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For $n\ge 1$ and $\alpha\in (-1,1)$, let $H^{\alpha,2}$ be the space ofharmonic functions $u$ on the upper half space $\mathbb{R}^{n+1}_+$ satisfying$$\displaystyle\sup_{(x_0,r)\in \mathbbR^{n+1}_+}r^{-(2\alpha+n)}\int_{B(x_0,r)}\int_0^r|\nabla_{x,t}u(x,t)|^2t\,dt\,dx<\infty,$$ and $\mathcal{L}_{2,n+2\alpha}$ be the Campanatospace on $\mathbb R^n$. We show that $H^{\alpha,2}$ coincide with$e^{-t\sqrt{-\Delta}}\mathcal{L}_{2,n+2\alpha}$ for all $\alpha\in (-1,1)$,where the case $\alpha\in [0,1)$ was originally discovered by Fabes, Johnsonand Neri [Indiana Univ. Math. J. 25 (1976), 159-170] and yet the case$\alpha\in (-1,0)$ was left open. Moreover, for the scaling invariant versionof $H^{\alpha,2}$, $\mathcal{H}^{\alpha,2}$, which comprises all harmonicfunctions $u$ on $\mathbb R^{n+1}_+$ satisfying $$\sup_{(x_0,r)\in\mathbbR^{n+1}_+}r^{-(2\alpha+n)}\int_{B(x_0,r)} \int_0^r|\nabla_{x,t}u(x,t)|^2\,t^{1+2\alpha} \,dt\,dx<\infty,$$ we show that$\mathcal{H}^{\alpha,2}=e^{-t\sqrt{-\Delta}}(-\Delta)^\frac{\alpha}{2}\mathcal{L}_{2,n+2\alpha}$,where $(-\Delta)^{\frac{\alpha}{2}}\mathcal{L}_{2,n+2\alpha}$ is the collectionof all functions $f$ such that $(-\Delta)^{-\frac{\alpha}{2}}f$ are in$\mathcal{L}_{2,n+2\alpha}$. Analogues for solutions to the heat equation arealso established. As an application, we show that the spaces$\big((-\Delta)^{\frac{\alpha}{2}}\mathcal{L}_{2,n+2\alpha}\big)^{-1}$ unify$Q_{\alpha}^{-1}$, ${\mathrm{BMO}}^{-1}$ and $\dot{B}^{-1,\infty}_\infty$naturally.

Towards spaces of harmonic functions with traces in square Campanato space and its scaling invariant