# Are absolutely continuous functions with values in a Hilbert space differentiable almost everywhere?

I have some questions regarding absolutely continuous Hilbert space valued functions defined on the reals. Define the function

$$x:[t_0, +\infty) \to \mathcal{H},$$

with $$\mathcal{H}$$ a real Hilbert space. I want to get a better feeling for absolute continuity in this setting. Here is the Definition for absolute continuity:

Definition 1:
The function $$x$$ is called absolutely continuous, if for all $$\varepsilon > 0$$ there exists $$\delta > 0$$ such that for all non-overlapping families of intervals $$([a_k, b_k])_{k = 1}^K$$ with $$\sum_{k=1}^K |b_k - a_k| < \delta \quad \text{ it holds that } \quad \sum_{k=1}^K ||x(b_k) - x(a_k)|| < \varepsilon.$$

The following Theorem holds for real-valued functions:

Theorem 2:
Let $$x:[t_0,+\infty) \to \mathbb{R}$$. The following are equivalent:

• $$x$$ is absolutely continuous.
• There exists a Lebesgue integrable function $$y:[t_0, +\infty) \to \mathbb{R}$$ with $$x(t) = x(t_0) + \int_{t_0}^t y(s) ds$$ for all $$t \in [t_0, +\infty)$$.
• $$x$$ is differentiable almost everywhere with derivative $$\dot{x}$$, the derivative is Lebesgue integrable, and $$x(t) = x(t_0) + \int_{t_0}^t \dot{x}(s) ds$$ for all $$t \in [t_0, +\infty)$$.

(This Theorem is sometimes called the "The fundamental theorem of calculus for the Lebesgue integral".)

In this setting I have the following Questions:

• Are absolutely continuous functions $$x:[t_0, +\infty) \to \mathcal{H}$$ differentiable almost everywhere? (Almost everywhere = everywhere expect on a set with Lebesgue measure 0.)
• Does Theorem 2 hold also for a function $$x:[t_0, +\infty) \to \mathcal{H}$$ where we replace the Lebesgue integral with the Bochner integral? If not are there other statements connecting absolute continuity with differentials and integrals in this setting?
• Does Theorem 2 hold also for a function $$x:[t_0, +\infty) \to \mathcal{X}$$, where $$\mathcal{X}$$ is a general Banach space? (I am more concerned with the Hilbert space setting but would be interested in this to get a more complete view.)

I am thankful for any sources or books on this topic. I could not find any work summarizing results on these questions. I only found this source treating vector-valued functions:https://www.math.ucdavis.edu/~hunter/pdes/ch6A.pdf. But since it is not peer-reviewed and does not hint at further sources I am a bit paranoid.

Thank you very much!