Let $\mathbb{T}$ be the torus and $D$ the unit disk. Let $h(D)$ be the space of the harmonic complex functions in the unit disk equipped with its natural topology, i.e. the topology of the uniform convergence on the compact subsets of $D$. We know that by convolving with Poisson kernel, we obtain a continuous linear embedding of $L^1(\mathbb{T})$ into $h(D)$. Also we know that, for $0<p<1$, $L^1(\mathbb{T})$ is dense in $L^p(\mathbb{T})$ equipped with its natural metric.
So the question: does exist a (unique, by density) continuous embedding of $L^p(\mathbb{T})$ into $h(D)$ whose restriction to $L^1(\mathbb{T})$ is the Poisson's embedding?
No. Let $P_r(\theta)$ be the Poisson kernel. For any $p<1$, we have $P_r\to 0$ in $L^p(\mathbb{T})$ as $r\to 1$; indeed, this is a sequence of functions of the same $L^1$ norm which become more and more concentrated; thus, applying a power $p<1$ significantly decreases the integral.
On the other hand, the harmonic extensions of $P_r$ do not converge to $0$ on compact subsets; indeed they are all equal to $1$ at the center.