Proof. We will define Lagrangian Rabinowitz Floer homology in the case where $q_0\neq q_1$ and then briefly explain the case where $q_0=q_1$ . Therefore, we first assume $q_0\neq q_1$ .

The first step is to show that for $q_0\neq q_1$ and generic $h\in \mathcal {H}$ , the Lagrangian Rabinowitz action functional $\mathcal {A}^{H+h}_{q_0,q_1}$ is Morse, that is, its Hessian $\nabla ^2\mathcal {A}^{H+h}_{q_0,q_1}(x)$ has trivial kernel for all $x \in \operatorname {Crit} \mathcal {A}^{H+h}_{q_0,q_1}$ . Observe that for $q_0\neq q_1$ , we we have $\operatorname {Crit} \mathcal {A}^{H+h}_{q_0,q_1} \cap ( \mathscr {H}_{q_0,q_1}\times \{0\})=\emptyset $ . By a standard Sard–Smale argument [Reference McDuff and Salamon10, Theorem A.51], there exists a residual subset $\mathcal {H}^{\,\operatorname {reg}}\subseteq \mathcal {H}\subseteq C_0^\infty (K)$ such that for all $h \in \mathcal {H}^{\,\operatorname {reg}}$ , we have

(2.6) $$ \begin{align} D \phi^\eta_{H+h} (T^*_{q_0}Q) \pitchfork T^*_{q_1}Q\quad\text{ for all}\ (v, \eta)\in \operatorname{Crit}\mathcal{A}^{H+h}_{q_0,q_1}, \end{align} $$

where $\phi ^t_{H+h}$ denotes the Hamiltonian flow of $X_{H+h}$ . The condition is equivalent to the triviality of the kernel of the Hessian $\nabla ^2 \mathcal {A}^{H+h}_{q_0,q_1}(x)$ for all $x \in \operatorname {Crit} \mathcal {A}^{H+h}_{q_0,q_1}$ . A straightforward consequence of the Morse property of the Lagrangian Rabinowitz action functional and continuous compactness of the critical set of $\mathcal {A}^{H}_{q_0,q_1}$ is that the critical set of $\mathcal {A}^{H+h}_{q_0,q_1}$ is finite for each $h \in \mathcal {H}^{\,\operatorname {reg}}$ .

Fix $h\in \mathcal {H}^{\,\operatorname {reg}}$ . Since $K \subseteq \mathcal {V}$ , by continuous compactness of the critical set of $\mathcal {A}^{H}_{q_0,q_1}$ in $(K,\mathcal {H})$ for every $J\in \mathcal {J}(\mathcal {V}, \mathbb {J})$ , we have

(2.7) $$ \begin{align} w(\mathbb{R}\times[0,1])\cap \mathcal{V}\neq \emptyset\quad \text{ for all}\ u=(w,\sigma)\in \mathscr{F}_{H+h,J}(x,y)\text{ for all}\ x,y \in \operatorname{Crit}\mathcal{A}^{H+h}_{q_0,q_1}. \end{align} $$

Therefore, we can apply the Sard–Smale argument in [Reference McDuff and Salamon10, Theorem A.51] to conclude that there exists a residual set $\mathcal {J}^{\,\operatorname {reg}}_h\subseteq \mathcal {J}(T^*Q,\omega ,\mathcal {V})$ such that for all $x,y\in \operatorname {Crit}\mathcal {A}^{H+h}_{q_0,q_1}$ and all $J \in \mathcal {J}^{\,\operatorname {reg}}_h$ , the space of Floer trajectories $\mathscr {F}_{H+h,J}(x,y)$ is a transversely cut out smooth manifold.

Let us now fix $h\in \mathcal {H}^{\,\operatorname {reg}}$ and $J\in \mathcal {J}^{\,\operatorname {reg}}_h$ . For every $x,y\in \operatorname {Crit}\mathcal {A}^{H+h}_{q_0,q_1}$ , there is a natural $\mathbb {R}$ -action on $\mathscr {F}_{H+h,J}(x,y)$ given by

$$ \begin{align*} \mathscr{F}_{H+h,J}(x,y) \times \mathbb{R} \ni (u, s) \longmapsto u(s + \cdot) \in \mathscr{F}_{H+h,J}(x,y). \end{align*} $$

Denote the quotient space $\overline {\mathscr {F}}_{H+h,J}(x,y):=\mathscr {F}_{H+h,J}(x,y)/ \mathbb {R}$ . By [Reference Merry11, Theorem 2.23], for all $x,y\in \operatorname {Crit}\mathcal {A}^{H+h}_{q_0,q_1}$ with $x\neq y$ , the quotient $\overline {\mathscr {F}}_{H+h,J}(x,y)$ is also a smooth manifold and its dimension is given by

$$ \begin{align*} \dim \overline{\mathscr{F}}_{H+h,J}(x,y)= \mu (y) - \mu (x) -1. \end{align*} $$

By condition (3) in Theorem 2.5, for all $x,y\in \operatorname {Crit}\mathcal {A}^{H+h}_{q_0,q_1}$ , the space of Floer trajectories $\mathscr {F}_{H+h,J}(x,y)$ is bounded in the $L^\infty $ -norm. Now, standard compactness arguments (see [Reference Floer7, Proposition 3b] or [Reference Audin and Damian2, Theorem 9.1.7]) imply that $\overline {\mathscr {F}}_{H+h,J}(x,y)$ is compact up to breaking in the sense of Floer. If $\mu (y)-\mu (x)=1$ , this means that $\overline {\mathscr {F}}_{H+h,J}(x,y)$ is a finite set of points [Reference Audin and Damian2, Theorem 9.2.1].

Now, we are ready to define Lagrangian Rabinowitz Floer homology. For $h\in \mathcal {H}^{\,\operatorname {reg}}$ and $k\in \mathbb {Z}$ , let $CF_k(\mathcal {A}^{H+h}_{q_0,q_1})$ be the $\mathbb {Z}_2$ -vector space of formal sums of the form $\sum _{x\in S}x$ , where $S\subseteq \operatorname {Crit}\mathcal {A}^{H+h}_{q_0,q_1}$ is a (possibly infinite) set satisfying $\mu (x)=k$ for all $x\in S$ and the Novikov finiteness condition

$$ \begin{align*} \# \{ x\in S\ |\ \mathcal{A}^{H+h}_{q_0,q_1}(x)>a\}<+\infty\quad\text{ for all}\ a\in\mathbb{R}. \end{align*} $$

Now, we fix an almost complex structure $J\in \mathcal {J}^{\,\operatorname {reg}}_h$ and turn $CF(\mathcal {A}^{H+h}_{q_0,q_1})$ into a chain complex. For $x,y\in \operatorname {Crit}\mathcal {A}^{H+h}_{q_0,q_1}$ with $\mu (y)-\mu (x)=1$ , we set

$$ \begin{align*} n(x,y):=\# \overline{\mathscr{F}}_{H+h,J}(x,y) \operatorname{mod}2\in \mathbb{Z}_2. \end{align*} $$

The Floer boundary operator $\partial _k^J: CF_k(\mathcal {A}^{H+h}_{q_0,q_1})\to CF_{k-1}(\mathcal {A}^{H+h}_{q_0,q_1})$ is the linear map defined on generators by

$$ \begin{align*} \partial_k^J y := \sum_x n(x,y)x. \end{align*} $$

Compactness up to breaking implies that $\partial _k^J\circ \partial _{k+1}^J=0$ , so we can define the Lagrangian Rabinowitz Floer homology

$$ \begin{align*} \mathrm{LRFH}_k(\mathcal{A}^{H+h}_{q_0,q_1};J) := \ker \partial_k^J / \operatorname{im}\partial_{k+1}^J. \end{align*} $$

A standard continuation argument shows that the Lagrangian Rabinowitz Floer homology does not depend on the choice of $h\in \mathcal {H}^{\,\operatorname {reg}}$ and $J\in \mathcal {J}^{\,\operatorname {reg}}_h$ . For this, let $h_i\in \mathcal {H}^{\,\operatorname {reg}}$ and $J_i\in \mathcal {J}^{\,\operatorname {reg}}_{h_i}$ for $i=0,1$ be given. Pick a homotopy $\Gamma = \{(H+h_s, J_s)\}_{s\in \mathbb {R}}$ connecting ${(H+h_0,J_0)}$ to $(H+h_1,J_1)$ as in condition (3) of Theorem 2.5. As before, it follows that for generic $\Gamma $ and any pair $(x,y)\in \operatorname {Crit} \mathcal {A}^{H+h_0}_{q_0,q_1} \times \operatorname {Crit} \mathcal {A}^{H+h_1}_{q_0,q_1}$ , the associated space of Floer trajectories $\mathscr {F}_\Gamma (x,y)$ is a smooth manifold of dimension $\mu (y)-\mu (x)$ which is compact up to breaking. In particular, if $\mu (x)=\mu (y)$ , then $\mathscr {F}_\Gamma (x,y)$ is a finite set of points and we set $m(x,y):= \# \mathscr {F}_\Gamma (x,y)\mod 2$ . We define a linear map $\phi ^\Gamma : (CF_*(\mathcal {A}^{H+h_1}_{q_0,q_1}),\partial ^{J_1})\to (CF(\mathcal {A}^{H+h_0}_{q_0,q_1}),\partial ^{J_0})$ on generators by

$$ \begin{align*} \phi^\Gamma(y) := \sum_{x\in \operatorname{Crit} \mathcal{A}^{H+h_0}_{q_0,q_1}}m(x,y)x. \end{align*} $$

Here, the Novikov finiteness condition (2.5) on $\Gamma $ ensures that $\phi ^\Gamma ( CF(\mathcal {A}^{H+h_1}_{q_0,q_1}))\subseteq CF(\mathcal {A}^{H+h_0}_{q_0,q_1})$ . Compactness up to breaking implies that $\phi ^\Gamma $ is a chain map, and composition with a homotopy in the opposite direction shows that $\phi ^\Gamma $ induces an isomorphism on homology (see [Reference Audin and Damian2, Ch. 11] for details on this standard argument).

Since $\mathrm {LRFH}_k(\mathcal {A}^{H+h}_{q_0,q_1};J)$ does not depend on $J\in \mathcal {J}^{\,\operatorname {reg}}_h$ , we can drop J from the notation and write it as $\mathrm {LRFH}_k(\mathcal {A}^{H+h}_{q_0,q_1})$ . Since this homology is the same vector space for all ${h\in \mathcal {H}^{\,\operatorname {reg}}}$ , we can unambiguously extend it as this vector space to all $h\in \mathcal {H}$ . This proves the theorem in the case where $q_0\neq q_1$ .

In the case where $q_0=q_1$ , by condition (1) for each $h\in \mathcal {H}$ , we have critical points $(v,0)\in \operatorname {Crit}\mathcal {A}^{H+h}_{q_0,q_0}$ , where v is a constant path in the non-empty transverse intersection $S_h:=T^*_{q_0}Q\cap (H+h)^{-1}(0)$ . If $S_h$ has positive dimension, then the functional $\mathcal {A}^{H+h}_{q_0,q_0}$ will only be Morse–Bott for generic $h\in \mathcal {H}$ . This phenomenon is well known in Rabinowitz Floer theory and can be dealt with by picking a Morse function on $S_h$ and counting gradient flow lines with cascades, see [Reference Cieliebak and Frauenfelder4]. With this understood, the rest of the argument works as in the previous case.

Proof. Denote by $\mathcal {H}^{\,\,\operatorname {reg}}$ the subset of $\mathcal {H}$ consisting of elements satisfying (2.6). If $h\in \mathcal {H}^{\,\operatorname {reg}}$ , then we can define $\mathrm {LRFH}_*(\mathcal {A}^{H+h}_{q_0,q_1})$ as explained in the proof of Theorem 2.5. In particular, the complex of $\mathrm {LRFH}_*(\mathcal {A}^{H+h}_{q_0,q_1})$ is generated by elements of $\operatorname {Crit}(\mathcal {A}^{H+h}_{q_0,q_1})$ . Moreover, by Theorem 2.5, $\mathrm {LRFH}_*(\mathcal {A}^{H+h}_{q_0,q_1})$ is isomorphic to $\mathrm {LRFH}_*(\mathcal {A}^{H}_{q_0,q_1})\neq 0$ . Hence,

$$ \begin{align*} \# \operatorname{Crit}(\mathcal{A}^H_{q_0,q_1}) \geq \sum_{k\in ({1}/{2})+\mathbb{Z}} \mathrm{LRFH}_k(\mathcal{A}^{H+h}_{q_0,q_1})\geq 1. \end{align*} $$

Suppose there exists $h\in \mathcal {H}$ such that $\operatorname {Crit}(\mathcal {A}^{H+h}_{q_0,q_1})= \emptyset $ . Then, it automatically satisfies condition (2.6), so by definition, it belongs to $\mathcal {H}^{\,\operatorname {reg}}$ . However, that brings us a contradiction by the argument presented above.

Proof. Let $h\in \mathcal {H}$ . Then, there exists $n\in \mathbb {N}$ such that $h\in \mathcal {H}_n$ . Since $\mathcal {H}$ is an open neighbourhood of $0$ in $\{h\in C^\infty (T^*Q)\ |\ dh\in C_c^\infty (T^*Q)\}$ , the set $\mathcal {H}_n$ is an open neighbourhood of $0$ in $\{h\in C^\infty (T^*Q)\ |\ dh\in C_0^\infty (K_n)\}$ . By assumption, the sets $K_n$ and $\mathcal {H}_n$ satisfy the assumptions of Theorem 2.5. By Theorem 2.5, for every $h\in \mathcal {H}_n$ , the Lagrangian Rabinowitz Floer homology $\mathrm {LRFH}_*(\mathcal {A}^{H+h}_{q_0,q_1})$ is well defined and isomorphic to $\mathrm {LRFH}_*(\mathcal {A}^{H}_{q_0,q_1})$ .

Proof. By assumption, there exists a Liouville vector field Y such that $dH(Y)>0$ along $H^{-1}(0)$ . Since K is compact,

$$ \begin{align*} \delta:=\inf\{dH(Y)(x)\ |\ x\in K\cap H^{-1}(0)\}> 0. \end{align*} $$

For $h\in \mathcal {H}$ small enough in the $C^1$ -norm, we then have

(2.13) $$ \begin{align} (H+h)^{-1}(0)\cap K \subseteq \lbrace x\in K d(H+h)(Y)(x)\geq\delta/2 \rbrace. \end{align} $$

Denote by $\mathcal {O}(K)$ the set of all $h\in \mathcal {H}$ satisfying (2.13). Then, $\mathcal {O}(K)$ is an open neighbourhood of $0$ in $\{h\in C^\infty (T^*Q)\ |\ dh\in C_0^\infty (K)\}$ . Consider $h\in \mathcal {O}(K)$ such that $\operatorname {Crit}^+ \mathcal {A}^{H+h}_{q_0,q_1}\neq \emptyset $ . If $(v,\eta )\in \operatorname {Crit}^+ \mathcal {A}^{H+h}_{q_0,q_1}$ , then $\eta>0$ in view of (2.10) applied to $H+h$ . By (2.11), we have $v([0,1])\subset K\cap (H+h)^{-1}(0)$ , so from (2.9) and (2.13), we obtain

$$ \begin{align*} |\mathcal{A}^{H+h}_{q_0,q_1}(v,\eta)| \geq \delta\eta/2. \end{align*} $$

However, from $v(i)\in T_{q_i}^*Q$ for $i=0,1$ and $\partial _sv=\eta X_{H+h}(v)$ , we deduce

$$ \begin{align*} 0 < \varepsilon \leq |v(1)-v(0)| \leq \eta\int_0^1|X_{H+h}(v(t))|\,dt \leq C\eta \end{align*} $$

with the positive constants

$$ \begin{align*} \varepsilon := \textrm{dist}(K\cap T_{q_0}^*Q,K\cap T_{q_1}^*Q),\quad C := \max\{|X_{H+h}(x)|\;|\;x\in K\}. \end{align*} $$

The two estimates combine to $|\mathcal {A}^{H+h}_{q_0,q_1}(v,\eta )| \geq \delta \varepsilon /2C>0$ .

Proof. By assumption, the critical set of $\mathcal {A}^H_{q_0,q_1}$ is continuously compact in $(K,\mathcal {O})$ . Denote by $\mathcal {O}^{\,\operatorname {reg}}$ the subset of $\mathcal {O}$ consisting of all Hamiltonian functions $h\in \mathcal {O}$ such that $\mathcal {A}^{H+h}_{q_0,q_1}$ is Morse. By the standard Sard–Smale argument, $\mathcal {O}^{\,\operatorname {reg}}$ is dense in $\mathcal {O}$ . Fix $h_0,h_1 \in \mathcal {O}^{\,\operatorname {reg}}$ . By assumption, there exist $J_i \in \mathcal {J}_{h_i}^{\,\operatorname {reg}}\subset \mathcal {J}(\mathcal {V}, \mathbb {J})$ for $i=0,1$ and a homotopy $\Gamma = \{(H+h_s, J_s)\}_{s\in \mathbb {R}}$ with $h_s\in \mathcal {O}$ and $J_s\in \mathcal {J}(\mathcal {V}, \mathbb {J})$ from $(H+h_0,J_0)$ to $(H+h_1,J_1)$ with the following properties:

1. (a) the homotopy $\Gamma $ satisfies the Novikov finiteness condition (2.5);

2. (b) for any pair $a,b\in \mathbb {R}$ , the space of Floer trajectories $\mathcal {M}^\Gamma (a,b)$ is bounded in the $L^\infty $ -norm;

3. (c) for every $x\in \operatorname {Crit}^+\mathcal {A}^{H+h_0}_{q_0,q_1}$ and every $y\in \operatorname {Crit}\mathcal {A}^{H+h_1}_{q_0,q_1}$ such that $\mathscr {F}_\Gamma (x,y)\neq \emptyset $ , we have $\mathcal {A}^{H+h_1}_{q_0,q_1}(y)>0$ .

From properties (a) and (b), we get a chain map $\phi ^\Gamma : CF_*(\mathcal {A}^{H+h_1}_{q_0,q_1})\to CF_*(\mathcal {A}^{H+h_0}_{q_0,q_1})$ . It satisfies $\phi ^\Gamma (CF_*^{\leq 0}(\mathcal {A}^{H+h_1}_{q_0,q_1})) \subseteq CF_*^{\leq 0}(\mathcal {A}^{H+h_0}_{q_0,q_1})$ , since otherwise, there would exist ${x\in \operatorname {Crit}^+\mathcal {A}^{H+h_0}_{q_0,q_1}}$ and $y\in \operatorname {Crit}\mathcal {A}^{H+h_1}_{q_0,q_1}$ such that $\mathscr {F}_\Gamma (x,y)\neq \emptyset $ and $\mathcal {A}^{H+h_1}_{q_0,q_1}(y)\leq 0$ , which contradicts property (c). Thus, $\phi ^\Gamma $ descends to a chain map $\phi ^\Gamma _+ : CF_*^+(\mathcal {A}^{H+h_1}_{q_0,q_1})\to CF_*^+(\mathcal {A}^{H+h_0}_{q_0,q_1})$ , and the usual argument using a homotopy in the opposite direction (see [Reference Audin and Damian2, Proposition 11.2.9]) shows that the induced map on homology $\Phi ^\Gamma _+ \colon \mathrm {LRFH}_*^+(\mathcal {A}^{H+h_1}_{q_0,q_1})\to \mathrm {LRFH}_*^+(\mathcal {A}^{H+h_0}_{q_0,q_1})$ is an isomorphism.

Proof. Using (1.1), we can compute that the Hamiltonian $H_0$ satisfies the following equality:

(3.4) $$ \begin{align} dH_0(p\partial_p) = p_1^2+p_2^2+q_2p_1-q_1p_2 =\tfrac{1}{2}|p|^2+H_0(p,q). \end{align} $$

Combining that with equality (2.9) for all $(q,p,\eta )=(v,\eta )\in \mathscr {H}_{q_0,q_1}$ , we obtain

$$ \begin{align*} \mathcal{A}^{H_0-h}(v,\eta) -d\mathcal{A}^{H_0-h}[p\partial_p,0] & =\eta\int_0^1(d(H_0-h)(p\partial_p)-H_0(v)+h(v))\nonumber\quad \\ & = \eta \bigg(\dfrac{1}{2}\|p\|_{L^2} +\int_0^1 (h(v)-dh(p\partial p))\bigg)\quad \end{align*} $$

(3.5) $$ \begin{align} &\qquad\quad \geq \eta \bigg(\dfrac{1}{2}\|p\|_{L^2}+c\bigg), \end{align} $$

where the last inequality comes from the fact that $c:=\inf \{h-dh(p\partial _p\}>0$ .

Consequently, for all $(q,p,\eta )=(v,\eta )\in \mathscr {H}_{q_0,q_1}$ such that $\|\nabla \mathcal {A}^{H_0-h}(v,\eta )\|_{L^2\times \mathbb {R}} <1$ , we obtain

$$ \begin{align*} |\eta| < \frac{|\mathcal{A}^{H_0-h}(v,\eta)|+\|p\|_{L^2}}{\frac{1}{2}\|p\|_{L^2}^2+c}. \end{align*} $$

Now, let us analyse the right-hand side as functions of $\|p\|_{L^2}$ :

$$ \begin{align*} \max_{x\geq 0}\bigg(\frac{1}{\tfrac{1}{2} x^2+c}\bigg) & =\dfrac{1}{c},\\ \dfrac{d}{dx}\bigg(\frac{x}{\tfrac{1}{2}x^2+c}\bigg) & = \dfrac{c-\tfrac{1}{2}x^2}{(\tfrac{1}{2}x^2+c)^2},\\ \max_{x\geq 0} \bigg(\dfrac{x}{\tfrac{1}{2}x^2+c}\bigg) & = \dfrac{1}{\sqrt{2c}}. \end{align*} $$

These estimates give the desired inequality.

Proof. Using the fact that $u\in \mathcal {M}^\Gamma (a,b)$ is a Floer trajectory, one can calculate the derivative of the action functional over s and obtain the following inequalities (see [Reference Pasquotto and Wiśniewska13, Proposition 3.3]):

(3.6) $$ \begin{align} \|\mathcal{A}^{H_0-h_{s}}( u)\|_{L^{\infty}} & \leq \max\{|a|,|b|\} + \|\eta\|_{L^{\infty}}\|\partial_{s}h_{s}\|_{L^{\infty}}, \end{align} $$

(3.7) $$ \begin{align} \|\nabla^{J_{s}} \mathcal{A}^{H_0-h_{s}}(u)\|_{L^{2}(\mathbb{R}\times [0,1])}^{2} & \leq \|J\|_{L^{\infty}} (b-a + \|\eta\|_{L^{\infty}}\|\partial_{s}h_{s}\|_{L^{\infty}}). \end{align} $$

In particular, the convergence of the integral

$$ \begin{align*} \|\nabla \mathcal{A}^{H_0-h_{s}}(u)\|_{L^{2}(\mathbb{R}\times [0,1])} \leq \|J\|_{L^{\infty}} \|\nabla^{J_{s}} \mathcal{A}^{H_0-h_{s}}(u)\|_{L^{2}(\mathbb{R}\times [0,1])} \end{align*} $$

implies that for small enough s, we have $\|\nabla \mathcal {A}^{H_0-h_{s}}(u(s))\|_{L^{2}\times \mathbb {R}}<1$ . This ensures that for all $s \in \mathbb {R}$ , the following value $\tau _{0}(s)$ is well defined and finite:

$$ \begin{align*} \tau_{0}(s): = \sup\{ \tau \leq s\ |\ \|\nabla \mathcal{A}^{H_0-h_{\tau}}(u(\tau))\|_{L^{2}\times\mathbb{R}}<1\}. \end{align*} $$

For $\tau \in [\tau _{0}(s),s]$ , we have $\|\nabla \mathcal {A}^{H_0-h_{\tau }}(u(\tau ))\|_{L^{2}\times \mathbb {R}}\geq 1$ , which allows us to estimate (see [Reference Pasquotto and Wiśniewska13, Proposition 3.3])

(3.8) $$ \begin{align} |s -\tau_0(s)| & \leq \|J\|_{L^{\infty}}^3 (b-a + \|\eta\|_{L^{\infty}}\|\partial_{s}h_{s}\|_{L^{\infty}}), \end{align} $$

(3.9) $$ \begin{align} |\eta(s) -\eta(\tau_0(s))| & \leq \|J\|_{L^{\infty}}^2 (b-a + \|\eta\|_{L^{\infty}}\|\partial_{s}h_{s}\|_{L^{\infty}}). \end{align} $$

Now, using the constant $c>0$ as in (3.3) together with the the estimates from Lemma 3.4 and (3.6), we can calculate

$$ \begin{align*} |\eta(s)|& \leq |\eta(\tau_0(s))|+|\eta(s) -\eta(\tau_0(s))|\\ & \leq \dfrac{1}{c}|\mathcal{A}^{H_0-h_s}(u(s))|+\dfrac{1}{\sqrt{2c}}+\|J\|_{L^{\infty}}^2 (b-a + \|\eta\|_{L^{\infty}}\|\partial_{s}h_{s}\|_{L^{\infty}})\\ & \leq \|\eta\|_{L^{\infty}}\|\partial_{s}h_{s}\|_{L^{\infty}} \bigg(\frac{1}{c}+\|J\|_{L^{\infty}}^2\bigg)+\frac{1}{c}\max\{|a|,|b|\}+\frac{1}{\sqrt{2c}}+ \|J\|_{L^{\infty}}^2 (b-a). \end{align*} $$

Since this inequality has to hold for all $s\in \mathbb {R}$ , it also has to hold for $\|\eta \|_{L^\infty }$ . Using (3.2), we obtain the following estimate:

(3.10) $$ \begin{align} \|\eta\|_{L^{\infty}} & \leq \frac{({1}/{c})\max\{|a|,|b|\}+{1}/{\sqrt{2c}}+\|J\|_{L^{\infty}}^2 (b-a)}{1-\|\partial_{s}h_{s}\|_{L^{\infty}}({1}/{c}+\|J\|_{L^{\infty}}^2)}\nonumber\\ &\leq \frac{3}{2} \bigg( \dfrac{1}{c}\max\{|a|,|b|\}+\dfrac{1}{\sqrt{2c}}+\|J\|_{L^{\infty}}^2 (b-a)\bigg)=:\mathfrak{y}. \end{align} $$

Now, using (3.2), (3.6), (3.7) and (3.10), we obtain the desired uniform bounds:

(3.11) $$ \begin{align} \|\mathcal{A}^{H_0-h_{s}-c}( u)\|_{L^{\infty}} & \leq \max\{|a|,|b|\} + \frac{c}{3}\mathfrak{y}=:\mathfrak{a}, \end{align} $$

(3.12) $$ \begin{align} \|\nabla^{J_{s}} \mathcal{A}^{H_0-h_{s}-c}(u)\|_{L^{2}(\mathbb{R}\times [0,1])}^{2} & \leq \|J\|_{L^{\infty}} \bigg(b-a + \frac{c}{3}\mathfrak{y}\bigg)=:\mathfrak{e}. \\[-25pt]\nonumber \end{align} $$

Proof. This proof follows arguments presented in [Reference Cieliebak and Frauenfelder4, Corollary 3.8]. We will prove the first inequality since the second is completely analogous.

(3.13) $$ \begin{align} \textrm{First, assume }\ 3\sqrt{\frac{c}{2}} & \leq a \quad \textrm{and} \quad |b|\leq a.\\ \textrm{Then, } \max\{|a|,|b|\} & = a \quad \textrm{and} \quad b-a\leq 0.\nonumber \end{align} $$

By (3.10) and (3.13), we get

$$ \begin{align*} \|\eta\|_{L^\infty}\leq \dfrac{3}{2}\bigg(\dfrac{a}{c}+\dfrac{1}{\sqrt{2c}}\bigg)=2\dfrac{a}{c}. \end{align*} $$

However, by (3.2), we have $\|\partial _sh_s\|_{L^\infty }\leq ({c}/{3})$ . This, together with (3.7), implies

$$ \begin{align*} b \geq a- \|\eta\|_{L^\infty}\|\partial_sh_s\|_{L^\infty} \geq a -2 \frac{a}{c}\cdot \frac{c}{3}=\frac{1}{3}a. \end{align*} $$

Now, assume that $ 3\sqrt {({c}/{2})} \leq a < |b|$ . To finish the proof, it suffices to exclude the case $a< - b$ .

$$ \begin{align*} \textrm{Then } \max\{|a|,|b|\} = -b \quad \textrm{and} \quad b-a\leq 0. \end{align*} $$

By (3.10) and (3.13), we get

$$ \begin{align*} \|\eta\|_{L^\infty}\leq \frac{3}{2}\bigg(\dfrac{-b}{c}+\dfrac{1}{\sqrt{2c}}\bigg)\leq \frac{3}{2}\bigg(\dfrac{-b}{c}+\dfrac{a}{3c}\bigg)\leq -2 \frac{b}{c}. \end{align*} $$

This, together with (3.2) and (3.7), gives us the following contradiction:

$$ \begin{align*} & b \geq a- \|\eta\|_{L^\infty}\|\partial_sh_s\|_{L^\infty}\geq a + 2 \frac{b}{c} \cdot \frac{c}{3}=a+\frac{2}{3}b,\\ & \qquad\qquad\quad 0 \geq \frac{1}{3} b \geq a \geq 3\sqrt{\frac{c}{2}}>0. \\[-42pt] \end{align*} $$

Proof. Using (3.5) for $(q,p,\eta ) \in \mathcal {B}_h (\mathfrak {a}, \mathfrak {y}, \varepsilon )$ , we obtain

$$ \begin{align*} \eta \bigg(\dfrac{1}{2}\|p\|_{L^2}^2+c\bigg) &\leq \mathcal{A}^{H_0-h}(v,\eta)-d\mathcal{A}^{H_0-h}[p\partial_p,0] \leq \mathfrak{a}+\varepsilon\|p\|_{L^2},\\ |\eta|\|p\|_{L^2}&\leq \frac{\mathfrak{a}\|p\|_{L^2}+\varepsilon\|p\|_{L^2}^2}{\tfrac{1}{2}\|p\|_{L^2}^2+c}. \end{align*} $$

Now, we would like to estimate the maximum of the function on the right-hand side of the inequality:

$$ \begin{align*} \frac{d}{dx}\bigg(\frac{\mathfrak{a}x+\varepsilon x^2}{\tfrac{1}{2}x^2+c}\bigg)& = -\frac{\tfrac{1}{2}\mathfrak{a}x^2-2\varepsilon c x -\mathfrak{a} c}{(\tfrac{1}{2}x^2+c)^2},\\ \quad\frac{d}{dx}\bigg(\frac{\mathfrak{a}x+\varepsilon x^2}{\tfrac{1}{2}x^2+c}\bigg)& = 0 \quad \iff \quad x=\frac{1}{\mathfrak{a}}(2\varepsilon c \pm \sqrt{4 \varepsilon^2 c^2 + 2 \mathfrak{a}^2 c}),\\ \quad\max_{x\geq 0} \bigg(\frac{\mathfrak{a}x+\varepsilon x^2}{\tfrac{1}{2}x^2+c}\bigg)&= \varepsilon+ \sqrt{\frac{\mathfrak{a}^2}{2c}+\varepsilon^2} \leq 2\varepsilon +\frac{\mathfrak{a}}{\sqrt{2c}}. \end{align*} $$

This gives us the bound on $|\eta | \|p\|_{L^2}$ , that is, the first inequality.

Let us denote

$$ \begin{align*} (X_{H_0-h})_p:= \bigg(p_2 +\frac{\partial h}{\partial_{q_1}}\bigg)\partial_{p_1}-\bigg(p_1-\frac{\partial h}{\partial_{q_2}}\bigg)\partial_{p_2}. \end{align*} $$

Then, we can estimate

$$ \begin{align*} \|\partial_tp\|_{L^2} & \leq \| \partial_t p-\eta (X_{H_0-h})_p\|_{L^2}+|\eta| \|(X_{H_0-h})_p\|_{L^2}\nonumber\\ & \leq \|\nabla\mathcal{A}^H(q,p,\eta)\|_{L^2\times \mathbb{R}}+|\eta|(\|p\|_{L^2}+\|\nabla h \|_{L^\infty})\\ & \leq 3\varepsilon +\frac{\mathfrak{a}}{\sqrt{2c}} + \mathfrak{y}\|\nabla h \|_{L^\infty}. \\[-42pt] \end{align*} $$

Proof. First, observe that

$$ \begin{align*} qdq(X_{H_0})=q_1(p_1+q_2)+q_2(p_2-q_1)=q_1 p_1 + q_2 p_2. \end{align*} $$

Therefore, for all $t\in [0,1]$ , we have

(3.15) $$ \begin{align} |q(t)|^2 - |q_0|^2 & =2\int_0^t qdq(\partial_tv - \eta X_{H_0-h})+2\eta \int_0^t (qdq(X_{H_0})-qdq(X_h))\nonumber\\ & \leq 2\|q\|_{L^2}\|\nabla \mathcal{A}^H(q,p,\eta)\|_{L^2\times \mathbb{R}}+2|\eta|\|q\|_{L^2}(\|p\|_{L^2} + \|\nabla h\|_{L^\infty})\nonumber\\ & \leq 2 \|q\|_{L^2}\bigg(3\varepsilon + \frac{\mathfrak{a}}{\sqrt{2c}}+\mathfrak{y}\|\nabla h\|_{L^\infty}\bigg), \end{align} $$

where the last inequality was obtained using the result from Lemma 3.8. Integrating both sides, we obtain

$$ \begin{align*} \|q\|_{L^2}^2\leq |q_0|^2+2 \|q\|_{L^2}\bigg(3\varepsilon + \frac{\mathfrak{a}}{\sqrt{2c}}+\mathfrak{y}\|\nabla h\|_{L^\infty}\bigg). \end{align*} $$

By solving this quadratic inequality, we obtain the following bound:

$$ \begin{align*} \|q\|_{L^2} & \leq 3\varepsilon + \frac{\mathfrak{a}}{\sqrt{2c}}+\mathfrak{y}\|\nabla h\|_{L^\infty} + \sqrt{\bigg(3\varepsilon + \frac{\mathfrak{a}}{\sqrt{2c}}+\mathfrak{y}\|\nabla h\|_{L^\infty}\bigg)^2+|q_0|^2}\\ & \leq |q_0|+2 \bigg(3\varepsilon + \frac{\mathfrak{a}}{\sqrt{2c}}+\mathfrak{y}\|\nabla h\|_{L^\infty}\bigg). \end{align*} $$

By repeating this procedure with the equation $|q(t)|^2=|q_1|^2-2\int _t^1 q\partial _tq$ , we obtain the bound on $\|q\|_{L^2}$ for which we were looking.

To obtain the bound for $\|q\|_{L^\infty }$ , we will use (3.15) again:

$$ \begin{align*} \|q\|_{L^\infty} & \leq \sqrt{|q_0|^2+ 2 \|q\|_{L^2}\bigg(3\varepsilon + \frac{\mathfrak{a}}{\sqrt{2c}}+\mathfrak{y}\|\nabla h\|_{L^\infty}\bigg)}\\ & \leq \sqrt{|q_0|^2+ 2 \bigg( |q_0|+2 \bigg(3\varepsilon + \frac{\mathfrak{a}}{\sqrt{2c}}+\mathfrak{y}\|\nabla h\|_{L^\infty}\bigg)\bigg)\bigg(3\varepsilon + \frac{\mathfrak{a}}{\sqrt{2c}}+\mathfrak{y}\|\nabla h\|_{L^\infty}\bigg)}\\ & \leq |q_0|+ 2\bigg(3\varepsilon + \frac{\mathfrak{a}}{\sqrt{2c}}+\mathfrak{y}\|\nabla h\|_{L^\infty}\bigg). \end{align*} $$

Analogously as before, we repeat this procedure with the equation $|q(t)|^2=|q_1|^2-2\int _t^1 q\partial _tq$ to obtain the bound on $\|q\|_{L^\infty }$ for which we were looking.

Proof. By (3.4), we have

$$ \begin{align*} \frac{1}{2}\|p\|_{L^2}^2 & = \int_0^1 (H_0-h)(q,p)\,dt -\int_0^1(p_1q_2-p_2q_1)\,dt +\int_0^1 h(q,p)\,dt\\ & \leq \|\nabla \mathcal{A}^{H_0-h-c}\|_{L^2\times \mathbb{R}}+\|p\|_{L^2}\|q\|_{L^2}+\|h\|_{L^\infty}+c\\ & \leq \varepsilon + \mathfrak{q}\|p\|_{L^2} +\|h\|_{L^\infty}+c, \end{align*} $$

where $\mathfrak {q}:=\min \{|q_0|,|q_1|\}+2 (3\varepsilon + {\mathfrak {a}}/{\sqrt {2c}}+\mathfrak {y}\|\nabla h\|_{L^\infty })$ as in Lemma 3.9. By solving this quadratic inequality, we obtain the following bound:

$$ \begin{align*} \|p\|_{L^2} & \leq \mathfrak{q} + \sqrt{\mathfrak{q}^2 + 2 ( \|h\|_{L^\infty}+c+\varepsilon)} \leq 2 \mathfrak{q}+\sqrt{2 ( \|h\|_{L^\infty}+c+\varepsilon)}\\ & = 2\min\{|q_0|,|q_1|\}+4 \bigg(3\varepsilon + \frac{\mathfrak{a}}{\sqrt{2c}}+\mathfrak{y}\|\nabla h\|_{L^\infty}\bigg)+\sqrt{2 ( \|h\|_{L^\infty}+c+\varepsilon)}. \end{align*} $$

To prove the second bound, first observe that there exists $t_0\in [0,1]$ such that $|p(t_0)|\leq 2 \mathfrak {q}+\sqrt {2 ( \|h\|_{L^\infty }+c+\varepsilon )}$ . Thus, we can estimate

$$ \begin{align*} \begin{aligned} |p(t)| & \leq |p(t_0)| + \int_{t_0}^t |\partial_tv|= |p(t_0)| + \int_{t_0}^t |\partial_tv- X_{H_0-h}|+\int_{t_0}^t |X_h|\\ & \leq |p(t_0)| +\|\nabla\mathcal{A}^{H_0-h-c}\|_{L^2\times \mathbb{R}} + \|\nabla h\|_{L^\infty} \leq |p(t_0)| +\varepsilon+\|\nabla h\|_{L^\infty}\\ & \leq 2\min\{|q_0|,|q_1|\}+13\varepsilon + 4\frac{\mathfrak{a}}{\sqrt{2c}}+(4\mathfrak{y}+1)\|\nabla h\|_{L^\infty} +\sqrt{2 ( \|h\|_{L^\infty}+c+\varepsilon)}. \end{aligned} \end{align*} $$

Proof. By Lemma 3.5, there exist $\mathfrak {a,e, y}>0$ such that for every $u:=(v,\eta )\in \mathcal {M}^\Gamma (a,b)$ , we have

$$ \begin{align*} \|\eta\|_{L^\infty} \leq \mathfrak{y}, \quad\|\mathcal{A}^{H_0-h_s}( u)\|_{L^{\infty}} &\leq\mathfrak{a} \quad \textrm{and}\quad \|\nabla^{J_s} \mathcal{A}^{H_0-h_s}(u)\|_{L^{2}(\mathbb{R}\times [0,1])}^{2} \leq \mathfrak{e}. \end{align*} $$

Moreover, by (3.12), the convergence of the integral

$$ \begin{align*} \|\nabla \mathcal{A}^{H_0-h_s}(u)\|_{L^{2}(\mathbb{R}\times [0,1])}^{2} \leq \mathfrak{e}\|J\|_{\infty}^2 \end{align*} $$

implies that for every $\varepsilon>0$ , there exists $s\in \mathbb {R}$ small enough such that $\|\nabla \mathcal {A}^{H_0-h_s} (u(s))\|_{L^{2}\times \mathbb {R}}<\varepsilon $ . This ensures that for all $s \in \mathbb {R}$ , the following value $\tau _\varepsilon (s)$ is well defined and finite:

$$ \begin{align*} \tau_\varepsilon(s): = \sup\lbrace \tau \leq s\ |\ u(\tau) \in \mathcal{B}_{h_\tau}(\mathfrak{a},\mathfrak{y}, \varepsilon)\rbrace. \end{align*} $$

Note that for all $\tau \in [\tau _\varepsilon , s]$ , we have $\|\nabla \mathcal {A}^{H_0-h_\tau -c}(u(\tau ))\|_{L^{2}\times \mathbb {R}}\geq \varepsilon $ ; thus, we can estimate

(3.16) $$ \begin{align} & \varepsilon^2 |s-\tau_\varepsilon(s)| \leq\|\nabla \mathcal{A}^{H_0-h_s}(u)\|_{L^{2}(\mathbb{R}\times [0,1])}^{2}\leq\mathfrak{e}\|J\|_{\infty}^2,\nonumber\\ & \quad |s-\tau_\varepsilon(s)|\leq\frac{ \mathfrak{e}\|J\|_{\infty}^2}{\varepsilon^2}. \end{align} $$

However, we know that

$$ \begin{align*} & c:=\inf_{s\in[0,1]}\{h_s-dh_s(p\partial_p)\}>0,\end{align*} $$

$$ \begin{align*} &\sup_{s\in[0,1]}\|h_s\|_{L^\infty} < +\infty \quad\textrm{and} \quad \sup_{s\in[0,1]}\|\nabla h_s\|_{L^2} < +\infty, \end{align*} $$

therefore, by Remark 3.12, we know that the set

(3.17) $$ \begin{align} \mathcal{B}^\Gamma(\mathfrak{a},\mathfrak{y},\varepsilon):=\bigcup_{s\in [0,1]}\mathcal{B}_{h_s}(\mathfrak{a},\mathfrak{y},\varepsilon)\subseteq \mathscr{H}_{q_0,q_1}\times \mathbb{R} \end{align} $$

is bounded in $L^\infty \times \mathbb {R}$ - and $L^2\times \mathbb {R}$ -norm. In fact, we can use the same bounds as in Lemmas 3.9 and 3.10, denoting

$$ \begin{align*} \mathfrak{h} & :=\sup_{s\in[0,1]}\|h_s\|_{W^{1,\infty}},\\ \mathfrak{q} & := \min\{|q_0|,|q_1|\}+2 \bigg(3\varepsilon + \frac{\mathfrak{a}}{\sqrt{2c}}+\mathfrak{y h}\bigg),\\ \mathfrak{p} & := 2\mathfrak{q}+\sqrt{2 (\mathfrak{h}+c+\varepsilon)}. \end{align*} $$

Now, using (3.16) together with Lemma 3.9, we obtain

(3.18) $$ \begin{align} |q(s,t)| & \leq |q(\tau_\varepsilon(s),t)|+\int_{\tau_\varepsilon(s)}^s |\partial_s q(\tau,t)|\,d\tau,\nonumber\\ & \leq |q(\tau_\varepsilon(s),t)|+\int_{\tau_\varepsilon(s)}^s |\partial_s v(\tau,t)|\,d\tau,\nonumber\\ \|q(s)\|_{L^2}& \leq \|q \circ \tau_0(s)\|_{L^2}+\bigg( \int_0^1\bigg| \int_{\tau_0(s)}^s\partial_sv\, d\tau\bigg|^2\,dt\bigg)^{\frac{1}{2}}\nonumber\\ & \leq \|q \circ \tau_0(s)\|_{L^2}+\sqrt{|s-\tau_0(s)|}\bigg( \int_0^1\int_{\tau_0(s)}^s|\partial_sv|^2\,d\tau \,dt\bigg)^{\frac{1}{2}}\nonumber\\ & \leq \|v \circ \tau_0(s)\|_{L^2}+\frac{\sqrt{\mathfrak{e}}\|J\|_\infty}{\varepsilon}\ \|\nabla^{J_s}\mathcal{A}^{H_0-h_s}\|_{L^2(\mathbb{R}\times[0,1])}\nonumber\\ & \leq \mathfrak{q}+\frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}. \end{align} $$

Analogously, using Lemma 3.10, we obtain

(3.19) $$ \begin{align} \|p(s)\|_{L^2}& \leq \mathfrak{p} +\frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}, \end{align} $$

(3.20) $$ \begin{align} \begin{aligned} \|v(s)\|_{L^2}&\leq\mathfrak{q}+\mathfrak{p}+\frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}. \end{aligned} \end{align} $$

Proof of Theorem 3.3

Let $\Gamma :={(h_s,J_s)}_{s\in \mathbb {R}}$ be a smooth homotopy of Hamiltonians ${h_s\in \mathcal {H}}$ and almost complex structures $J_s\in C^\infty ([0,1]\times \mathbb {R}, \mathcal {J}(\mathcal {V},\mathbb {J}))$ satisfying (3.2). Here, $\mathcal {V}\subseteq T^*\mathbb {R}^2$ is the open, but pre-compact subset defined in Theorem 3.3 with the property that for all $s\in [0,1]$ , $\operatorname {supp} dh_s\subseteq \mathcal {V}$ and $J_s|_{T^*\mathbb {R}^2\setminus \mathcal {J}}\equiv \mathbb {J}$ .

Let $c>0$ be as in (3.3). If we fix $a,b \in \mathbb {R}$ , then by Lemma 3.5 we know that there exists $\mathfrak {y,a,e}>0$ , which depend only on $a,b,c$ and $\|J\|_\infty $ such that

(3.23) $$ \begin{align} \begin{aligned} &\qquad\quad\!\sup\lbrace|\eta(s)|\ |\ (v,\eta)\in \mathcal{M}^\Gamma(a,b), \ s\in \mathbb{R}\rbrace \leq \mathfrak{y},\\ &\qquad\quad\ \kern3pt\sup\lbrace|\mathcal{A}^{H_0-h_s-c}(u)|\ |\ u\in \mathcal{M}^\Gamma(a,b)\rbrace \leq \mathfrak{a},\\ &\sup\lbrace\|\nabla^{J_s}\mathcal{A}^{H_0-h_s-c}(u)\|_{L^2(\mathbb{R}\times[0,1]) \times\mathbb{R}}\ |\ u\in \mathcal{M}^\Gamma(a,b)\rbrace \leq \mathfrak{e}. \end{aligned} \end{align} $$

However, if we fix $\varepsilon>0$ , then by Proposition 3.7 and Remark 3.12, we know that the set $\mathcal {B}^\Gamma (\mathfrak {a},\mathfrak {y},\varepsilon )$ , defined in (3.17), is bounded in the $L^\infty \times \mathbb {R}$ norm and the bounds depend only on $a,b,\varepsilon $ and $\Gamma $ . Consequently, there exists a compact subset $K_\varepsilon \subseteq T^*\mathbb {R}^2$ such that

(3.24) $$ \begin{align} v([0,1])\subseteq K_\varepsilon\quad\textrm{for all}\; (v,\eta)\in \mathcal{B}^\Gamma(\mathfrak{a},\mathfrak{y},\varepsilon). \end{align} $$

Without loss of generality, we assume that $\overline {\mathcal {V}} \subseteq K_\varepsilon $ .

Let us fix a Floer trajectory $u=(v,\eta )\in \mathcal {M}^\Gamma (a,b)$ and denote a connected component

(3.25) $$ \begin{align} \Omega \subseteq (\mathbb{R} \times [0,1])\setminus v^{-1}(K_\varepsilon). \end{align} $$

By assumption, $\operatorname {supp} dh_s \subseteq \overline {V} \subseteq K_\varepsilon $ ; hence, $dh_s \circ v |_{\Omega }\equiv 0$ for all $s\in [0,1]$ . This means that instead of $H-h_s$ , we can use the Hamiltonian $H_0-\mathbf {c}_s$ for some positive function $\mathbf {c}_s>0$ on $v(\Omega )$ , which will simplify our computations. More precisely, for $(s,t)\in \Omega $ , the Floer trajectory $(v,\eta )$ satisfies

(3.26) $$ \begin{align} \partial_sv = \mathbb{J}(\partial_t v - \eta X_{H_0}). \end{align} $$

Naturally, $\mathfrak {c}_s\leq \sup _{s\in [0,1]}\|h_s\|_{L^\infty }$ .

By the convergence of the integral

$$ \begin{align*}\|\nabla\mathcal{A}^{H_0-h_s}(u)\|_{L^2(\mathbb{R}\times[0,1])\times\mathbb{R}} \leq \|J\|_{L^\infty}\sqrt{\mathfrak{e}},\end{align*} $$

we know that $\lim _{s\to \pm \infty }u(s)\in \mathcal {B}^\Gamma (\mathfrak {a},\mathfrak {y},\varepsilon )$ . Therefore,

$$ \begin{align*}v(\partial\Omega)\subseteq K_\varepsilon\cup T^*_{q_0}\mathbb{R}^2\cup T^*_{q_1}\mathbb{R}^2.\end{align*} $$

Consequently, we have $\sup _{\partial \Omega }|q(s,t)|\leq \sup _{K_\varepsilon }|q|$ . Unfortunately, we do not have a uniform bound on $\sup _{\partial \Omega }|p|$ . Therefore, we will need to treat the cases of the position and momentum coordinates separately. We introduce the following functions on $T^*\mathbb {R}^2$ :

(3.27) $$ \begin{align} F_1:= \tfrac{1}{2}|q|^2\quad \textrm{and}\quad F_2:=\tfrac{1}{2}|p|^2. \end{align} $$

Both of the functions $F_1$ and $F_2$ are plurisubharmonic, since their corresponding gradients $q\partial _q$ and $p\partial _p$ are Liouville vector fields (see Remark 3.16).

In Lemma 3.18, we will show that on $\Omega $ , the functions $F_1\circ v$ and $F_2\circ v$ satisfy

$$ \begin{align*} \triangle (F_1 \circ v) &\geq -\tfrac{1}{2}\eta^2 F_1\circ v+ \partial_{s}\eta (F_2\circ v -H_0\circ v)=:f_1(s,t),\\ \triangle (F_2 \circ v)&\geq -\tfrac{1}{2}\eta^2F_1\circ v-\partial_s\eta(F_2\circ v+H_0\circ v)=:f_2(s,t). \end{align*} $$

In Lemmas 3.22 and 3.24, we will prove that $f_1$ and $f_2$ , respectively, are bounded in the $W^{1,1}$ -norm on $\Omega $ and the bounds depend only on $a,b,c,\varepsilon $ and $\Gamma $ , but not on our choice of the Floer trajectory $u\in \mathcal {M}^\Gamma (a,b)$ or our choice of the connected component $\Omega \subseteq (\mathbb {R}\times [0,1] )\setminus v^{-1}(K_\varepsilon )$ . By the continuity of the Sobolev embedding $W^{1,1}\hookrightarrow L^2$ , we obtain $L^2$ -bounds on $f_1$ and $f_2$ .

Since we know that $\sup _{\partial \Omega }|q(s,t)|\leq \sup _{K_\varepsilon }|q|$ , we can now apply the Alexandrov maximum principle to obtain

$$ \begin{align*} \sup_{(s,t)\in\mathbb{R}\times[0,1]}F_1\circ v \leq \sup_{K_\varepsilon}f_1\circ v+C\|f_1\|_{L^2(\Omega)}, \end{align*} $$

and the right-hand side of the inequality does not depend on our choice of the Floer trajectory $u\in \mathcal {M}^\Gamma (a,b)$ or our choice of the connected component $\Omega \subseteq (\mathbb {R}\times [0,1] )\setminus v^{-1}(K_\varepsilon )$ . In other words, we obtain that the set

$$ \begin{align*} \{\pi \circ v \ |\ u=(v,\eta)\in\mathcal{M}^\Gamma(a,b)\} \end{align*} $$

is bounded in $\mathbb {R}^2$ .

Now, we would like to establish the $L^\infty $ -bounds on the p-variable of the Floer trajectories. To establish uniform bounds on $F_2 \circ v$ for the Floer trajectory $u=(v,\eta )$ , we extend the domain of the Floer trajectory $v:\mathbb {R}\times [0,1]\to T^*\mathbb {R}^2$ to a cylinder ${\mathbb {R}\times ([-1,1]/-1\sim 1)}$ in the following way:

(3.28) $$ \begin{align} \bar{v}(s,t):=\begin{cases} v(s,-t) & \textrm{for } t\leq 0,\\ v(s,t) & \textrm{for } t\geq 0. \end{cases} \end{align} $$

If we extend the domain $\Omega $ to the cylinder $\mathbb {R}\times ([-1,1]/-1\sim 1)$ in the following way:

(3.29) $$ \begin{align} \Theta:=\Omega\cup\{(s,-t)\ |\ (s,t)\in \Omega\}\quad \textrm{then}\quad \bar{v}(\partial\Theta)\subseteq K_\varepsilon\quad\textrm{and}\quad v(\Omega) =\bar{v}(\Theta). \end{align} $$

In particular, we have that $\sup _{\Omega }f\circ v=\sup _\Theta f\circ \bar {v}$ and $\|f\circ \bar {v}\|_{L^2(\Theta )}=2\|f\circ v\|_{L^2(\Omega )}$ for any smooth function $f\in T^*\mathbb {R}^2$ .

After proving in Lemma 3.19 that $F_2\circ \bar {v}$ is in $C^2(\Theta )$ , we can apply the Aleksandrov maximum principle and obtain

$$ \begin{align*} \sup_{(s,t)\in\mathbb{R}\times[0,1]}F_2\circ v & \leq \sup_{\partial\Theta}F_2\circ\bar{v}+C\|f_2\circ \bar{v}\|_{L^2(\Theta)}\\ & \leq \sup_{K_\varepsilon}f_2\circ v+2C\|f_2\circ v\|_{L^2(\Omega)}. \end{align*} $$

Since the established bounds do not depend on the choice of $u\in \mathcal {M}^\Gamma (a,b)$ or the choice of the connected component $\Omega \subseteq (\mathbb {R}\times [0,1] )\setminus v^{-1}(K_\varepsilon )$ , this concludes the proof that $\mathcal {M}^\Gamma (a,b)$ is bounded in the $L^\infty \times \mathbb {R}$ -norm.

Proof. Using (1.1), we can calculate that

(3.30) $$\begin{align}X_{H_0} & = (p_1+q_2)\partial_{q_1}+(p_2-q_1)\partial_{q_2}+p_2\partial_{p_1}-p_1\partial_{p_2},\nonumber\\ dF_1 (X_{H_0}) & = q_1p_1+q_2p_2,\nonumber\\ d^{\mathbb{C}}(dF_1(X_{H_0})) & = qd^{\mathbb{C}}p+pd^{\mathbb{C}}q=pdp-qdq=dF_2-dF_1,\nonumber\\dF_2 (X_{H_0}) & = 0,\end{align} $$

(3.31) $$ \begin{align} d^{\mathbb{C}} F_1 (X_{H_0}) & = qdp(X_{H_0})=q_1p_2-q_2p_1 = F_2-H_0 ,\qquad \end{align} $$

(3.32) $$ \begin{align} d^{\mathbb{C}} F_2 (X_{H_0}) & =-(|p|^2 +q_1p_2-q_2p_1) = -F_2-H_0,\qquad\ \ \qquad\\d(dF_1 (X_{H_0}))(X_{H_0}) & = |p|^2=2F_2,\nonumber\\ (dH_0+d(d^{\mathbb{C}} F_1 (X_{H_0}))&+d^{\mathbb{C}} (dF_1(X_{H_0}))) = 2dF_2-dF_1,\nonumber\\ (dH_0+d(d^{\mathbb{C}} F_2 (X_{H_0}))&+d^{\mathbb{C}} (dF_2(X_{H_0}))) = -dF_1.\nonumber \end{align} $$

Plugging these qualities into (3.22) for a Floer trajectory $u=(v,\eta )$ satisfying the relation (3.26), we can calculate

$$ \begin{align*} \triangle (F_1 \circ v) & = \|\partial_{s}v\|^{2}+\eta (dH_0 +d(d^{\mathbb{C}}F_1(X_{H_0}))+d^{\mathbb{C}}(dF_1(X_{H_0})))(\partial_{s}v)\\ & \quad +\eta^2 d(dF_1(X_{H_0}))(X_{H_0})+\partial_{s}\eta\ d^{\mathbb{C}}F_1(X_{H_0})\\ & = \|\partial_{s}v\|^{2}+\eta (2pdp-qdq)(\partial_{s}v)+\eta^2|p|^2+ \partial_{s}\eta (\tfrac{1}{2}|p|^2-H_0\circ v )\\ & \geq -\eta^2 \tfrac{1}{4}|q|^2+ \partial_{s}\eta (\tfrac{1}{2}|p|^2-H_0 )\\ & = -\tfrac{1}{2}F_1\circ v+ \partial_{s}\eta (F_2\circ v-H_0\circ v ),\\ \triangle (F_2 \circ v) & = \|\partial_{s}v\|^{2}+\eta (dH_0 +d(d^{\mathbb{C}}F_2(X_{H_0}))+d^{\mathbb{C}}(dF_2(X_{H_0})))(\partial_{s}v)\\ &\quad +\eta^2 d(dF_2(X_{H_0}))(X_{H_0})+\partial_{s}\eta\ d^{\mathbb{C}}F_2(X_{H_0})\\ & = \|\partial_{s}v\|^{2}-\eta qdq(\partial_sv)-\partial_s\eta(\tfrac{1}{2}|p|^2+H_0)\\ & \geq -\tfrac{1}{4}\eta^2|q|^2-\partial_s\eta(\tfrac{1}{2}|p|^2+H_0)\\ & = -\tfrac{1}{2}\eta^2F_1\circ v-\partial_s\eta(F_2\circ v+H_0\circ v).\\[-38pt] \end{align*} $$

Proof. First, observe that $\bar {v}$ is continuous. Moreover, $\bar {v}$ is smooth on $\mathbb {R}\times ((-1,0)\cup (0,1))$ . Therefore, $F_2\circ \bar {v}$ is also everywhere continuous and smooth on $\mathbb {R}\times ((-1,0)\cup (0,1))$ . What is left to check is that $F_2\circ \bar {v}$ is $C^2$ on $\mathbb {R}\times \{-1,0,1\}$ . Note that

$$ \begin{align*} \frac{d}{dt} (F_2\circ \bar{v})(s,t)=\begin{cases} dF_2( \partial_tv)(s,t) & \textrm{for } t\geq 0,\\ -dF_2(\partial_tv)(s,-t) & \textrm{for } t\leq 0. \end{cases} \end{align*} $$

Hence, the function $F_2\circ \bar {v}$ is $C^1$ if and only if

(3.33) $$ \begin{align} dF_2( \partial_tv)(s,0)=dF_2( \partial_tv)(s,1)=0\quad \text{ for all } s\in \mathbb{R}. \end{align} $$

By assumption, $v: \mathbb {R} \to \mathscr {H}_{q_0,q_1}$ , so $v(s,0)\equiv q_0$ and $v(s,1)\equiv q_1$ for all $s\in \mathbb {R}$ . Consequently,

(3.34) $$ \begin{align} dq_i(\partial_sv)(s,0)=dq_i(\partial_sv)(s,1)=0\quad \textrm{for } i=0,1 \ \textrm{and} \text{ for all } s\in \mathbb{R}. \end{align} $$

Now, using the Floer equation (3.26) for the Hamiltonian $H_0$ and the relation (3.30), we can calculate for $t\geq 0$ ,

$$ \begin{align*} dF_2( \partial_tv) & = d^{\mathbb{C}}F_2(\partial_sv)+\eta dF_2(X_{H_0})=-pdq(\partial_sv). \end{align*} $$

Plugging (3.34) into the equation above, we obtain (3.33), which proves that $F_2\circ \bar {v}$ is $C^1$ everywhere.

To prove that $F_2\circ \bar {v}$ is $C^2$ , we first calculate:

$$ \begin{align*} \frac{d^2}{dt^2}F_2\circ v & = D^2F_2(\partial_t v,\partial_t v)+dF_2(\partial_{tt}v)= \eta^2 D^2F_2(X_{H_0},X_{H_0})+dF_2(\partial_{tt}v)\\ &=\eta^2|p|^2+dF_2(\partial_{tt}v). \end{align*} $$

Obviously, $|p|^2$ does not depend on the sign of $\partial _tv$ . However, the function $\partial _{tt}\bar {v}= \partial _{tt}v$ is continuous, since the two negative signs cancel each other. Therefore, the sum of these two functions is also continuous on the whole cylinder $\mathbb {R}\times ([-1,1]/-1\sim 1)$ .

Now, $({d^2}/{dsdt})F_2\circ \bar {v}$ is well defined and continuous on the whole $\mathbb {R}\times ((-1,0)\cup (0,1))$ . However, by (3.33), we obtain that $({d^2}/{dsdt})F_2\circ v (s,0)= ({d^2}/{dsdt})F_2\circ v (s,1)=0$ ; thus, $({d^2}/{dsdt})F_2\circ \bar {v}$ is a continuous function on the whole cylinder $\mathbb {R}\times ([-1,1]/-1\sim 1)$ .

Finally, we observe that $({d^2}/{ds^2})(F_2\circ v (s,t))=({d^2}/{ds^2})(F_2\circ v (s,-t))$ as it does not depend on the sign of the t-variable.

Proof. By Lemma 3.5, we obtain uniform bounds on the action, energy and the $\eta $ -parameter for all $u\in \mathcal {M}^\Gamma (a,b)$ . By the convergence of the integral

$$ \begin{align*}\|\nabla\mathcal{A}^{H_0-h_s}(u)\|_{L^2(\mathbb{R}\times[0,1])\times\mathbb{R}} \leq \|J\|_{L^\infty}\sqrt{\mathfrak{e}},\end{align*} $$

we know that $\lim _{s\to \pm \infty }u(s)\in \mathcal {B}^\Gamma (\mathfrak {a},\mathfrak {y},\varepsilon )$ . For the compact subset $K_\varepsilon \subseteq T^*\mathbb {R}^2$ defined in (3.24), we have $v([0,1])\subseteq K_\varepsilon $ for all $(v,\eta )\in \mathcal {M}^\Gamma (a,b)$ . Since $\Omega \subseteq (\mathbb {R} \times [0,1])\setminus v^{-1}(K_\varepsilon )$ , the values $s_0$ and $s_1$ as in (3.35) are well defined and finite. Moreover, for all $s\in (s_0,s_1)$ , $u(s)\notin \mathcal {B}^\Gamma (\mathfrak {a},\mathfrak {y}, \varepsilon )$ ; therefore, by (3.16), we can estimate

$$ \begin{align*} |s_1-s_0|\leq \frac{\mathfrak{e}\|J\|^2_\infty}{\varepsilon^2}, \end{align*} $$

where $\mathfrak {e}=\|J\|_\infty (b-a+{c}/{3}\mathfrak {y})$ as in Lemma 3.5.

Proof. Let $s_0$ and $s_1$ be as in (3.35). Denote $\overline {\Omega }:= [s_0,s_1]\times [0,1]$ . Naturally, $\Omega \subseteq \overline {\Omega }$ . Before we estimate $\|f_1\|_{L^1(\Omega )}$ , we first use the Cauchy–Schwartz inequality together with (3.12), (3.18), (3.19) and (3.36) to do the following estimates:

(3.37) $$ \begin{align} \|\partial_s\eta\|_{L^1(\Omega)}& \leq \|\partial_s\eta\|_{L^1(\overline{\Omega})}\leq \sqrt{|s_1-s_0|}\|\nabla^{J_s} \mathcal{A}^{H_0-h_s}(u(s))\|_{L^2\times\mathbb{R}}\nonumber\\&\leq \frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}, \end{align} $$

(3.38) $$ \begin{align} \|\partial_s\eta(H_0-\mathbf{c}_s)\|_{L^1(\Omega)}&\leq \|\partial_s\eta (H_0-h_s)\circ v\|_{L^1(\overline{\Omega})}=\|\partial_s\eta\|_{L^2(\overline{\Omega})}^2\nonumber\\& \leq \|\nabla^{J_s} \mathcal{A}^{H_0-h_s}(u(s))\|_{L^2\times\mathbb{R}}^2\leq \mathfrak{e},\end{align} $$

(3.39) $$ \begin{align} \|q\|_{L^2(\Omega)} & \leq \|q \|_{L^2(\overline{\Omega})}\leq \sqrt{|s_1-s_0|} \sup_{s\in\mathbb{R}}\|q(s)\|_{L^2([0,1])}\nonumber\\& \leq \frac{\sqrt{\mathfrak{e}}\|J\|_\infty}{\varepsilon} \bigg(\mathfrak{q}+\frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg), \end{align} $$

(3.40) $$ \begin{align} \|p\|_{L^2(\Omega)} & \leq \frac{\sqrt{\mathfrak{e}}\|J\|_\infty}{\varepsilon} \bigg(\mathfrak{p}+\frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg),\end{align} $$

(3.41) $$ \begin{align} \|\partial_s\eta\ p^2\|_{L^1(\Omega)} & \leq \|\partial_s\eta\|_{L^1(\overline{\Omega})} \sup_{s\in\mathbb{R}}\|p(s)\|_{L^2([0,1])}^2 \leq \frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg(\mathfrak{p} + \frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg)^2. \end{align} $$

Using the bounds obtained above and the inequality (3.10), we can calculate

$$ \begin{align*} \begin{aligned} \|f_1\|_{L^1(\Omega)}& \leq \frac{1}{4}\mathfrak{y}^2 \|q\|_{L^2(\Omega)}^2+\frac{1}{2}\|\partial_s\eta\ p^2\|_{L^1(\Omega)}+\|\partial_s\eta(H_0-\mathbf{c}_s)\|_{L^1(\Omega)}+\|\mathbf{c}_s\partial_s\eta\|_{L^1(\Omega)}\\ &\leq \frac{\mathfrak{e}\mathfrak{y}^2\|J\|_\infty^2}{4\varepsilon^2} \bigg(\mathfrak{q}+\frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg)^2+ \frac{\mathfrak{e}\|J\|_\infty^2}{2\varepsilon^2}\bigg( \mathfrak{p}+ \frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg)^2+ \mathfrak{e}\bigg(\frac{\mathfrak{h}\|J\|_\infty}{\varepsilon}+1\bigg). \end{aligned} \end{align*} $$

Proof. To prove that $f_1\in W^{1,1}(\Omega )$ , we first need to calculate its derivatives:

$$ \begin{align*} \partial_sf_1 & = - \eta\partial_s\eta\ F_1+\partial_{ss}\eta\ d^{\mathbb{C}}F_1(X_{H_0}) +(\partial_s\eta\ d(F_2-H_0)-\tfrac{1}{2}\eta^2 dF_1)(\partial_sv)\\ \partial_tf_1 & = (\partial_s\eta\ d(F_2-H_0)-\tfrac{1}{2}\eta^2 dF_1)(\mathbb{J}\partial_sv+\eta X_{H_0})\\ & = (\partial_s\eta\ d^{\mathbb{C}}(F_2-H_0)-\tfrac{1}{2}\eta^2 d^{\mathbb{C}} F_1)(\partial_sv)-\tfrac{1}{2}\eta^3dF_1(X_{H_0}), \end{align*} $$

where the last equality comes from the fact that $dF_2(X_{H_0})(q,p)=0$ . Using (3.18), (3.20) and (3.36), we can estimate

(3.42) $$ \begin{align} \|dF_1(X_{H_0})\|_{L^1(\Omega)}& = \|q_1p_1+q_2p_2\|_{L^1(\Omega)}\leq\frac{1}{2}\| v\|_{L^2(\Omega)}^2\leq \frac{1}{2}|s_1-s_0|\sup_{s\in\mathbb{R}}\|v(s)\|_{L^2([0,1])}^2\nonumber\\[5pt] & \leq \frac{\mathfrak{e}\|J\|_\infty^2}{2\varepsilon^2}\bigg( \mathfrak{q}+ \mathfrak{p}+\frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg)^2, \end{align} $$

(3.43) $$ \begin{align} \|dF_1(\partial_sv)\|_{L^1(\Omega)}& \leq \|\nabla F_1\|_{L^2(\Omega)}\|\partial_sv\|_{L^1(\Omega)}\leq \|q\|_{L^2(\overline{\Omega})} \|\nabla^{J_s}\mathcal{A}^{H_0-h_s}(u)\|_{L^2\times\mathbb{R}}\nonumber\\[5pt]& \leq \frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg(\mathfrak{q}+ \frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg), \end{align} $$

(3.44) $$ \begin{align} \|&\partial_s\eta\ d(F_2 -H_0)(\partial_sv)\|_{L^1(\Omega)} \leq \int_{s_0}^{s_1}|\partial_s\eta|\int_0^1|\nabla (F_2-H_0)||\partial_sv|\,dt\,ds\nonumber\\ &= \int_{s_0}^{s_1}|\partial_s\eta|\int_0^1|v||\partial_sv|\,dt\,ds \leq \sup_{s\in\mathbb{R}} \|v(s)\|_{L^2([0,1])}\int_{s_0}^{s_1}|\partial_s\eta|\|\partial_sv(s)\|_{L^2([0,1])}\,ds\nonumber\\ & \leq\sup_{s\in\mathbb{R}}\|v(s)\|_{L^2([0,1])}\|\nabla^{J_s}\mathcal{A}^{H_0-h_s}(u) \|_{L^2\times\mathbb{R}}^2 \leq \mathfrak{e}\bigg( \mathfrak{q}+\mathfrak{p}+\frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg). \end{align} $$

Similarly, we have

(3.45) $$ \begin{align} &\|d^{\mathbb{C}} F_1(\partial_sv)\|_{L^1(\Omega)}\leq \frac{\mathfrak{e}\|J\|_\infty}{\varepsilon} \bigg(\mathfrak{q}+ \frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg), \end{align} $$

(3.46) $$ \begin{align} \|\partial_s\eta\ d^{\mathbb{C}}(F_2-H_0)(\partial_sv)\|_{L^1(\Omega)} &\leq \mathfrak{e}\bigg(\mathfrak{q}+\mathfrak{p}+\frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg). \end{align} $$

Using the estimates above and (3.10), we can combine them to obtain a uniform $L^1$ -bound on $\partial _tf_1$ :

$$ \begin{align*} \|\partial_tf_1\|_{L^1(\Omega)} &\leq \|\partial_s\eta d^{\mathbb{C}}( F_2-H_0)(\partial_sv)\|_{L^1(\Omega)}\\ & \quad +\frac{1}{2}\|\eta^2 d^{\mathbb{C}} F_1(\partial_sv)\|_{L^1(\Omega)}+\frac{1}{2}\|\eta^3 dF_1(X_{H_0})\|_{L^1(\Omega)}\\ & \leq \mathfrak{e}\mathfrak{p}+ \mathfrak{e}\bigg(1+\frac{\mathfrak{ey}^2\|J\|_\infty}{2\varepsilon} \bigg)\bigg(\mathfrak{q}+\frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg)+ \frac{\mathfrak{ey}^3\|J\|_\infty^2}{4\varepsilon^2}\bigg( \mathfrak{q}+\mathfrak{p}+\frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg)^2. \end{align*} $$

This gives us a bound on $\|\partial _tf_1\|_{L^1(\Omega )}$ , which does not depend on the choice of the Floer trajectory u or the connected component $\Omega $ .

To estimate $\|\partial _sf_1\|_{L^1(\Omega )}$ , we first use (3.18) and (3.37) to estimate

(3.47) $$ \begin{align} \|\partial_s\eta\ F_1\circ v\|_{L^1(\Omega)} & \leq \|\partial_s\eta\ F_1\circ v\|_{L^1(\overline{\Omega})} = \frac{1}{2}\int_{s_0}^{s_1}|\partial_s\eta|\|q(s)\|_{L^2([0,1])}^2\,ds\nonumber\\ & \leq \frac{1}{2}\|\partial_s\eta\|_{L^1(\overline{\Omega})} \sup_{s\in\mathbb{R}}\|q(s)\|_{L^2([0,1])}^2\nonumber\\ &\leq \frac{\mathfrak{e}\|J\|_\infty}{2\varepsilon} \bigg(\mathfrak{q}+\frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg)^2. \end{align} $$

Second, we analyse $\partial _{ss}\eta $ :

$$ \begin{align*} \partial_{ss}\eta = -\int_0^1d(H_0-h_s)(\partial_sv)\,dt-\int_0^1 \partial_sh_s (v)\,dt. \end{align*} $$

Using (3.2), (3.19) and (3.20), we can estimate

(3.48) $$ \begin{align} \|\partial_{ss}&\eta\|_{L^1(\Omega)} \nonumber\\ & = \int_{\Omega}\bigg|\int_0^1(d(H_0-h_s)(\partial_sv)+\partial_sh_s(v))\,dt\bigg|\,dt \,ds\nonumber\\ &\leq\|d(H_0-h_s)(\partial_sv)\|_{L^1(\overline{\Omega})}+\|\partial_sh_s\|_{L^\infty}\nonumber\\ &\leq (\|\nabla H_0\|_{L^2(\overline{\Omega})}+\|\nabla h_s\|_{L^2(\overline{\Omega})}) \|\partial_sv\|_{L^2(\overline{\Omega})}\|\partial_sh_s\|_{L^\infty}\nonumber\\ & \leq (\|v+p\|_{L^2(\overline{\Omega})}+\|\nabla h_s\|_{L^2(\overline{\Omega})}) \|\nabla^{J_s}\mathcal{A}^{H_0-h_s}(u)\|_{L^2\times\mathbb{R}}+\|\partial_sh_s\|_{L^\infty}\nonumber\\ &\leq \sqrt{\mathfrak{e}|s_1-s_0|}\Big(\sup_{s\in\mathbb{R}}\|v(s)\|_{L^2([0,1])}+ \sup_{s\in\mathbb{R}}\|p(s)\|_{L^2([0,1])}+\sup_{s\in\mathbb{R}}\|\nabla h_s\|_{L^\infty}\Big)+\|\partial_sh_s\|_{L^\infty}\nonumber\\ &\leq \frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg(\mathfrak{h}+ \mathfrak{q}+ 2\mathfrak{p}+\frac{2\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg)+\frac{c}{3}. \end{align} $$

Now, we will use the Floer equations, the Cauchy–Schwartz inequality together with (3.12), (3.16), (3.18), (3.19) and (3.31) to calculate the following:

(3.49) $$ \begin{align} \|\partial_{ss}\eta\ d^{\mathbb{C}}F_1(X_{H_0})\|_{L^1(\Omega)} & \leq \|\partial_{ss}\eta\ d^{\mathbb{C}}F_1(X_{H_0})\|_{L^1(\overline{\Omega})} =\int_{s_0}^{s_1}|\partial_{ss}\eta|\int_0^1|q_1p_2-q_2p_1|\,dt\,ds\nonumber\\ & \leq \frac{1}{2}\int_{s_0}^{s_1}|\partial_{ss}\eta|(\|q(s)\|_{L^2([0,1])}^2 +\|p(s)\|_{L^2([0,1])}^2)\,ds\nonumber\\ &\leq \frac{1}{2}\|\partial_{ss}\eta\|_{L^1(\overline{\Omega})} \sup_{s\in\mathbb{R}}\|v(s)\|_{L^2([0,1])}^2\nonumber\\ & \leq \bigg(\frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg( \frac{1}{2}(\mathfrak{h}+ \mathfrak{q})+\mathfrak{p}+\frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg)+\frac{c}{3}\bigg) \bigg(\mathfrak{q}+\mathfrak{p}+\frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg)^2. \end{align} $$

Using the estimates above together with (3.10), (3.43), (3.44) and (3.47), we can estimate

$$ \begin{align*} \|\partial_sf_1\|_{L^1(\Omega)} & \leq \mathfrak{y}\|\partial_s\eta\ F_1\circ v\|_{L^1(\Omega)}+\|\partial_{ss}\eta d^{\mathbb{C}}F_1(X_{H_0})\|_{L^1(\Omega)}+\frac{1}{2}\mathfrak{y}^2\|d F_1(\partial_sv)\|_{L^1(\Omega)}\\ &\quad +\|\partial_s\eta\ d(F_2-H)(\partial_sv)\|_{L^1(\Omega)} \\ & \leq \frac{\mathfrak{e y}\|J\|_\infty}{2\varepsilon}\bigg(\mathfrak{q}+ \frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg)^2 + \frac{\mathfrak{e y}^2\|J\|_\infty}{2\varepsilon}\bigg(\mathfrak{q}+\frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg)\\ & + \bigg(\frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg( \frac{1}{2}(\mathfrak{h}+\mathfrak{q})+\mathfrak{p}+\frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg)+ \frac{c}{3}\bigg)\bigg(\mathfrak{q}+\mathfrak{p}+\frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg)^2\\ &+\mathfrak{e}\bigg(\mathfrak{q}+\mathfrak{p}+\frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg). \end{align*} $$

This gives us a bound on $\|\partial _sf_1\|_{L^1(\Omega )}$ , which does not depend on the choice of the Floer trajectory u or the connected component $\Omega $ .

Proof. To estimate $\|f_2\|_{L^1(\Omega )}$ , we use the bounds obtained in (3.10), (3.37), (3.38), (3.39) and (3.41) to calculate

$$ \begin{align*} \begin{aligned} \|f_2\|_{L^1(\Omega)} & \leq \frac{\mathfrak{y}^2}{4}\|q\|^2_{L^2(\Omega)} + \frac{1}{2}\|\partial_s\eta\ p^2\|_{L^2(\Omega)}+\|\partial_s\eta (H_0-\mathfrak{c}_s)\|_{L^2(\Omega)}+\|\mathfrak{c}_s\partial_s\eta\|_{L^1(\Omega)} \\ &\leq\frac{\mathfrak{ey}^2\|J\|_\infty^2}{4\varepsilon^2}\bigg(\mathfrak{q}+ \frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg)^2+ \frac{\mathfrak{e}}{2\varepsilon} \bigg(\mathfrak{p}+\frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg)^2+\mathfrak{e} \bigg(\frac{c}{\varepsilon}+\|J\|_\infty^2\bigg). \end{aligned} \end{align*} $$

Proof. To prove that $f_2\in W^{1,1}(\Omega )$ , we first need to calculate its derivatives:

$$ \begin{align*} \partial_sf_2 & = - \eta\partial_s\eta F_1 + \partial_{ss}\eta\ d^{\mathbb{C}}F_2(X_{H_0})-(\partial_s\eta\ d(F_2+H_0)+\tfrac{1}{2}\eta^2 dF_1)(\partial_sv)\\ \partial_tf_2 & = -(\partial_s\eta\ d(F_2+H_0)+\tfrac{1}{2}\eta^2 dF_1)(\mathbb{J}\partial_sv+\eta X_{H_0})\\ & = -(\partial_s\eta\ d^{\mathbb{C}}(F_2+H_0)+\tfrac{1}{2}\eta^2 d^{\mathbb{C}} F_1)(\partial_sv)-\tfrac{1}{2}\eta^3 dF_1(X_{H_0}), \end{align*} $$

where the last equality comes from (3.30).

Let $s_0$ and $s_1$ be as in (3.35). Denote $\overline {\Omega }:= [s_0,s_1]\times [0,1]$ . Naturally, $\Omega \subseteq \overline {\Omega }$ . Before we estimate $\|\partial _t f_2\|_{L^1(\Omega )}$ , we will first recall (3.31) and (3.32), which give us the following relation:

(3.50) $$ \begin{align} d^{\mathbb{C}}F_1(X_{H_0})=F_2-H_0=2F_2+d^{\mathbb{C}}F_2(X_{H_0}). \end{align} $$

Further on, we use (3.12), (3.19), (3.40) and (3.44) to calculate the following bounds:

$$ \begin{align*} \|\partial_s\eta\ dF_2(\partial_sv)\|_{L^1(\Omega)}& \leq \|\partial_s\eta\ dF_2(\partial_sv)\|_{L^1(\overline{\Omega})}\\ & \leq \int_{s_0}^{s_1}|\partial_s\eta|\int_0^1|p(s,t)||\partial_sv(p,s)|\,dt\,ds\\ &\leq \int_{s_0}^{s_1}|\partial_s\eta|\|p(s)\|_{L^2([0,1])}\|\partial_s v(s)\||_{L^2([0,1])}ds\\ &\leq \sup_{s\in \mathbb{R}}\|p(s)\|_{L^2([0,1])}\|\partial_s\eta\|_{L^2(\mathbb{R})}\| \partial_sv\|_{L^2(\mathbb{R}\times[0,1])}\\ &\leq \mathfrak{e}\bigg(\mathfrak{p}+\frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg). \end{align*} $$

This, together with (3.50), gives us

(3.51) $$ \begin{align} \|\partial_s\eta\ d(F_2+H_0)(\partial_sv)\|_{L^1(\Omega)} & \leq 2\|\partial_s\eta\ dF_2(\partial_sv)\|_{L^1(\Omega)}+\|\partial_s\eta\ d(F_2-H_0)(\partial_sv)\|_{L^1(\Omega)}\nonumber\\ &\leq \mathfrak{e}\bigg( \mathfrak{q}+3\mathfrak{p}+2\frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg). \end{align} $$

Similarly, we have

$$ \begin{align*} \|\partial_s\eta\ d^{\mathbb{C}}(F_2+H_0)(\partial_sv)\|_{L^1(\Omega)}\leq \mathfrak{e}\bigg( \mathfrak{q}+3\mathfrak{p}+2\frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg). \end{align*} $$

Using the bounds obtained above together with (3.10), (3.42) and (3.45), we can estimate

$$ \begin{align*} \|\partial_tf_2\|_{L^1(\Omega)} & \leq \|\partial_s\eta\ d^{\mathbb{C}}(F_2+H_0)(\partial_s v)\|_{L^1(\Omega)}\\ &\quad +\frac{1}{2}\mathfrak{y}^2 (\|d^{\mathbb{C}} F_1(\partial_sv)\|_{L^1(\Omega)}+\mathfrak{y} \|dF_1(X_{H_0})\|_{L^1(\Omega)})\\ &\leq \mathfrak{e}\bigg( \mathfrak{q}+3\mathfrak{p}+ 2\frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg)+\frac{\mathfrak{ey}^2\|J\|_\infty}{2\varepsilon} \bigg(\mathfrak{q}+\frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg)\\ &\quad +\frac{\mathfrak{ey}^3\|J\|_\infty^2}{4\varepsilon^2}\bigg( \mathfrak{q}+\mathfrak{p}+\frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg)^2. \end{align*} $$

Before we estimate $\|\partial _sf_2\|_{L^2}$ , we first use (3.19), (3.48) and (3.49) to calculate the following bounds:

$$ \begin{align*} \|\partial_{ss}\eta\ F_2\circ v \|_{L^1(\Omega)} & \leq \|\partial_{ss}\eta\ F_2\circ v \|_{L^1(\overline{\Omega)}}=\frac{1}{2}\int_{s_0}^{s_1}|\partial_{ss}\eta|\int_0^1|p(s,t)|^2\,dt\,ds\\ &\leq \frac{1}{2}\|\partial_{ss}\eta\|_{L^1(\overline{\Omega})}\sup_{s\in\mathbb{R}}\|p(s)\|_{L^2([0,1])}^2\\ & \leq \bigg(\frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg(\frac{1}{2}(\mathfrak{h}+ \mathfrak{q})+\mathfrak{p}+\frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg)+ \frac{c}{3}\bigg)\bigg(\mathfrak{p}+\frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg)^2\\ \|\partial_{ss}\eta\ d^{\mathbb{C}}F_2(X_{H_0})\|_{L^1(\Omega)} & = \|\partial_{ss}\eta(d^{\mathbb{C}}F_1(X_{H_0})-2F_2)\|_{L^1(\Omega)}\\ & \leq \|\partial_{ss}\eta\ d^{\mathbb{C}}F_1(X_{H_0})\|_{L^1(\Omega)}+2\|\partial_{ss}\eta\ F_2\circ v\|_{L^1(\Omega)}\\ & \leq \bigg(\frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg( \frac{1}{2}(\mathfrak{h}+ \mathfrak{q})+\mathfrak{p}+\frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg)+\frac{c}{3}\bigg)\\ &\quad \times \bigg(\bigg(\mathfrak{q}+\mathfrak{p}+\frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg)^2 +2\bigg(\mathfrak{p}+\frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg)^2\bigg). \end{align*} $$

Finally, using the estimates above together with (3.10), (3.41), (3.44), (3.47) and (3.51), we can calculate

$$ \begin{align*} \|\partial_sf_2\|_{L^1(\Omega)} & \leq \mathfrak{y}\|\partial_s\eta\ F_1\|_{L^1(\Omega)}+\|\partial_s\eta\ d(F_2+H_0)(\partial_sv)\|_{L^1(\Omega)}\\ &\quad +\frac{\mathfrak{y}^2}{2} \|dF_1(\partial_sv)\|_{L^1(\Omega)}+\|\partial_{ss}\eta(F_2+H_0)\|_{L^1(\Omega)}\\ & \leq \frac{\mathfrak{ey}\|J\|_\infty}{2\varepsilon} \bigg(\mathfrak{q}+\frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg)^2+\mathfrak{e}\bigg( \mathfrak{q}+3\mathfrak{p}+2\frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg)\\ &\quad +\frac{\mathfrak{ey}^2}{2}\bigg( \mathfrak{q}+\mathfrak{p}+\frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg)\\ & \quad +\bigg(\frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg( \frac{1}{2}(\mathfrak{h}+ \mathfrak{q})+\mathfrak{p}+\frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg)+\frac{c}{3}\bigg)\\ &\quad \times\bigg(\bigg(\mathfrak{q}+\mathfrak{p}+\frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg)^2 +2\bigg(\mathfrak{p}+\frac{\mathfrak{e}\|J\|_\infty}{\varepsilon}\bigg)^2\bigg). \end{align*} $$

This concludes the proof that $f_2$ is bounded in the $W^{1,1}$ -norm and the bounds do not depend on the choice of the Floer trajectory $u\in \mathcal {M}^\Gamma (a,b)$ or the connected component $\Omega \subseteq \mathbb {R}\times [0,1]\setminus v^{-1}(K_\varepsilon )$ , but only on the constants $a,b, \varepsilon $ and the smooth homotopy  $\Gamma $ .

Proof of Theorem 1.1

Let $H_0:T^*\mathbb {R}^2\to \mathbb {R}$ be the Hamiltonian defined in (1.1) and let $\mathcal {H}$ be the set of perturbations as defined in (1.2). Fix a pair $q_0,q_1\in \mathbb {R}^2$ . Let $\{K_{n,m}\}_{n,m\in \mathbb {N}}$ be the sequence of compact sets as in Lemma 3.2 such that

(3.52) $$ \begin{align} \begin{aligned} K_{n,m}\subseteq K_{n+1,m}\quad &\textrm{and} \quad \bigcup_{n\in\mathbb{N}}K_{n,m}=T^*\mathbb{R}^2\quad \text{for all}\ m\in \mathbb{N},\\ K_{n,m}\subseteq K_{n,m+1} \quad &\textrm{and} \quad T_{q_0}^*\mathbb{R}^2, T_{q_1}^*\mathbb{R}^2\subseteq \bigcup_{m\in\mathbb{N}}K_{n,m}\quad \text{for all}\ n\in\mathbb{N}. \end{aligned} \end{align} $$

Fix a Hamiltonian $h_0\in \mathcal {H}$ and denote $c(h_0):=\inf (h-dh(p\partial _p))$ . Naturally, ${c(h_0)>0}$ . Let $K_{n,m}$ be any of the compact sets in the sequence such that $\operatorname {supp} dh_0\subseteq K_{n,m}$ and $\|h_0\|_{L^\infty }<m$ . Denote

$$ \begin{align*} \mathcal{H}_{h_0}(K_{n,m}):=\left\lbrace\!\!\begin{array}{c|c} & dh \in C_0^\infty(K_{n,m}), \quad \|h\|_{L^\infty}<\dfrac{c(h_0)}{50},\\ h \in C^\infty(T^*\mathbb{R}^2) & h_0+h \in \mathcal{H}, \quad \|h_0+h\|_{L^\infty}<m,\\ & \inf((h+h_0)-d(h+h_0)(p\partial_p))>\dfrac{1}{2} c(h_0). \end{array}\!\!\right\rbrace. \end{align*} $$

This way, $\mathcal {H}_{h_0}(K_{n,m})$ is an open neighbourhood of $0$ in $\{h\in C^\infty (T^*\mathbb {R}^2)\ |\ dh\in C_0^\infty (K_{n,m})\}$ . By Lemma 3.2, the critical set of the action functional $\mathcal {A}^{H-h_0}_{q_0,q_1}$ is continuously compact in $(K_{n,m}, \mathcal {H}_{h_0}(K_{n,m}))$ . However, by construction, if $h\in \mathcal {H}_{h_0}(K_{n,m})$ , then $h_0+h\in \mathcal {H}$ ; hence, by Lemma 3.1, every Hamiltonian $H_0-h_0-h$ satisfies the condition (2.7). This proves that the compact set $K_{n,m}$ and the set of perturbations $\mathcal {H}_{h_0}(K_{n,m})$ satisfy assumptions (1) and (2) of Theorem 2.5 for the Hamiltonian $H_0-h_0$ .

Now, if we take any open, pre-compact set $\mathcal {V}\subseteq T^*\mathbb {R}^2$ such that $K_{n,m}\subseteq \mathcal {V}$ , any two almost complex structures $J_0,J_1\in \mathcal {J}(\mathcal {V},\mathbb {J})$ and any $h\in \mathcal {H}_{h_0}(K_{n,m})$ , then every smooth homotopy $\Gamma :=\{(h_0+h,J_s)\}_{s\in \mathbb {R}}$ , with $J_s\in C^\infty ([0,1]\times \mathbb {R},\mathcal {J}(\mathcal {V},\mathbb {J}))$ such that

(3.53) $$ \begin{align} J_s=\begin{cases} J_0 & \textrm{for } s\leq 0,\\ J_1 & \textrm{for } s\geq 1, \end{cases} \end{align} $$

will automatically satisfy condition (3.2). Consequently, by Theorem 3.3 for every pair $(a,b)\in \mathbb {R}^2$ , the corresponding space $\mathcal {M}^\Gamma (a,b)$ of Floer trajectories is bounded in $L^\infty $ -norm. Therefore, we can apply Theorem 2.5 and conclude that there exists a residual set $\mathcal {H}^{\,\operatorname {reg}}_{h_0}(K_{n,m})\subseteq \mathcal {H}_{h_0}(K_{n,m})$ such that for every $h\in \mathcal {H}^{\,\operatorname {reg}}_{h_0}(K_{n,m})$ , the Lagrangian Rabinowitz Floer homology $\mathrm { LRFH}_*(\mathcal {A}^{H_0-h_0-h}_{q_0,q_1})$ is well defined and independent of the choice of the almost complex structure.

Now, if we fix $h_1,h_2\in \mathcal {H}^{\,\operatorname {reg}}_{h_0}(K_{n,m})$ , then we can construct a homotopy $\Gamma $ satisfying (3.2). Indeed, let $J_0\in \mathcal {J}^{\,\operatorname {reg}}_{h_0}$ and $J_1\in \mathcal {J}^{\,\operatorname {reg}}_{h_2}$ will be two almost complex structures close enough, so that there exists a smooth homotopy $\{J_s\}_{s\in \mathbb {R}}$ with $J_s\in C^\infty ([0,1]\times \mathbb {R},\mathcal {J}(\mathcal {V},\mathbb {J}))$ satisfying (3.53) and $\sup _{s\in \mathbb {R}}\|J_s\|_{L^\infty }<({1}/{\sqrt {6c(h_0)}})$ . Moreover, let $\chi \in C^\infty (\mathbb {R})$ be a smooth function satisfying $\|\chi '\|_{L^\infty } \leq 2$ and

(3.54) $$ \begin{align} \chi(s)=\begin{cases} 0 & \textrm{for } s\leq 0,\\ 1 & \textrm{for } s\geq 1. \end{cases} \end{align} $$

Define $h_s:=h_0+h_1(1-\chi (s))+\chi (s)h_2$ . Then, the smooth homotopy $\Gamma =\{(h_s,J_s)\}_{s\in \mathbb {R}}$ satisfies

$$ \begin{align*} \|\partial_sh_s\|\bigg(\dfrac{4}{c(h_0)}+\|J\|_{L^\infty}^2\bigg) &\leq \|\chi'\|_{L^\infty}\|h_1-h_2\|_{L^\infty}\bigg(\dfrac{4}{c(h_0)}+\dfrac{1}{6c(h_0)}\bigg)\\ &< 2\cdot \frac{c(h_0)}{25}\cdot \frac{25}{6c(h_0)}=\frac{1}{3}. \end{align*} $$

In other words, $\Gamma $ satisfies (3.2) and thus by Theorem (3.3), for every pair $(a,b)\in \mathbb {R}^2$ , the corresponding space $\mathcal {M}^\Gamma (a,b)$ of Floer trajectories is bounded in $L^\infty $ -norm. Therefore, we can apply Theorem 2.5 and conclude that for every $h\in \mathcal {H}_{h_0}(K_{n,m})$ , the Lagrangian Rabinowitz Floer homology is well defined and isomorphic to $\mathrm {LRFH}_*(\mathcal {A}^{H_0-h_0}_{q_0,q_1})$ .

Taking all the sets of the form $\mathcal {H}_{h_0}(K_{n,m})$ with $h_0\in \mathcal {H}$ and $n,m\in \mathbb {N}$ gives us an open cover of $\mathcal {H}$ . Since $\mathcal {H}$ is path connected and the Lagrangian Rabinowitz Floer homology is constant on every open set $\mathcal {H}_{h_0}(K_{n,m})$ , a basic topological argument gives us that for any pair $h_0,h_1\in \mathcal {H}$ , the respective homologies $\mathrm {LRFH}_*(\mathcal {A}^{H_0-h_0}_{q_0,q_1})$ and $\mathrm { LRFH}_*(\mathcal {A}^{H_0-h_1}_{q_0,q_1})$ are isomorphic.

Proof. The following proof is an adjustment of the proof of [Reference Cieliebak and Frauenfelder4, Corollary 3.8] to our setting, and it is similar to the proof of Lemma 3.6 where we have proven the Novikov finiteness condition.

Fix $x\in \operatorname {Crit}^+\mathcal {A}^{H_0-h_0}_{q_0,q_1}$ and $y\in \operatorname {Crit}\mathcal {A}^{H_0-h_1}_{q_0,q_1}$ and abbreviate

$$ \begin{align*} a:= \mathcal{A}^{H_0-h_0}_{q_0,q_1}(x) \quad\textrm{and}\quad \mathcal{A}^{H_0-h_1}_{q_0,q_1}(y). \end{align*} $$

By assumption, $a\geq \delta (h_0)$ . Let $u=(v,\eta )\in \mathscr {F}_\Gamma (x,y)$ . Since $h_0,h_1\in \mathcal {H}_c$ and $\Gamma $ satisfies (3.2), our setting satisfies the assumptions of Lemma 3.5. Suppose $|b|\leq a$ . Then, ${b-a\leq 0}$ and (3.10) gives

(4.2) $$ \begin{align} \|\eta\|_{L^\infty} &\leq \frac{3}{2} \bigg(\frac{2}{c(h_0)}\max\{|a|,|b|\}+\frac{1}{\sqrt{c(h_0)}}+\|J\|_{L^{\infty}}^2 (b-a)\bigg)\nonumber\\ &\leq\frac{3}{c(h_0)}\bigg(a+\frac{\sqrt{c(h_0)}}{2}\bigg)\leq \frac{3a}{c(h_0)}\bigg(1+\frac{\sqrt{c(h_0)}}{2\delta(h_0)}\bigg). \end{align} $$

However, by (3.7), we have

$$ \begin{align*} \|\nabla^{J_{s}} \mathcal{A}^{H_0-h_{s}}(u)\|_{L^{2}(\mathbb{R}\times [0,1])}^{2} \leq \|J\|_{L^{\infty}} (b-a + \|\eta\|_{L^{\infty}}\|\partial_{s}h_{s}\|_{L^{\infty}}). \end{align*} $$

Combined with (4.1) and (4.2), we obtain the following estimate:

$$ \begin{align*} b\geq a-\|\eta\|_{L^{\infty}}\|\partial_{s}h_{s}\|_{L^{\infty}}\geq a\bigg(1-\frac{3}{c(h_0)} \bigg(1+\frac{\sqrt{c(h_0)}}{2\delta(h_0)}\bigg)\|\partial_{s}h_{s}\|_{L^{\infty}}\bigg)\geq \frac{1}{2}a. \end{align*} $$

In particular, $\mathcal {A}^{H_0-h_1}_{q_0,q_1}(y)=b\geq \tfrac 12\delta (h_0)$ . By assumption, $h_1 \in \mathcal {O}_{h_0}(K)$ , and hence, by Lemma 2.8, we infer that $\mathcal {A}^{H_0-h_1}_{q_0,q_1}(y)\geq \delta (h_0)$ . This proves the claim under the assumption $|b|\leq a$ .

Now, suppose $b< -a\leq -\delta (h_0)$ . Then, $b-a\leq 0$ and by (3.10), we have

$$ \begin{align*} \|\eta\|_{L^\infty} &\leq \frac{3}{2} \bigg( \frac{2}{c(h_0)}\max\{|a|,|b|\}+\frac{1}{\sqrt{c(h_0)}}+\|J\|_{L^{\infty}}^2 (b-a)\bigg)\nonumber\\ &\leq \frac{3}{c(h_0)}\bigg(\frac{\sqrt{c(h_0)}}{2}-b\bigg) <\frac{-3b}{c(h_0)}\bigg(1+\frac{\sqrt{c(h_0)}}{2\delta(h_0)}\bigg). \end{align*} $$

Combining it with (3.7) and (4.1), we obtain the following inequality:

$$ \begin{align*} a \leq b + \|\eta\|_{L^{\infty}}\|\partial_{s}h_{s}\|_{L^{\infty}} \leq b\bigg(1-\frac{3b}{c(h_0)}\bigg(1+\frac{\sqrt{c(h_0)}}{2\delta(h_0)}\bigg) \|\partial_{s}h_{s}\|_{L^{\infty}}\bigg)\leq \frac{1}{2}b<-\frac{1}{2}a, \end{align*} $$

which contradicts the assumption $a\geq \delta (h_0)>0$ . That excludes the case $b<-a$ and proves the lemma.

Proof. Fix two points $q_0,q_1 \in \mathbb {R}^2$ and let $\{K_{n,m}\}_{n,m\in \mathbb {N}}$ be the sequence of compact sets as in Lemma 3.2 satisfying (3.53). Fix $h_0\hspace{-1pt}\in\hspace{-1pt} \mathcal {H}^{\,\operatorname {reg}}$ and denote $c(h_0)\hspace{-1pt}:=\hspace{-1pt}\inf (h_0\hspace{-1pt}-\hspace{-1pt}dh_0(p\partial _p))\hspace{-1pt}>\hspace{-1pt}0$ . Let $K_{n,m}\subseteq T^*\mathbb {R}^2$ be any compact set from the sequence such that $\operatorname {supp} dh_0 \subseteq K_{n,m}$ and $\|h_0\|_{L^\infty }<m$ . By Lemmas 2.8 and 3.2, there exists an open neighbourhood $\mathcal {O}_{h_0}(K_{n,m})$ of $0$ in $\{h\in C^\infty (T^*\mathbb {R}^2)\ |\ dh\in C_0^\infty (K_{n,m})\}$ such that:

1. (1) $\{h_0+h\ |\ h\in \mathcal {O}_{h_0}(K_{n,m})\}\subseteq \mathcal {H}$ ;

2. (2) the critical set of $ \mathcal {A}^{H_0-h_0}$ is continuously compact in $(K_{n,m}, \mathcal {O}_{h_0}(K_{n,m}))$ ;

3. (3) for all $h\in \mathcal {O}_{h_0}(K_{n,m})$ , we have $\inf ((h_0+h)-d(h_0+h)(p\partial _p))>\tfrac 12c(h_0)$ ;

4. (4) there exists $\delta (h_0)>0$ such that for all $h\in \mathcal {O}_{h_0}(K_{n,m})$ and all $x\in \operatorname {Crit}^+ \mathcal {A}^{H_0-h_0-h}_{q_0,q_1}$ , we have $\mathcal {A}^{H_0-h_0-h}_{q_0,q_1}(x)\geq \delta (h_0)$ ;

5. (5) for all $h\in \mathcal {O}_{h_0}(K_{n,m})$ , we have

   $$ \begin{align*} \|h_0-h\|_{L^\infty}<\frac{1}{2}c(h_0)\inf \bigg\lbrace\frac{1}{13}, \frac{1}{6}\bigg(1+\frac{\sqrt{c(h_0)}}{2\delta(h_0)}\bigg)^{-1}\bigg\rbrace. \end{align*} $$

We will show that for every $h\in \mathcal {O}_{h_0}(K_{n,m})$ , the positive Lagrangian Rabinowitz Floer homology $\mathrm { LRFH}_*^+(\mathcal {A}^{H_0-h_0-h}_{q_0,q_1})$ is well defined and isomorphic to $\mathrm {LRFH}_*^+(\mathcal {A}^{H_0-h_0}_{q_0,q_1})$ .

Fix $h_1 \in \{h_0+h\ |\ h\in \mathcal {O}_{h_0}(K_{n,m})\}\cap \mathcal {H}^{\,\operatorname {reg}}$ . Let $J_0\in \mathcal {J}^{\,\operatorname {reg}}_{h_0}$ and $J_1\in \mathcal {J}^{\,\operatorname {reg}}_{h_1}$ be two almost complex structures close enough, so that there exists a smooth homotopy $\{J_s\}_{s\in \mathbb {R}}$ with ${J_s\in C^\infty ([0,1]\times \mathbb {R},\mathcal {J},\mathbb {J}))}$ satisfying (3.54) and $\sup _{s\in \mathbb {R}}\|J_s\|_{L^\infty }<({1}/{\sqrt {3c(h_0)}})$ . Moreover, let ${\chi \in C^\infty (\mathbb {R})}$ be a smooth function satisfying $\|\chi '\|_{L^\infty } \leq 2$ and (3.54). Define $h_s:=h_0(1-\chi (s))+\chi (s)h_1$ . Then, the smooth homotopy $\Gamma =\{(h_s,J_s)\}_{s\in \mathbb {R}}$ satisfies (4.1).

In particular, the homotopy $\Gamma $ satisfies (3.2); thus, by Lemma 3.6, it satisfies the Novikov finiteness condition. Moreover, by Theorem 3.3 for any pair $a,b\in \mathbb {R}$ , the space of Floer trajectories $\mathcal {M}^\Gamma (a,b)$ is bounded in $L^\infty $ -norm. Finally, by Lemma 4.1, we know that for every $x\in \operatorname {Crit}^+\mathcal {A}^{H_0-h_0}_{q_0,q_1}$ and every $y\in \operatorname {Crit}\mathcal {A}^{H_0-h_1}_{q_0,q_1}$ such that $\mathscr {F}_\Gamma (x,y)\neq \emptyset $ , we have $\mathcal {A}^{H-h_1}_{q_0,q_1}(y)>0$ . This means that for every $h_1 \in \{h_0+h\ |\ h\in \mathcal {O}_{h_0}(K_{n,m})\}\cap \mathcal {H}^{\,\operatorname {reg}}$ , there exists a homotopy $\Gamma $ satisfying all three conditions of Corollary 2.9. Hence, for every $h \in \mathcal {O}_{h_0}(K_{n,m})$ , the positive Lagrangian Rabinowitz Floer homology $\mathrm {LRFH}_*^+(\mathcal {A}^{H_0-h_0-h}_{q_0,q_1})$ is well defined and isomorphic to $\mathrm { LRFH}_*^+(\mathcal {A}^{H_0-h_0}_{q_0,q_1})$ .

Taking all the sets of the form $\mathcal {O}_{h_r}(K_{n,m})$ with $h_r\in \mathcal {H}^{\,\operatorname {reg}}$ and $n,m\in \mathbb {N}$ gives us an open cover of $\mathcal {H}$ . Since $\mathcal {H}$ is path connected and the positive Lagrangian Rabinowitz Floer homology is constant on every open set $\mathcal {O}_{h_r}(K_{n,m})$ , a basic topological argument gives us that for any pair $h_0,h_1\in \mathcal {H}$ , the respective homologies $\mathrm {LRFH}_*^+(\mathcal {A}^{H_0-h_0}_{q_0,q_1})$ and $\mathrm {LRFH}_*^+(\mathcal {A}^{H_0-h_1}_{q_0,q_1})$ are isomorphic.

Proof. For clarity of the argument, we will divide the proof of this proposition into a sequence of lemmas. First, recall that by (2.10), we have

$$ \begin{align*} \operatorname{Crit}^\pm\mathcal{A}^{H}_{q_0,q_1}=\lbrace (v,\eta)\in \operatorname{Crit}\mathcal{A}^{H}_{q_0,q_1}\ |\ \pm \eta>0\rbrace. \end{align*} $$

In Lemma 4.4, we will show that for each $c>0$ and each pair $(q_0,q_1)\in \mathbb {R}^4$ , there exists a smooth function $f\in C^\infty (\mathbb {R})$ such that $(v,\eta )\in \operatorname {Crit}\mathcal {A}^{H_0-c}_{q_0,q_1}$ if and only if $f(\eta )=0$ . We also show that for $q_0\neq q_1$ , the critical set of $\mathcal {A}^{H_0-c}_{q_0,q_1}$ is discrete. Denote by $Z(f)$ the set of roots of the function f, $Z(f):=\{\eta \in \mathbb {R} \mid f(\eta )=0\}$ . In Lemma 4.5, we will consider pairs $(q_0,q_1)\in \mathbb {R}^4$ such that $|q_0||q_1|\leq 2c$ and show that for $q_0\neq q_1$ , the corresponding function f satisfies $\#\{\eta>0\ |\ f(\eta )=0\}=\#\{\eta <0\ |\ f(\eta )=0\}=1$ , whereas for $q_0=q_1$ , we have $Z(f)=\{0\}$ .

Denote $\Delta :=\{(q,q)\ |\ q\in \mathbb {R}^2\}\subseteq \mathbb {R}^4$ . In Lemma 4.6, we prove that for a fixed ${c>0}$ , there exists a residual set $\mathcal {Q}^c\subseteq \mathbb {R}^4\setminus \Delta $ such that for all pairs $(q_0,q_1)\in \mathcal {Q}^c$ , the corresponding function f satisfies

(4.3) $$ \begin{align} f'(\eta)\neq 0\quad \text{for all } \eta \in Z(f) \quad \textrm{where } Z(f):=\{\eta \in \mathbb{R}\ |\ f(\eta)=0\}. \end{align} $$

In Lemma 4.7, we show that for a fixed pair $(q_0,q_1)\in \mathbb {R}^4, q_0\neq q_1$ , there exists a residual set $\mathcal {I}_{q_0,q_1}\subseteq \mathbb {R}_+$ such that for all $c\in \mathcal {I}_{q_0,q_1}$ , the corresponding function f satisfies property (4.3).

In Lemma 4.8, we prove that if $c>0$ and $(q_0, q_1)\in \mathbb {R}^4, q_0\neq q_1$ are such that the corresponding function f satisfies (4.3), then f has an odd number of roots on each positive and negative half-line. This, together with the bijection established in Lemma 4.4, concludes the proof.

Proof. By definition, a pair $(v,\eta )\in \operatorname {Crit}\mathcal {A}^{H_0-c}_{q_0,q_1}$ if and only if it satisfies the following three conditions.

1. (1) The curve v starts in the Lagrangian $T^*_{q_0}\mathbb {R}^2$ and ends in $T^*_{q_1}\mathbb {R}^2$ .

2. (2) The curve v is a Reeb chord and $\eta $ is its period.

3. (3) The curve v lies on the level set $H_0^{-1}(c)$ .

Note that if $q_0\neq q_1$ , we have $\operatorname {Crit}\mathcal {A}^{H_0-c}_{q_0,q_1}\cap (\mathscr {H}_{q_0,q_1}\times \{0\})=\emptyset $ . When $q_0= q_1$ , we always have a submanifold of constant solutions