Fredholm property about $L^p$-extension $(p\neq 2)$ of differential operators

Question

The following is a well-known result for elliptic operators.

Theorem. Let $P: \Gamma(E)\to \Gamma(F)$ be an elliptic operator of order $m$ between vector bundles $E$ and $F$ over a compact manifold $X$. Then $P$ extends to a Fredholm map $P: W^{k,2}(E) \to W^{k-m,2}(F)$ whose index is independent of $k$ (cf. p193. Lawson-Michelsohn's book 'Spin Geometry')

This kind of extension involves Hilbert spaces and thus is relatively not hard to work with. However, concerning Floer theory (or pseudoholomorphic curvers theory), the linearized maps we concern have to be between non-Hilbert Banach spaces, such as $W^{1,p}(\mathbb R \times S^1, u^*TM)$ or $L^p(S^2, \Lambda^{0,1} \otimes_J u^*TM)$, where $p$ is always assumed to be greater than 2.

In the case $p\neq 2$, it seems much harder to show any Fredholm properties, and the proofs are usually not natural but very technical. You may agree to me if you refer to Audin-Damian's book(section 8.7) or McDuff-Salamon's book(Proposition 3.1.11 which requires Theorem C.2.3) respectively.

So, I would like to ask does there exist any general theorem concerning Fredholm properties, which is similar to the above one? If exist, what could be a general principle to show the Fredholm properties? In the above theorem, can we relax the condition that the base manifold $X$ is compact, as in Floer theory, the base manifold $\mathbb R \times S^1$ is non-compact? Any great reference?

Edit: I recently found a paper (cf. (0.1) in the very beginning) which states that (if I haven't misunderstood): let $E$ and $F$ be two vector bundles over a compact manifold $X$ and let $P: \Gamma(E)\to \Gamma(F)$ is an elliptic differential operator of order $m$, then the paper claims that it is well-known that it has a Fredholm extension $$ P:W^{k+m,p}(E) \to W^{k,p}(F)$$ for every $1<p<\infty$. This should be a really good result but I have trouble in finding a serious reference to confirm this.

Answer 1

See Chapter 3 of Matthias Schwarz' thesis or Chapter 3 of Donaldson's book on Floer homology. Both do L^p theory on cylindrical end manifolds (following Maz'ya etc.)

Answer 2

Too long for a comment:

The paper you linked of Lockhart-McOwen shows you the faults for noncompact manifolds (independent of $p\ne 2$). Look up the Atiyah–Patodi–Singer spectral boundary-value problem. We need asymptotic conditions (or boundary conditions, there is a way to pass between the two) otherwise the kernel of our operator is infinite-dimensional. I've learned the most from reading the analysis chapters in Kronheimer-Mrowka's "Monopoles and 3-manifolds", specifically Chapter 17.

About your question for $p>2$ and $X$ compact, I don't know in general. We can use the Rellich–Kondrachov embedding theorem to get compactness for $W^{k+m,\,p}\to W^{k,\,p}$ with $m\ge0$. It then suffices to prove (what I call) this "Calderon–Zygmund" inequality to get Fredholmness: $||s||_{k+m,\,p} \le C(||Ps||_{k,\,p} + ||s||_{k,\,p})$. To my understanding, this is a rephrasal of finding a parametrix (Green's operator) for $P$ such that $QP-1$ (and $PQ-1$) extends to $W^{k,\,p}$. In other words, we know that an operator is Fredholm when it is invertible up to compact operators, and an elliptic operator $P$ admits a parametrix $Q$ with $QP-1$ and $QP-1$ "smoothing operators". Working over compact manifolds, the hope is that these smoothing operators have a continuous extension to $W^{k,\,p}$. Is that granted by the property "smoothing"? This is probably basic (i.e. the proof using $p=2$ might allow freedom in $p$).

There is a trick in Hormander's "Analysis of linear PDO's III", Corollary B.1.6, which shows how you can take knowledge of your (linear) elliptic operator on $W^{k,2}$ and get the same knowledge over certain Besov spaces $B^{p,k}$ for any $1\le p\le\infty$, but I don't think this is good enough for $W^{k,\,p}$ (that is, I don't think the norms are equivalent (and if it were then we'd get knowledge for $p=1,\infty$ which seems too strong)). But I can very easily be wrong, maybe this gets us what we want... in either case there should be a more direct route.