Decompositions of Lp and Hardy Spaces of Polyharmonic Functions

Let Hkp, q, a 0-p, qF`, y`-a -`.denote the space of those polyharmonic functions f of order k on the unit n-ball for which the function r¬ 1yr.ay1rqMp f,r.belongs to Lq 0, 1.. Our main result is that, when kG2 and a)y1, the operator f¬ Pf, D f., where Pf is the Poisson integral of f, acts as an isomorphism of Hkp, q, a onto the direct sum of H1p, q, a and Hkpy, q1, aq2. Another decomposition theorem, closely related to the Almansi representation theorem, is also given.