Let M be a smooth hypersurface in R^{2n}. Suppose that on each side of M, there is a complex structure which is smooth up to M. Assume that the difference of the two complex structures is sufficiently small on M. We show that if a continuous function is holomorphic with respect to both complex structures, the function must be smooth from both sides of M. The regularity proof makes essential use of J-holomorphic curves and Fourier transform on families of curves.

This is joint work with Florian Bertrand and Jean-Pierre Rosay.