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Zbl 0583.32016
Carmichael, Richard D.
Holomorphic extension of generalizations of $H\sp p$ functions.
(English)
[J] Int. J. Math. Math. Sci. 8, 417-424 (1985). ISSN 0161-1712; ISSN 1687-0425/e

A kind of edge of the wedge theorem is studied, generalizing the Hardy spaces on tube domains. For an open subset $B$ of ${\bbfR}\sp n$, $0<p<\infty$ and $A\ge 0$, let $d(y)=\inf \{\vert y-x\vert;\quad x\not\in B\}$ and define the space $S\sp p\sb A(T\sp B)$, $T\sp B={\bbfR}\sp n+iB,$ by the set of all functions f which are holomorphic on $T\sp B$ and satisfy, for some r, $s>0$, $$\Vert f(\cdot +iy)\Vert\sb{L\sp p}\le M(1+d(y)\sp{-r})\sp s \exp (2\pi A\vert y\vert)$$ for $y\in B$. Then the main theorem is as follows: Let C be an open cone in ${\bbfR}\sp n$ which is the union of a finite number of open convex cones $C\sb j$, such that $(O(C))\sp*$ contains interior points and a basis in ${\bbfR}\sp n$. Here O(C) is the convex hull of C and * denotes the operation of taking dual cone. Suppose $1<p\le 2$, $A\ge 0$, $f\in S\sp p\sb A(T\sp C)$ and the boundary values of $f(x+iy)$ in ${\cal S}'$ (as $y\to 0$ in $C\sb j)$, corresponding to each connected component $C\sb j$ of C are equal in ${\cal S}'$. Then there is an F which is holomorphic on $T\sp{O(C)}$ and $F(z)=f(z)$ in $T\sp C$, where F has the form $F(z)=P(z)H(z)$, $z\in T\sp{O(C)}$, with P(z) being a polynomial in z and $$H(z)\in S\sp 2\sb{A\rho\sb C}(T\sp{O(C)})\cap S\sp q\sb{A\rho\sb C}(T\sp{O(C)}),\quad 1/p+1/q=1,$$ $\rho$ ${}\sb C:$ a constant depending on C.
[K.Yabuta]
MSC 2000:
*32A35 $H^p$-spaces (several complex variables)
32D15 Continuation of analytic objects (several variables)
32A07 Special domains in $C^n$

Keywords: holomorphic extension of generalizations of $H\sp p$ functions in tube domains; edge of the wedge theorem

Cited in: Zbl 0617.32005

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