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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 n , 0<p< and A0, let d(y)=inf{|y-x|;xB} and define the space S A p (T B ), T B = n +iB, by the set of all functions f which are holomorphic on T B and satisfy, for some r, s>0,

f(·+iy) L p M(1+d(y) -r ) s exp(2πA|y|)

for yB. Then the main theorem is as follows: Let C be an open cone in 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 n . Here O(C) is the convex hull of C and * denotes the operation of taking dual cone. Suppose 1<p2, A0, fS A p (T C ) and the boundary values of f(x+iy) in 𝒮 ' (as y0 in C j ), corresponding to each connected component C j of C are equal in 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), zT O(C) , with P(z) being a polynomial in z and

H(z)S Aρ C 2 (T O(C) )S Aρ C q (T O(C) ),1/p+1/q=1,

ρ C : a constant depending on C.

Reviewer: K.Yabuta

32A35H p -spaces, Nevanlinna spaces (several complex variables)
32D15Continuation of analytic objects (several variables)
32A07Special domains in n (Reinhardt, Hartogs, circular, tube)