Let $U:=\{z\in\mathbb{C}\ |\ \operatorname{Im}(z)>0\}$. Let $h(U,\mathbb{R})$ be the set of real harmonic functions defined on $U$. Define $$h_0(U,\mathbb{R}):=\left\{u\in h(U,\mathbb{R})\ |\ \left(\forall y>0, \sup_{x\in\mathbb{R}}|u(x+iy)|<\infty\right) \land \left(\lim_{y\rightarrow+\infty}\sup_{x\in\mathbb{R}}|u(x+iy)| =0\right) \right\}; \\ \forall p\in[1,\infty), h^p(U,\mathbb{R}):=\{u\in h(U,\mathbb{R})\ |\ \|u\|_{h^2(U)}^2:=\sup_{y>0}\int_{\mathbb{R}}|u(x+iy)|^2\operatorname{d}x <+\infty \}.$$ By the standard estimate: $$\forall p\in[1,\infty),\forall u\in h^p(U,\mathbb{R}), \forall x\in\mathbb{R}, \forall y>0, |u(x+iy)|\le \left(\frac{4}{\pi y}\right)^{1/p} \|u\|_{h^p(U)}$$ we have that $$\forall p \in [1,\infty), h^p(U,\mathbb{R})\subset h_0(U,\mathbb{R})$$ Equip $h_0(U,\mathbb{R})$ with the topology of uniform convergence on the family of upper half plane given by $$U_\varepsilon:=\{z\in\mathbb{C}\ |\ \operatorname{Im}(z)>\varepsilon\}$$ where $\varepsilon$ varies in $(0,\infty)$. It seems clear that $h_0(U,\mathbb{R})$ with this topology is a Frechet space.
What is the closure of $h^2(U,\mathbb{R})$ in $h_0(U,\mathbb{R})$?
Actually, I suspect that the closure can't be the whole $h_0(U,\mathbb{R})$, but it's just a guess and even if so, I'm seeking for a characterization of this closure.
The reason for this question is that I want to find a space that plays a role analogous to the role played by $h(D,\mathbb{R})$ with respect to $h^p(D,\mathbb{R})$, where $D$ is the disk and $h(D)$ is the set of harmonic real functions in the disk and $h^p(D,\mathbb{R})$ are the harmonic Hardy spaces in the disk. In particular, I'm looking for a subspace of $h_0(U,\mathbb{R}$) that contains all $h^p(U,\mathbb{R})$ for $1\le p <\infty$ where we can positively answer to this question, and my shot is that this space could be the closure of $h^2(U,\mathbb{R})$ in $h_0(U,\mathbb{R})$.