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..)

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Morrey spaceswere used by Cliff Taubes (and Karen Uhlenbeck and probably others). You'll find these used in most papers of Taubes related to J-holomorphic curves, solving for kernel of the deformation operators $D_u$ (Morrey spaces require particular decay over all small balls in $u$). $\endgroup$ – Chris Gerig Oct 10 at 15:11