On Bell's duality theorem for harmonic functions

Abstract

[EN] Define h(infinity)(E) as the subspace of C-infinity((B) over bar, E) consisting of all harmonic functions in B, where B is the ball in the n-dimensional Euclidean space and E is any Banach space. Consider also the space h(-infinity)(E*) consisting of all harmonic E*-valued functions g such that (1 - \x\)(m)f is bounded for some m > 0. Then the dual h(infinity)(E)* is represented by h(-infinity)(E*) through [f, g](0) = lim(r-1) integral(B)[f(rx), g(x)] dx, f is an element of h(-infinity)(E*), g is an element of h(infinity)(E). This extends the results of S. Bell in the scalar case.