Let $(M, d\theta, \theta, Z)$, be an exact Liouville domain, where $Z$ is the Liouville vector field and $\theta$ is the primitive of the symplectic form. Let $\bar{M}$, be the symplectic completion and $\rho$ the canonical coordinate on the symplectization(which in some conventions is denoted by $e^r$). There are many foundational papers constructing symplectic cohomology groups for functions which are "asymptotically quadratic" or "asymptotically linear" as functions of $\rho$. For example Floer-Hofer's original:

http://link.springer.com/article/10.1007%2FBF02571699#page-1 or Oancea's

http://arxiv.org/abs/math/0503193

A prototypical example is e.g. a function of the form $ \rho^2 + f$, where $f$ is a sufficiently small function pulled back from $\partial M$. Ultimately, such approaches rely on some delicate $C^0$ estimates (that frankly go a bit over my head, hence my question) for solutions to Floer's equation.

In the foundational paper(s) of Abouzaid-Seidel and Abouzaid, wrapped Floer cohomology groups $WF^*(L_0,L_1)$ are constructed for $L_0$ and $L_1$ are "conical at $\infty$" Lagrangians, e.g. satisfy $\theta|_{L_i}=0$ for large $\rho$ (and its $A_\infty$ enhancements though I am not concerned with these for the moment) using Hamiltonians which only depend of the coordinate $\rho$. For such Hamiltonians, the above analysis is slightly simplified due to the existence of a maximum principle.

General Question: Is there any treatment in the literature of wrapped Floer cohomology (again only additively or maybe with product but no $A_\infty$ operations) for Hamiltonian functions which allows one to use Hamiltonians which are only asymptotically linear or quadratic?

The closest to what I would want is in the work of Abbondandolo-Schwarz.

http://arxiv.org/pdf/0810.1995v4.pdf http://arxiv.org/pdf/math/0408280v2.pdf

There, the technical setup is a little different, but still of interest to me. Let $M$ be a compact oriented Riemannian manifold. They consider just the case of a cotangent bundle $T^*M$ and use almost complex structures which are $C^0$ close to the Levi-Civita almost complex structure $J_g$. Let $S_i$ be embedded compact, oriented, submanifolds. They then define wrapped Floer cohomology groups $WF^*(N^*S_0, N^*S_1)$ along with compositions using asymptotically quadratic Hamiltonians using the type of $C^0$ estimates I mention above (according to the authors, the estimates here are even more refined than those above).

Concrete question: Does their analysis for $C^0$ estimates of solutions to Floer's equation on strips or (a very carefully constructed) pair of pants still go through if we assume that the Lagrangians $L_i$ are exact and conical at $\infty$? For example, when each $L_i$ is some compactly supported Hamiltonian isotopy of a cotangent fiber?