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Show that $F : L^2 \to L^2$ is continuous

Let $F : L^2 \to L^2$ be the linear map $F(g) = P_0(g)- P_1(g)$ where $P_0(g)$ and $P_1(g)$ are the constant-coefficient and variable-coefficient Poisson kernels, respectively.
I need to show that $F$ is continuous, but I have no idea how to show this. I have tried to use the following equality:
$$P_0(g) – P_1(g) = \frac{1}{2} \sum_{n \in \mathbb{Z}} \int_0^\infty e^{ -2 \pi^2 |n|^2 r^2} P_0(g)(x,r) – P_1(g)(x,r) dr$$
but that doesn’t seem to be helpful at all. Another idea would be showing that $F$ is uniformly continuous, but I haven’t been able to get a clue as to how to do that. Any help will be greatly appreciated.

A:

If the space $X$ is a Hilbert space, and $T\colon X\to X$ is a linear operator, then
$$\|T\| = \sup_{\|x\|=1} \|T x\|$$
is a norm. It is easy to check that this quantity is finite iff $T$ is bounded.
Another idea for a uniform continuity is to start with the following estimate for the Poisson Kernel:
 |P_
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