Generalizations of Sard-Smale Theorem

187 Views Asked by At

Sard-Smale theorem holds for Fredholm maps $f:M\rightarrow B$ between separable Banach manifolds $M,N$. There are some constrains relating the Fredholm index $\operatorname{ind}(f)$ of $f$ to its differentiablity class. More precisely, we need to require $f\in C^{r}$, where $r>\max{(\operatorname{ind}(f),0)}$.

I would like to know what are the generalizations of this theorem into two directions:

  1. assuming weaker regularity on $f$;
  2. assuming weaker structure on the spaces $M,N$.

For instance, I know that the result is valid for bounded Fréchet manifolds in the case where $f$ is Lipchitz (see here). The result is also valid for some domains with empty interior (see here).

I would also like to know if the above (or others) generalizations are maximal, in the sense that there are counterexamples in the case of weaker requirements.

Any help is welcome.

Edit: I posted the question on MathOverflow.