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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 ${ℝ}^{n}$, $0 and $A\ge 0$, let $d\left(y\right)=inf\left\{|y-x|;\phantom{\rule{1.em}{0ex}}x\notin B\right\}$ and define the space ${S}_{A}^{p}\left({T}^{B}\right)$, ${T}^{B}={ℝ}^{n}+iB,$ by the set of all functions f which are holomorphic on ${T}^{B}$ and satisfy, for some r, $s>0$,

${\parallel f\left(·+iy\right)\parallel }_{{L}^{p}}\le M{\left(1+d{\left(y\right)}^{-r}\right)}^{s}exp\left(2\pi A|y|\right)$

for $y\in B$. Then the main theorem is as follows: Let C be an open cone in ${ℝ}^{n}$ which is the union of a finite number of open convex cones ${C}_{j}$, such that ${\left(O\left(C\right)\right)}^{*}$ contains interior points and a basis in ${ℝ}^{n}$. Here O(C) is the convex hull of C and * denotes the operation of taking dual cone. Suppose $1, $A\ge 0$, $f\in {S}_{A}^{p}\left({T}^{C}\right)$ and the boundary values of $f\left(x+iy\right)$ in ${𝒮}^{\text{'}}$ (as $y\to 0$ in ${C}_{j}\right)$, corresponding to each connected component ${C}_{j}$ of C are equal in ${𝒮}^{\text{'}}$. Then there is an F which is holomorphic on ${T}^{O\left(C\right)}$ and $F\left(z\right)=f\left(z\right)$ in ${T}^{C}$, where F has the form $F\left(z\right)=P\left(z\right)H\left(z\right)$, $z\in {T}^{O\left(C\right)}$, with P(z) being a polynomial in z and

$H\left(z\right)\in {S}_{A{\rho }_{C}}^{2}\left({T}^{O\left(C\right)}\right)\cap {S}_{A{\rho }_{C}}^{q}\left({T}^{O\left(C\right)}\right),\phantom{\rule{1.em}{0ex}}1/p+1/q=1,$

$\rho$ ${}_{C}:$ a constant depending on C.

Reviewer: K.Yabuta

##### MSC:
 32A35 ${H}^{p}$-spaces, Nevanlinna spaces (several complex variables) 32D15 Continuation of analytic objects (several variables) 32A07 Special domains in ${ℂ}^{n}$ (Reinhardt, Hartogs, circular, tube)