Tame transverse-strip coercivity and actual weighted end inverse

Choose a fixed Euclidean norm on $V$. Compactness of the interval/unit sphere gives $c>0$, $M<\mathrm{\infty }$ with $\omega (v,J_{t}v)\ge c\mathrm{\mid }v{\mathrm{\mid }}^2$ and $\mathrm{\mid }{J}_t\mathrm{\mid }+\mathrm{\mid }\omega \mathrm{\mid }\le M$. There is a boundary Poincaré constant $B$: the map $F_0\oplus F_1\to V$, $(a,b)\mapsto b-a$, is an isomorphism. For a path $v(t)$ with endpoints in the two planes, $\mathrm{\mid }v(0)\mathrm{\mid }\le B\mathrm{\mid }v(1)-v(0)\mathrm{\mid }\le B\mathrm{\parallel }{v}_t{\mathrm{\parallel }}_2$, and the fundamental theorem then gives $\mathrm{\parallel }v{\mathrm{\parallel }}_2\le (B+1)\mathrm{\parallel }{v}_t{\mathrm{\parallel }}_2$. The same estimates hold after complexification, applied to real and imaginary parts.

First take a smooth section compactly supported in $s$ and satisfying the real boundary traces. The radial primitive ${\lambda }_V=\frac{1}{2}\omega (x,dx)$ restricts to zero on each linear Lagrangian plane. Stokes on a sufficiently long finite strip gives ${\int }_S\omega ({\xi }_s,{\xi }_t)=0$; the horizontal-edge integrals vanish and the two remote vertical-edge integrals are zero. If $f=P\xi$, it follows exactly that ${\int }_S\omega ({\xi }_t,J_t{\xi }_t)=-{\int }_S\omega (f,{\xi }_t)\mathrm{.}$ Hence $c\mathrm{\parallel }{\xi }_t{\mathrm{\parallel }}_2^2\le M\mathrm{\parallel }f{\mathrm{\parallel }}_2\mathrm{\parallel }{\xi }_t{\mathrm{\parallel }}_2$. Thus $\mathrm{\parallel }{\xi }_t{\mathrm{\parallel }}_2\le (M\mathrm{/}c)\mathrm{\parallel }f{\mathrm{\parallel }}_2$, $\mathrm{\parallel }{\xi }_s{\mathrm{\parallel }}_2\le (1+M^2\mathrm{/}c)\mathrm{\parallel }f{\mathrm{\parallel }}_2$, and step 1.1 controls $\mathrm{\parallel }\xi {\mathrm{\parallel }}_2$. Therefore $\mathrm{\parallel }\xi {\mathrm{\parallel }}_{H^1}\le C\mathrm{\parallel }P\xi {\mathrm{\parallel }}_2$, with no spectral or selfadjointness assumption.

This estimate holds for every $H^1$ section with these traces. For density, near edge $j$ extend across it by the real linear reflection $2{\mathrm{\Pi }}_{F_j}-I$. The boundary trace condition makes the extension $H^1$ with matching values. Use an even normal mollifier so its smoothed boundary value remains fixed by the reflection, hence belongs to $F_j$. A real partition near the two disjoint edges and a compact $s$ cutoff give the required smooth approximants. Their $H^1$ convergence passes step 2.1 to the limit.

To prove surjectivity, Fourier transform only in $s$. For $\tau \in \mathbb{R}$, consider $D_{\tau }\psi =i\tau \psi +J_t{\psi }^{\mathrm{\prime }},\psi (0)\in (F_0{)}_{\mathbb{C}},\psi (1)\in (F_1{)}_{\mathbb{C}}\mathrm{.}$ For $\tau \mathrm{\ne }0$, apply step 2.1 to the REAL periodic map $\mathrm{Re}(e^{i\tau s}\psi (t))$ on one $s$-period. Its vertical-side integrals cancel by periodicity, so the estimate is unchanged, independently of the period length. Average the squared norms over that period. It gives $\mathrm{\parallel }\psi {\mathrm{\parallel }}_{H_t^1}+\mathrm{\mid }\tau \mathrm{\mid }\mathrm{\parallel }\psi {\mathrm{\parallel }}_2\le C\mathrm{\parallel }{D}_{\tau }\psi {\mathrm{\parallel }}_2\mathrm{.}$ For $\tau =0$, apply the same calculation separately to its two real parts, constant in $s$. Thus this uniform estimate holds for ALL real $\tau$.

The one-dimensional boundary operator $D_{\tau }$ is onto. Solve ${\psi }^{\mathrm{\prime }}=i\tau J_t\psi -J_{t}f$ with an initial vector in $(F_0{)}_{\mathbb{C}}$. The fundamental matrix exists by its convergent iterated-integral series, whose $r$-th term is bounded by $(\mathrm{\mid }\tau \mathrm{\mid }M{)}^r\mathrm{/}r!$ on the unit interval. Variation of constants gives all $H^1$ solutions for an $L^2$ forcing. Their endpoint condition is a square linear system from the $n$-dimensional complex initial plane to the $n$-dimensional quotient $V_{\mathbb{C}}\mathrm{/}(F_1{)}_{\mathbb{C}}$. Its kernel is zero by step 4.1, so its matrix is invertible. The solution is unique, depends continuously on $\tau$, and has precisely the uniform bound in step 4.1.

For $f\in L^2(S,V)$, set $\widehat{\xi }(\tau )=D_{\tau }^{-1}\widehat{f}(\tau )$. Continuity in $\tau$ gives measurability; the uniform estimate and Plancherel give $\xi ,{\xi }_s,{\xi }_t\in L^2$. The endpoint conditions hold in the $H_t^1$ trace and therefore in the full strip trace. Reality follows from conjugating the equation at $\tau$ to that at $-\tau$. Thus $\xi \in H^1(S,V;F_0,F_1)$ and $P\xi =f$. Step3 gives uniqueness. This is an actual onto inverse, rather than a coercive estimate falsely treated as surjectivity.

For higher orders, $P$ commutes with ${\mathrm{\partial }}_s$. Step6 applied to ${\mathrm{\partial }}_s^{j}f$ controls ${\mathrm{\partial }}_s^j\xi$ in $H^1$ for $j\le k-1$. Recover every additional normal derivative from ${\xi }_t=J_t{\xi }_s-J_{t}f$. Differentiate this identity and induct on the number of $t$ derivatives, using the bounded derivatives of $J_t$. Every total derivative through order $k$ is controlled by the derivatives of $f$ through order $k-1$. This proves the stated $H^k$ inverse and smoothness up to the two edges for smooth forcing, without imposing artificial boundary compatibility on $f$.

Conjugation by $w=e^{\delta \chi }$ gives $wP{w}^{-1}=P-\delta {\chi }^{\mathrm{\prime }}$. The bounded zeroth-order perturbation has norm at most $\mathrm{\mid }\delta \mathrm{\mid }$ from $H^1$ to $L^2$. If $\mathrm{\mid }\delta \mathrm{\mid }\mathrm{\parallel }{P}^{-1}{\mathrm{\parallel }}_{L^2\to H^1}<1\mathrm{/}2$, its inverse is the norm-convergent Neumann series with norm at most twice that of $P^{-1}$. Commuting derivatives gives higher estimates as in step 7.1; derivatives of ${\chi }^{\mathrm{\prime }}$ are bounded, and the lower derivatives are already controlled inductively. This proves the weighted assertion. All constants depend continuously on the finite-dimensional plane angle/tameness bounds and finitely many coefficient derivatives, so are uniform on the stated compact families.

Along a Hamiltonian chord $\gamma (t)={\mathrm{\Phi }}_H^t(p)$, the symplectic linearized flow sends the two endpoint tangent conditions to the fixed planes $T_p{L}_0$ and $(D{\mathrm{\Phi }}_H^1{)}^{-1}{T}_{\gamma (1)}{L}_1$. They are transverse exactly by chord nondegeneracy. Conjugating the actual frozen Floer linearization by this flow removes its Hamiltonian zeroth-order evolution term, and sends $J_t$ to the tame structure $(D{\mathrm{\Phi }}_H^t{)}^{-1}{J}_{t}D{\mathrm{\Phi }}_H^t$. Its principal part is precisely the operator above. Smooth bounded flow matrices on the compact interval preserve all stated norms. This supplies the actual weighted end inverse required by the forthcoming regular polygon collar construction.