Injectivity of the slope operator on the cylinder

Let $1 < p < \infty$, $a, b \in (0, 2\pi)$, and $\chi : \mathbb{R} \to \mathbb{R}$ smooth with $\chi'(s) = a$ for $s \le -1$ and $\chi'(s) = b$ for $s \ge 1$. Consider on the cylinder $\mathbb{R}\times S^1$ (functions on $\mathbb{R}^2$, $1$-periodic in $t$) the Cauchy-Riemann type operator

i.e. crApply with the zero-order term $S(s,t) = \chi'(s)\,\mathbb{1}$, acting on $W^{1,p}$ (W1p p 1) with values in $L^p$ (LpCyl 1 p).

Then $D$ is injective: every $w \in W^{1,p}$ with $Dw = 0$ almost everywhere is $0$.

Role. Conjugating by the weight $e^{\chi}$ turns this operator into $\partial_s + J_0\partial_t$ on exponentially weighted spaces whose weights have slopes $a$ and $b$ at the two ends. The statement covers weights that differ at the two ends, so it applies to the tangential part of perturbation problems for holomorphic curves with a removable point at one end and a nondegenerate asymptotic end at the other.