 Harmonic Hardy space of the upper half-space

For the upper half-space $H^{d+1}={\mathbb{R}}^d\times (0,\infty )$, this is the space of harmonic functions with finite norm $\Vert u{\Vert }_{h^1}={\sup }_{t>0}{\int }_{{\mathbb{R}}^d}|u(x,t)|dx$. It differs from an analytic Hardy space: no holomorphic function condition is imposed, and its boundary object can be a singular finite Borel measure.