A proof of a result of L. Boutet de Monvel (Q2419137)

This paper deals with a detailed proof of a theorem of \textit{L. Boutet de Monvel} [C. R. Acad. Sci., Paris, Sér. A 287, 855--856 (1978; Zbl 0392.35043)]. Other proofs of the same result are given by \textit{S. Zelditch} and \textit{M. B. Stenzel} [Manuscr. Math. 144, No. 1--2, 253--276 (2014; Zbl 1297.32021)]. Denote by \(\Delta_g\) the Laplace operator on the compact, connected, analytic Riemannian manifold \((M,g)\) of dimension \(m\). As it is known \(-\Delta_g\) possesses an orthonormal basis \(\{e_j\}^\infty_0\subset L^2(M,d_g x)\) of real analytic eigenfunctions on \(M\) having the eigenvalues \(\omega^2-j\) with \(0=\omega_0<\omega_1\le \omega_2\le\cdots\), \(\omega_j\to\infty\) for \(j\to\infty\). Let \(X\supset M\) be the complexification of \(M\) and \(d(x,y)\) be the distance function on \(M\times M\). As \(d^2(x,y)\) is analytic near the diagonal Diag\(_M\) it can be extended as an holomorphic function in a complex neighbourhood of Diag\(_M\) in \(X\times X\). The author introduces the function \(\phi(z)=\frac{1}{2} \sup_{y\in M} \text{Re}(-d^2(z,y))\) on \(X\) for \(z\in X\) close to \(M\) and proves that \(\phi\) is real analytic, strictly plurisubharmonic, \(\phi|_M= d\phi|_M=0\) and the signature of the Hessian of \(\phi\) is \((m,0)\). Thus, \(\phi(z)\ge 0\), \(\phi(z)=0\; \Leftrightarrow\; z\in M\). For each \(\varepsilon>0\), \(\varepsilon\) small enough, denote by \(B_\varepsilon= \{z\in X,\ \phi(z)<\frac{\varepsilon^2}{2}\}\) the tubular neighbourhood of \(M\) in \(X\) and by \(\mathcal O(B_\varepsilon)\) the space of holomorphic functions on \(B_\varepsilon\). The boundary value \(f|_{\partial B_\varepsilon}\) of \(f\in\mathcal O(B_\varepsilon)\) is a hyperfunction. Moreover, \(f|_{\partial B_\varepsilon})\) is a classical distribution iff \(|f(z)|\le C\text{\,dist}(z,\partial B_\varepsilon)^{-N}\). The Hardy space \(H(B_\varepsilon)\) is the Hilbert space of functions \(f\in\mathcal O(B_\varepsilon)\) s.t. \(f|_{\partial B_\varepsilon}\in L_2(\partial B_\varepsilon)\). A family \(H\supset \{u_j\}^\infty_0\) forms a Riesz basis of the Hilbert space \(H\) if any \(x\in H\) can be written in a unique way as the sum \[x= \sum c_j(x) u_j,\ \Vert x\Vert^2_H= \sum|c_j|^2.\] Theorem. For \(0<\varepsilon\ll 1\) and \[\{e^{-\varepsilon\omega_j}\langle\omega_j\rangle^{\frac{m-1}{y}} e_j(z)\}^\infty_0\] is a Riesz basis of \(H(B_\varepsilon)\). Each \(f\in H(B_\varepsilon)\) can be written as \[f(z)= \sum^\infty_0 a_j e_j(z),\quad a_j= \int_M fe_j d_g x,\] the series being convergent in \(H(B_\varepsilon)\) and uniformly convergent on any compact subset of \(B_\varepsilon\). The proof here proposed is not complicated and is based on noncharacteristic deformation techniques and calculus of the Hadamard type parametrix for the Poisson operator. For the entire collection see [Zbl 1412.32002].