Towards spaces of harmonic functions with traces in square Campanato spaces and their scaling invariants

For n >= 1 and alpha is an element of (-1, 1), let H-alpha,H- 2 be the space of harmonic functions u on the upper half-space R-+(n+1) satisfying sup((x0, r)is an element of R+n+1) r(-(2 alpha+ n)) integral(B(x0, r)) integral(r)(0) vertical bar del(x, t)u(x, t)vertical bar(2)t dt dx < infinity, and L-2, (n+ 2 alpha) be the Campanato space on R-n. We show that H-alpha,H-2 coincides with e(-t root-Delta) L-2,L-n+2 alpha for all alpha is an element of(-1, 1), where the case alpha is an element of[0, 1) was originally discovered by Fabes, Johnson and Neri [E. B. Fabes, R. L. Johnson and U. Neri, Spaces of harmonic functions representable by Poisson integrals of functions in BMO and L-p,L-lambda, Indiana Univ. Math. J. 25 (1976) 159-170] and yet the case alpha is an element of(-1, 0) was left open. Moreover, for the scaling invariant version of H-alpha,H-2, H-alpha,H-2, which comprises all harmonic functions u on R-+(n+1) satisfying sup((x0, r)is an element of R+n+1) r(-(2 alpha+ n)) integral(B(x0, r)) integral(r)(0) vertical bar del(x, t)u(x, t)vertical bar(2)t(1+2 alpha) dt dx < infinity, we show that H-alpha,H-2 = e(-t root-Delta)(-Delta)(alpha/2) L-2,L- n+ 2 alpha, where (-Delta)(alpha/2) L-2,L-n+ 2 alpha is the collection of all functions f such that (-Delta)(-alpha/2) f are in L-2,L-n+2 alpha. Analogues for solutions to the heat equation are also established. As an application, we show that the spaces ((-Delta)(alpha/2) L-2,L- n+2 alpha)(-1) unify naturally Q(alpha)(-1), BMO-1 and. B-infinity(-1,infinity) which can be effectively adapted and applicable to suit handling the well/ill-posedness of the incompressible Navier-Stokes system on R-+(3+1).