let Y be a contact (3) manifold and X be its symplectization. let's say the Reeb dynamics is at least Morse Bott. let $u: \Sigma \rightarrow X$ be a $J$ holomorphic curve. I know the usual model for a neighbourbood for it is the weighted sobolev space $W^{1,p,d}(u^*TX)$ plus maybe some asymptotic vectors. Do people know if we can replace this space with some Holder space (say C^{2,0}). Is it still true the linearized Cauchy Riemann operator is Fredholm of the same index? (we already the kernel and cokernel if we use the Sobolev spaces have exponential decay, can we say it doesn't have any new kernel and cokernel?) (I would really like some way to get rid of the exponential weights..)
Edit: I would like to note this wonderful paper discusses why we cannot use $C^k$ spaces. https://arxiv.org/pdf/2207.08509.pdf It seems to suggest we can indeed use holder spaces, but due to my ignorance I don't know where this is written down (this might be obvious to the more analytically inclined).
