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

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