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

Abstract

For $n\ge 1$ and $\alpha \in (-1,1)$, let $H^{\alpha ,2}$ be the space of harmonic functions $u$ on the upper half space ${\mathbb{R}}_{+}^{n+1}$ satisfying $$\underset{(x_0,r)\in {\mathbb{R}}_{+}^{n+1}}{\sup }{r}^{-(2\alpha +n)}{\int }_{B(x_0,r)}{\int }_0^r|{\nabla }_{x,t}u(x,t){|}^{2}tdtdx<\infty ,$$ and ${\mathcal{L}}_{2,n+2\alpha }$ be the Campanato space 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, Johnson and 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 version of $H^{\alpha ,2}$, ${\mathcal{H}}^{\alpha ,2}$, which comprises all harmonic functions $u$ on ${\mathbb{R}}_{+}^{n+1}$ satisfying $$\underset{(x_0,r)\in {\mathbb{R}}_{+}^{n+1}}{\sup }{r}^{-(2\alpha +n)}{\int }_{B(x_0,r)}{\int }_0^r|{\nabla }_{x,t}u(x,t){|}^2{t}^{1+2\alpha }dtdx<\infty ,$$ 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 collection of 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 are also established. As an application, we show that the spaces $((-\Delta {)}^{\frac{\alpha }{2}}{\mathcal{L}}_{2,n+2\alpha }{)}^{-1}$ unify $Q_{\alpha }^{-1}$, ${\mathrm{BMO}}^{-1}$ and ${\dot{B}}_{\infty }^{-1,\infty }$ naturally.

arXiv comment21 pages