Linear topological properties of the
harmonic Hardy spaces h^p for 0<p<1.

 by Joel H. Shapiro


Illinois J. Math. 29 (1985)  311--339. MR 86f:46023

 Abstract: h^p is the space of functions harmonic on the unit disc that obey the same integral growth condition as the corresponding Hardy spaces H^p of analytic functions . The theme of this paper is that, while h^p is well-behaved for p >= 1, and H^p is comparatively well-behaved for p > 0, such is not the case for h^p with 0 < p < 1. For example, if p < 1 then hp is not separable, the harmonic conjugation operator does not map h^p to any h^q, and the subspace (h^p)_ 0, consisting of functions whose L^p means on the circles of radius r tend to zero as r-->1-, is nontrivial, closed, and weakly dense in h^p(P), the closure in h^p of the harmonic polynomials. The space (h^p)_0 is rotation invariant, and it turns out that no rotation invariant subspace of it is locally convex. Nevertheless, every infinite-dimensional closed subspace contains a subspace isomorphic to the (Banach) sequence space c_0. This leads to some of the pathology of the conjugation operator. The quotient space h^p(P)/(h^p)_0 is identified as L^p via an isomorphism constructed using the Poisson kenrel. This method leads also to the approximation theorem that (h^p)_0 is spanned by rotates of the Poisson kernel.


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