Fraction of Lipschitz functions among absolutely continuous ones

Question

Is it true that the space of Lipschitz functions on $S^1$ is a $G_\delta$ subset of the space of absolutely continuous functions on $S^1$?

In which topologies ($L^p$, uniform, $C^k$, etc) it is true?

Answer 1

I'll work on $[0,1]$ for convenience. The natural norm for the absolutely continuous functions on $[0,1],$ let's call this space $\text {AC},$ is

$$\|f\|_{\text {AC}} = |f(0)| + \int_0^1|f'|.$$

In this norm $\text {AC}$ is a Banach space.

It's a nice exercise to show the space of Lipschitz functions, let's call this $\mathcal {L},$ is of the first category in $\text {AC}.$ Also, $\mathcal {L}$ is dense in $\text {AC}.$ If $\mathcal {L}$ were a $G_\delta$ in $\text {AC},$ then we would have produced a dense $G_\delta$ set that is of the first category in $\text {AC}.$ Since $\text {AC}$ is a Banach space, we have a contradiction by Baire's theorem.

Now the topology of $\text {AC}$ is strictly stronger than that inherited from any $L^p,$ including $p=\infty.$ Thus $\mathcal {L}$ cannot be a $G_\delta$ in any of these spaces. You mentioned $C^k,$ but since there are functions in $\text {AC}$ that are not $C^1,$ I'm not sure what you mean.