Holomorphic extension of generalizations of \(H^ p\) functions. (English) Zbl 0583.32016

A kind of edge of the wedge theorem is studied, generalizing the Hardy spaces on tube domains. For an open subset \(B\) of \({\mathbb{R}}^ n\), \(0<p<\infty\) and \(A\geq 0\), let \(d(y)=\inf \{| y-x|;\quad x\not\in B\}\) and define the space \(S^ p_ A(T^ B)\), \(T^ B={\mathbb{R}}^ n+iB,\) by the set of all functions f which are holomorphic on \(T^ B\) and satisfy, for some r, \(s>0\), \[ \| f(\cdot +iy)\|_{L^ p}\leq M(1+d(y)^{-r})^ s \exp (2\pi A| y|) \] for \(y\in B\). Then the main theorem is as follows: Let C be an open cone in \({\mathbb{R}}^ n\) which is the union of a finite number of open convex cones \(C_ j\), such that \((O(C))^*\) contains interior points and a basis in \({\mathbb{R}}^ n\). Here O(C) is the convex hull of C and * denotes the operation of taking dual cone. Suppose \(1<p\leq 2\), \(A\geq 0\), \(f\in S^ p_ A(T^ C)\) and the boundary values of \(f(x+iy)\) in \({\mathcal S}'\) (as \(y\to 0\) in \(C_ j)\), corresponding to each connected component \(C_ j\) of C are equal in \({\mathcal S}'\). Then there is an F which is holomorphic on \(T^{O(C)}\) and \(F(z)=f(z)\) in \(T^ C\), where F has the form \(F(z)=P(z)H(z)\), \(z\in T^{O(C)}\), with P(z) being a polynomial in z and \[ H(z)\in S^ 2_{A\rho_ C}(T^{O(C)})\cap S^ q_{A\rho_ C}(T^{O(C)}),\quad 1/p+1/q=1, \] \(\rho\) \({}_ C:\) a constant depending on C.
Reviewer: K.Yabuta


32A35 \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables
32D15 Continuation of analytic objects in several complex variables
32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010)
Full Text: DOI EuDML