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

© 2016 World Scientific Publishing Company. For n≥1 and α ϵ (-1, 1), let Hα,2 be the space of harmonic functions u on the upper half-space R+n+1 satisfying sup(x0,r) ϵR+n+1r-(2α+n)∫B(x0,r)∫0r|∇ x,tu(x,t)| 2tdtdx < ∞, and L2,n+2α be the Campanato space on Rn. We show that Hα,2 coincides with e-t-ΔL2,n+2α for all α ϵ (-1, 1), where the case α ϵ [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 Lp,λ, Indiana Univ. Math. J. 25 (1976) 159-170] and yet the case α ϵ (-1, 0) was left open. Moreover, for the scaling invariant version of Hα,2, Hα,2, which comprises all harmonic functions u on R+n+1 satisfying sup(x0,r) ϵR+n+1r-(2α+n)∫B(x0,r)∫0r|∇ x,tu(x,t)|2t1+2α dt dx < ∞, we show that Hα,2 = e-t-Δ(-Δ)α2L2,n+2α, where (-Δ)α 2L2,n+2α is the collection of all functions f such that (-Δ)- α2f are in L2,n+2α. Analogues for solutions to the heat equation are also established. As an application, we show that the spaces (-Δ)α 2L2,n+2α-1 unify naturally Qα -1, BMO-1 and B∞-1,∞ which can be effectively adapted and applicable to suit handling the well/ill-posedness of the incompressible Navier-Stokes system on R+3+1.