Extension and restriction of holomorphic functions on convex finite type domains (Q648754)

Let \(D\) be a convex domain of finite type in \(\mathbb C^n\) with smooth boundary \(bD\) and \(S\) a non-singular hypersurface in \(D\). The article deals with the problem of characterizing those holomorphic functions on \(S\) which can be extended to \(p\)-integrable holomorphic functions on \(D\). It is assumed that \(S=A\cap D\), where \(A:=\{z\in U\mid g(z)=0\}\) is defined by a regular holomorphic function \(g\) in a neighborhood \(U\) of \(\overline{D}\), and that \(A\) intersects \(bD\) transversally. The extension theorems of the paper are proved under additional assumptions (if \(n\geq 3\)) concerning specific non-degeneracy properties of \(g\). A central result is the following: It exists a finite positive Borel measure supported on S with the property that the restriction operator \(R_S:\text{Hol}(D)\rightarrow \)Hol\((S)\) induces an epimorphism \[ H^p(D)=\left\{f\in \text{Hol} (D)\;\Big|\; \int_D|f|^pdV<\infty\right\}\rightarrow H^p(S,\mu)=\left\{f\in \text{Hol}(S)\;\Big|\; \int_S|f|^pd\mu<\infty\right\}. \] Moreover there is a linear operator \(E_S:H^p(S,\mu)\rightarrow H^p(D)\) with \(R_S\circ E_S=\)id, hence a splitting of the Banach space \(H^p(D)=\big(\text{Ker}R_S\cap H^p(D)\big)\oplus H^p(S,\mu)\), \(1\leq p<\infty\). Another result of the article, which is valid for \(n>1\) in the above setting without additional assumptions, gives a necessary and sufficient condition for a finite Borel measure supported on \(S\) such that \(R_S\big(H^2(D)\big)\subset H^2(S,\mu)\). This result is closely related to the main topic of the author's paper [Manuscr. Math. 133, No. 1--2, 1--18 (2010; Zbl 1233.32006)].