= Poisson representation of harmonic h1 by finite measures
{c}
{title2=$u(x,t)=P_t*\mu(x),\quad\|u\|_{h^1}=\|\mu\|_{\mathrm{TV}}$}
The <Poisson kernel for the upper half-space> gives an isometric bijection from finite real or complex <Borel measures>, with their <total variation norm of a measure>, onto the <harmonic Hardy space of the upper half-space>. The kernel has unit mass, so <Tonelli theorem> gives the upper norm bound. Its <approximate identity> property gives weak-star convergence to the original measure and hence the reverse norm bound. For surjectivity, the <mean value property for harmonic functions> bounds each shifted harmonic function; bounded harmonic uniqueness gives $u(\cdot,s+t)=P_t*u(\cdot,s)$. A weak-star limit of the uniformly bounded measures $u(\cdot,s)\,dx$ then gives the required boundary measure. In particular $|u(x,t)|\leq c_dt^{-d}\|u\|_{h^1}$, where $c_d=\Gamma((d+1)/2)/\pi^{(d+1)/2}$.
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