D_{x'},x_n,D_{x_n})=\sum_{|\alpha|\leq m}a_\alpha(x)D_{x'}^{\alpha'}(x_nD_{x_n})^{\alpha_n},NEWLINE\]NEWLINE assumed to be the restriction to \(M\) of a holomorphic differential operator (of the same order and form) defined on \(X.\) Write \({\mathcal A}_V^2\) for the sheaf of the 2-analytic functions (that is, microfunctions with holomorphic parameters), and denote by \({\mathcal C}^{\mathbb R}_{Y|X}\) the sheaf of microfunctions on \(Y.\) The author first proves the following theorem. NEWLINENEWLINENEWLINETheorem. Suppose \(a_{(0,\ldots,0,m)}(x_0)\not=0.\) Then NEWLINE\[NEWLINE\text{Ker}\Bigl(P\colon{\mathcal A}_V^2\rightarrow{\mathcal A}_V^2\Bigr)_{\gamma_0} \subset{\mathcal C}^{\mathbb R}_{Y|X,\gamma_0}.NEWLINE\]NEWLINE He then proves the following solvability result. NEWLINENEWLINENEWLINETheorem. Let, for \(r>0,\) \(D_r^{n-1}=\{z\in{\mathbb C}^{n-1}\); \(|z_j-x_{0,j}|< r\), \(1\leq j\leq n-1\},\) \(U_r=\{z_n\in{\mathbb C}\); \(|z_n|<r\), \(\text{Im}(\xi_{0,n}z_n)>0\}.\) Let \(f\) be a germ in \({\mathcal A}_{V,\gamma_0}^2\) represented as boundary value of holomorphic function by \(f(x)=b_{D_{r_0}^{n-1}\times U_{r_0}}(F(z)),\) with \(F\) holomorphic in \(D_{r_0}^{n-1}\times U_{r_0}\) for some \(r_0>0,\) and satisfying the growth condition \(|F(z)|\leq C|\text{Im} z_n|^{-p}\), \(z\in D_{r_0}^{n-1}\times U_{r_0}\), for positive constants \(C,p>0\) with \(p<1\). Suppose \(a_{(m,0,\ldots,0)}(x_0)\not=0\), \(a_{(0,\ldots,0,m)}(x_0)\not=0\). Then one can find a solution \(u\in{\mathcal A}_{V,\gamma_0}^2\) of \(Pu=f\).