Weighted normal Cauchy–Riemann operators with a winding window are bijective

Let $2 < p < \infty$. Let $S_\infty : \mathbb{R} \to \mathbb{R}^{2\times 2}$ be a smooth symmetric loop (IsSymLoop, entries $C^\infty$). Let $\delta \in \mathbb{R}$ be a weight such that $\delta$ is not an eigenvalue of the asymptotic operator $A_\infty = -J_0\partial_t - S_\infty$ on loops, every eigenvalue $\mu < \delta$ has eigenfunctions of winding $\le 0$, and every eigenvalue $\mu > \delta$ has eigenfunctions of winding $\ge 1$ (IsLoopEigen, HasLoopEigenWind). Let $S : \mathbb{R}^2 \to \mathbb{R}^{2\times 2}$ have $C^\infty$ entries, satisfy $S(s, t+1) = S(s,t)$, be exponentially asymptotic to $S_\infty$ at some rate $d > 0$ (IsExpAsymptoticTo S Sinf d), and be the pull-back of a smooth zero-order term on the plane (IsPlaneCoeff S). Let $\chi$ be a smooth weight with $\chi' = \pi$ on $(-\infty, -1]$ and $\chi' = \delta$ on $[1, \infty)$ (IsSlopeWeight χ π δ).

Then the operator

(crApply with zero-order term $S + \chi'\,\mathbb{1}$) is a bijection from $W^{1,p}$ (W1p p 1) onto $L^p$ (LpCyl 1 p) on the cylinder: if $w \in W^{1,p}$ and $Dw = 0$ a.e., then $w = 0$; and every $f \in L^p$ equals $Dw$ a.e. for some $w \in W^{1,p}$.

Role. After conjugation by $e^{\chi}$ this is the normal Cauchy–Riemann operator along a holomorphic plane, on the space of sections that vanish to first order at $z = 0$ and decay like $e^{\delta s}$ at $\infty$. The winding window at $\delta$ is the condition under which it is an isomorphism.