Estimate for the adjoint of a Cauchy–Riemann operator with a large antilinear term and positive zeros

Let $S^\pm$ be continuous loops of symmetric $2\times2$ matrices and let $S:\mathbb R\times S^1\to\mathbb R^{2\times2}$ be continuous with $S(s,\cdot)\to S^\pm$ uniformly as $s\to\pm\infty$. Let $\beta:\mathbb R\times S^1\to\mathbb C$ be smooth with bounded first derivative, such that $|\beta|$ and $1/|\beta|$ are bounded for large $|s|$, and whose zeros form a finite set $Z^+$ with the following normal form: for some $r>0$ and every $\zeta\in Z^+$, on the disc $|z|<r$ around $\zeta$ one has $S=0$ and $\beta(\zeta+z)=z$. For $\sigma\in\mathbb R$ let $D_\sigma\eta=\partial_s\eta+J_0\partial_t\eta+S\eta+\sigma\beta\bar\eta$ and let

be its formal adjoint, where $A(\beta)$ is the matrix of $z\mapsto\beta\bar z$. Then there is $\sigma_0$ such that for all $\sigma\ge\sigma_0$ and every test section $\varphi$,

This is the injectivity estimate for the adjoint of $D_\sigma$ in Wendl's § 5.5 when all zeros of $\beta$ are positive; it gives the surjectivity of $D_\sigma$ in Proposition 5.24.

Formalization Note. $S^1=\mathbb R/\mathbb Z$; maps on the cylinder are maps of $(s,t)\in\mathbb R\times\mathbb R$ that are $1$-periodic in $t$, and integrals over $\mathbb R\times S^1$ are over the strip $\mathbb R\times[0,1)$ (cylMeasure). Test sections are IsTestSection (smooth, $1$-periodic in $t$, zero for large $|s|$); $\partial_s,\partial_t$ are partialS, partialT, and matrices act through Matrix.toEuclideanLin. $\mathbb C=\mathbb R^2$ via $(x,y)\mapsto x+iy$, $A(\beta)$ is antilinearMat β, and $D_\sigma$ is $D_{S_\sigma}$ with $S_\sigma=S+\sigma\cdot$antilinearMat ∘ β. Points of the cylinder are pairs $(s,t)$ and $\zeta+z$ means $(\zeta_s+\operatorname{Re}z,\zeta_t+\operatorname{Im}z)$; $\beta$ is $1$-periodic in $t$, the zeros are listed in $\mathbb R\times[0,1)$, and every zero of $\beta$ is a $t$-translate by an integer of a listed one. The norms are squared and integrated over cylMeasure.