For n ≥ 1 and α ∈(-1,1),let Hα,2 be the space of harmonic functions u on the upper half space Rn+1 + satisfying Sup(x0,r)∈Rn+1+r-(2α+n)∫B(x0,r)∫r0|▽x,tu(x,t)|2tdtdx＜∞,and L2,n+2α be the Campanato space on Rn.We show that Hα,2 coincide with e-t√?△L2,n+2α for all α∈(-1,1),where the case α∈ [0,1)was originally discovered by Fabes,Johnson and Neri [Indiana Univ.Math.J.25(1976),159-170] and yet the case α∈(-1,0)was left open.Moreover,for the scaling invariant version of Hα,2,Hα,2,which comprises all harmonic functions u on Rn+1 + satisfying Sup(x0,r)∈Rn+1+r-(2α+n)∫B(x0,r)∫r0|▽x,tu(x,t)|2t1+2αdtdx＜∞,we show that Hα,2 = e-√-△(-△)α/2 L2,n+2α,where(-△)α/2L2,n+2α is the collection of all functions f such that(-△)-α/2 f are in L2,n+2α.Analogues for solutions to the heat equation are also established.As an application,we show that the spaces((-△)α/2 L2,n+2α)-1 unify Q-1 α,BMO-1 and ˙B-1,∞∞ naturally.