Corollary (Poisson Formula)

Let $f\in \mathscr{L}^1(G)$. Suppose that the following conditions are satisfied:

(i) The restriction of $\mathscr{F}_G(f)$ to $H^{\bot}$ is integrable;

(ii) For every $x\in G$, the function $y\mapsto f(xy)$ on H is integrable;

(iii) The mapping $x\mapsto \int_Hf(xy)d\alpha (y)$ is continuous on G.

Then one has

Indeed, with the notation of the proof of the preceding proposition, the functions $f^{\flat}$ and $\overline{\mathscr{F}}_{\widehat{G}/H}(\mathscr{F}_{G/H}(f^{\flat}))$ on $G/H$ are continuous and equal almost everywhere. They are therefore equal everywhere and in particular at $e$, which gives formula (34).