For $1<q<2$, an $L^q$ map orthogonal to the image of a nondegenerate translation-invariant Cauchy–Riemann operator on test sections vanishes

Let $1<q<2$ and let $S_\infty$ be a continuous loop of symmetric $2n\times2n$ matrices whose asymptotic operator $-J_0\partial_t-S_\infty(t)$ is nondegenerate. If $g\in L^q(\mathbb R\times S^1,\mathbb R^{2n})$ satisfies

for every smooth $u:\mathbb R\times S^1\to\mathbb R^{2n}$ that vanishes for large $|s|$, then $g=0$.

This is the uniqueness of $L^q$ weak solutions of the formal adjoint equation $-\partial_s g+J_0\partial_t g+S_\infty(t)g=0$ for $1<q<2$. It is the dual-exponent counterpart of the density of the range of $\partial_s+J_0\partial_t+S_\infty(t)$ on test sections in $L^p$ for $p>2$ (Wendl, Theorem 4.11, case $k=1$).

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 $L^q(\mathbb R\times S^1)$ is LpCyl n q, i.e. $L^q$ of the strip $\mathbb R\times[0,1)$ (cylMeasure). Test sections are IsTestSection (smooth, $1$-periodic in $t$, zero for large $|s|$). The operator is crOp with $S(s,t)=S_\infty(t)$. Nondegeneracy is IsNondegenerate ($\det(1-\Psi(1))\ne0$ for the fundamental solution $\Psi'=J_0S_\infty\Psi$, $\Psi(0)=1$). $q$ is in ℝ≥0∞ with Fact (1 ≤ q), $1<q$ and $q<2$.