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The paper you linked of Lockhart-McOwen shows you the faults for noncompact manifolds (independent of $p\ne 2$). 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. Look up the Atiyah–Patodi–Singer spectral boundary-value problem. I've learned the most from reading the analysis chapters in Kronheimer-Mrowka's "Monopoles and 3-manifolds", specifically Chapter 17.

See my comment (my lack of confidence) about answering your question for $p>2$ and $X$ compact. There is a trick I found in Hormander's "The analysis of linear partial differential operators III", Corollary B.1.6, that gets you what you want when you already have your linear elliptic operator on $W^{k,2}$ (and hence the Fredholmness for all $k$): If your continuous linear operator $W^{k,\,2}\to W^{k,\,2}$ restricts to a continuous linear operator $W^{k',\,2}\to W^{k',\,2}$ for $k'>k$ then it also restricts to a continuous linear operator $W^{k'',\,p}\to W^{k'',\,p}$ for $k'>k''>k$ and any $p$. <--- Not quite, I am not sure the "Besov spaces" Hormander uses are the Sobolev spaces (i.e. I don't know if the norms are equivalent).

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.

The paper you linked of Lockhart-McOwen shows you the faults for noncompact manifolds (independent of $p\ne 2$). 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. Look up the Atiyah–Patodi–Singer spectral boundary-value problem. I've learned the most from reading the analysis chapters in Kronheimer-Mrowka's "Monopoles and 3-manifolds", specifically Chapter 17.

See my comment (my lack of confidence) about answering your question for $p>2$ and $X$ compact. There is a trick I found in Hormander's "The analysis of linear partial differential operators III", Corollary B.1.6, that gets you what you want when you already have your linear elliptic operator on $W^{k,2}$ (and hence the Fredholmness for all $k$): If your continuous linear operator $W^{k,\,2}\to W^{k,\,2}$ restricts to a continuous linear operator $W^{k',\,2}\to W^{k',\,2}$ for $k'>k$ then it also restricts to a continuous linear operator $W^{k'',\,p}\to W^{k'',\,p}$ for $k'>k''>k$ and any $p$.

The paper you linked of Lockhart-McOwen shows you the faults for noncompact manifolds (independent of $p\ne 2$). 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. Look up the Atiyah–Patodi–Singer spectral boundary-value problem. I've learned the most from reading the analysis chapters in Kronheimer-Mrowka's "Monopoles and 3-manifolds", specifically Chapter 17.

See my comment (my lack of confidence) about answering your question for $p>2$ and $X$ compact. There is a trick I found in Hormander's "The analysis of linear partial differential operators III", Corollary B.1.6, that gets you what you want when you already have your linear elliptic operator on $W^{k,2}$ (and hence the Fredholmness for all $k$): If your continuous linear operator $W^{k,\,2}\to W^{k,\,2}$ restricts to a continuous linear operator $W^{k',\,2}\to W^{k',\,2}$ for $k'>k$ then it also restricts to a continuous linear operator $W^{k'',\,p}\to W^{k'',\,p}$ for $k'>k''>k$ and any $p$. <--- Not quite, I am not sure the "Besov spaces" Hormander uses are the Sobolev spaces (i.e. I don't know if the norms are equivalent).

The paper you linked of Lockhart-McOwen shows you the faults for noncompact manifolds (independent of $p\ne 2$). 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. Look up the Atiyah–Patodi–Singer spectral boundary-value problem. I've learned the most from reading the analysis chapters in Kronheimer-Mrowka's "Monopoles and 3-manifolds", specifically Chapter 17.

See my comment (my lack of confidence) about answering your question for $p>2$ and $X$ compact. There is a trick I found in Hormander's "The analysis of linear partial differential operators III", Corollary B.1.6. This gets you what you want when your elliptic operator is linear: If your continuous linear operator $W^{k,\,2}\to W^{k,\,2}$ restricts to a continuous linear operator $W^{k',\,2}\to W^{k',\,2}$ for $k'>k$ then it also restricts to a continuous linear operator $W^{k'',\,p}\to W^{k'',\,p}$ for $k'>k''>k$ and any $p$.

The paper you linked of Lockhart-McOwen shows you the faults for noncompact manifolds (independent of $p\ne 2$). 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. Look up the Atiyah–Patodi–Singer spectral boundary-value problem. I've learned the most from reading the analysis chapters in Kronheimer-Mrowka's "Monopoles and 3-manifolds", specifically Chapter 17.

See my comment (my lack of confidence) about answering your question for $p>2$ and $X$ compact. There is a trick I found in Hormander's "The analysis of linear partial differential operators III", Corollary B.1.6, that gets you what you want when you already have your linear elliptic operator on $W^{k,2}$ (and hence the Fredholmness for all $k$): If your continuous linear operator $W^{k,\,2}\to W^{k,\,2}$ restricts to a continuous linear operator $W^{k',\,2}\to W^{k',\,2}$ for $k'>k$ then it also restricts to a continuous linear operator $W^{k'',\,p}\to W^{k'',\,p}$ for $k'>k''>k$ and any $p$.