Is there an intuitive (or geometric) reason why we need to consider $L^p$ Sobolev spaces instead of just considering $L^2$ when studying $J$-holomorphic curves?
The only reason I know is when proving some properties of the local linearized Cauchy-Riemann equation we need to use $L^p$ for $p>2$ to ensure some inequalities to be correct. But is there any geometric fact that will not be true for $L^2$ but true for $L^p$?