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 But since it is not peer-reviewed and does not hint at further sources I am a bit paranoid.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!
Best Answer
Thanks for the reply @Bruno B, I have not found this post.
In the post there is the same question regarding functions with values in a Banach Space. A reply to this question hints at the book "Vector Measures by Diestel and Uhl" and I got myself a copy from our univesity library this morning.
In the second paragraph of the introduction was already an answer with a hint to "Uniformly convex spaces by James A. Clarkson" (https://www.jstor.org/stable/pdf/1989630.pdf?refreqid=excelsior%3Af52aa5290759f14902853a420d2564fa&ab_segments=&origin=&initiator=&acceptTC=1)
Theorem 7 states everything I need. The answers to question 1 and 2 is yes.
The answer to question 3 is no. As stated in the beginning of the paper by Clarkson there a counter examples due to Bochner.