Exponential decay in s gives L^4 on the cylinder

Let $E$ be a normed group, $\kappa > 0$, $C \in \mathbb{R}$, and $f : \mathbb{R}\times\mathbb{R} \to E$ almost everywhere strongly measurable for Lebesgue measure on the strip $\mathbb{R}\times[0,1)$ (cylMeasure). If $\|f(s,t)\| \le C e^{-\kappa|s|}$ for all $s, t$, then $f \in L^4(\mathbb{R}\times[0,1))$.

Role. A convenient membership test for $L^4$ on the cylinder $\mathbb{R}\times S^1$, used for coefficients and weights that decay exponentially at both ends.