Fredholmness of multipliers on Hardy-Sobolev spaces

This paper discusses the Fredholmness of multipliers on Hardy-Sobolev Spaces and obtains an index formula for the multipliers with some special symbols. Our results show that Hardy-Sobolev spaces have richer properties than classical holomorphic function spaces, and the behavior of the operators on these spaces is complex. Some methods of Hardy or Bergman spaces fail in the case of the Hardy-Sobolev space.