## 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

### MSC:

 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)
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