DECOMPOSITIONS OF L-P AND HARDY-SPACES OF POLYHARMONIC FUNCTIONS

Let H-k(p,q,alpha) (0 < p, q less than or equal to infinity, -infinity < alpha < infinity) denote the space of those polyharmonic functions f of order k on the unit n-ball for which the function r --> (1 - r)M- alpha-1/q(p)(f,r) belongs to L-q(0,1). Our main result is that, when k greater than or equal to 2 and alpha > -1, the operator f --> (Pf, De lta f), where Pf is the Poisson integral of f, acts as an isomorphism of H-k(p,q,alpha) onto the direct sum of H-1(p,q,alpha) and H-k-1(p,q, alpha+2). Another decomposition theorem, closely related to the Almans i representation theorem, is also given. (C) 1997 Academic Press.