# Does anyone know if it's possible to construct Moduli space of J holomorphic curves using Holder spaces?

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..)

• Among other things you want some form of the Calderón-Zygmund inequality and also to be able to use the Contraction Mapping Theorem. Morrey spaces were 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 $u$). – Chris Gerig Oct 10 at 15:11
• I know Hutchings & TAubes used Morrey spaces - they used it over the normal bundle where the asymptotic operators are already non degenerate - could they have gotten away with it if they used it over the entire pull back of the tangent bundle $u^*(TX)$ where the asymptotic operators have degeneracies? Alternatively, if the orbits are only Morse-Bott, could they have used Morrey spaces over section of Normal bundle without exponential weights? – Yuan Yao Oct 10 at 21:46