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**Definitions and some motivation:**

For $k \geq 1$, we say a function $f: [0, 1] \to \mathbb R$ has *bounded $k$-variation* if $\sup \sum_{i = 1}^n |f(x_i) - f(x_{i-1})|^k <  \infty$, where the supremum is taken over all finite partitions $0 = x_0 < \dots < x_n = 1$ of  [0, 1].

Note that bounded $1$-variation corresponds to bounded variation in the usual sense.

We start with a characterisation of functions of bounded variation on $[0, 1]$, which follows rather directly from the general theory of functions of bounded variation as functions whose weak derivatives are Radon measures.

Let $\mu$ be a positive Radon measure on $[0, 1]$.

We say a function $f: [0, 1] \to \mathbb R$ is *absolutely continuous* with respect to $\mu$ if for all $\varepsilon > 0$, there exists $\delta > 0$ such that $\sum_{i=1}^n |f(x_i) - f(y_i)| < \varepsilon$ whenever $x_1, \dots, x_n$ and $y_1, ..., y_n$ are such that $\sum_{i = 1}^n \mu([x_i, y_i]) < \delta.$

We say a function $f: [0, 1] \to \mathbb R$ is *$1$-regular* if there exists a Radon measure $\mu$, and a decomposition $f = g + h$ of $f$ into functions $g, h$; with $g$ absolutely continuous with respect to Lebesgue measure, and $h$ absolutely continuous with respect to $\mu$.

We then have the following theorem:

> **Theorem:** A function $f$ is of bounded variation if and only if it is $1$-regular.

A quick sketch is as follows - the “if” part is immediate by direct calculation. For the “only if” part, take $\mu$ to be the total variation $|Df_s|$ of the singular part of $Df$, where $Df$ is the derivative of $f$ in the sense of Radon measures. We then have the decomposition $f = g + h$, with $g(x) := \int_{[0, x]} Df_s$, and $h := f - g$.

I was wondering if functions of bounded higher order variation - quadratic for instance - could be characterised in a similar way as well. The idea is that we might be able to detect the higher order variations by progressively “renormalising/zooming into” $f$. To this end, let us introduce a few definitions.

**Problem set up:**

Let $\mu$ be a non atomic probability measure on $[0, 1]$, and denote by $F_\mu$ it’s cumulative distribution function.

The following definition is central - for a function $f :[0, 1] \to \mathbb R$ we define the *normalisation* $f^\mu$ of $f$ (by $\mu$) by $f^\mu (x) := f(F_\mu^{-1} (x))$.

For integer $k \geq 2$, we say that $f$ is *$k$-regular* if there exists a non atomic Radon probability measure $\mu$, and a decomposition $f = g + h$ of $f$, where $g$ is $k-1$-regular, $h$ is constant on every connected component of $[0, 1] \setminus \text{supp} \ \mu$, and $h^\mu$ is $k-1$-regular.

The definition of normalisation is motivated by the fact that a function $h$ is absolutely continuous with respect to $\mu$ if and only if it is constant on every connected component of $[0, 1] \setminus \text{supp} \ \mu$, and $h^\mu$ is absolutely continuous in the usual sense with respect to Lebesgue measure.

> **Question:** Let $f$ be a continuous function. For integer $k \geq 2$, is it true that $f$ is of bounded $k$-variation if and only if $f$ is $k$-regular?

Wed, 07 Jul 2021 11:07 GMT