Given $F(x) = \int_a^xf(t)dt$, for $f \in L^p([a,b])$, prove a uniform bound of $\sum_{i=0}^{n-1} \frac{|F(x_{i+1})-F(x_i)|^p}{(x_{i+1}-x_i)^{p-1}}$

Question

I'm working through the following problem

Let $1 < p < \infty$ and $f\in L^p([a,b])$. Now, if $F(x) = \int_a^x f(t)\,dt$, show that there exists $K \in \mathbb{R}$ such that for every partition $\pi = \{x_i\}_{0\leq i\leq n}$ of $[a,b]$,

$$ \sum_{i=0}^{n-1} \frac{|F(x_{i+1})-F(x_i)|^p}{(x_{i+1}-x_i)^{p-1}} \leq K. $$

My argument is as follows: let $\pi = \{x_i\}_{0\leq i\leq n}$ be a partition. By density of $C_0^{\infty}([a,b])$ in $L^p([a,b])$, there exists a sequence $(g_m)_{m \in \mathbb{N}} \subseteq C_0^{\infty}([a,b])$ such that

$$g_m \xrightarrow{L^p} f.$$

Moreover, $g_m \in L^1([a,b]) \cap L^p([a,b])$ and if $G_m(x) := \int_a^xg_m(t)dt$,

$$ |F(x) - G_m(x)| \leq \int_a^x|f(t)-g_m(t)|dt \leq \int_a^b|f(t)-g_m(t)|dt \leq \|f-g_m\|_{1} $$

Now, let's see that we can obtain a uniform bound for the sum when replacing $f$ by $g_m$. By the intermediate value theorem for $G_m$, there exist $\xi_0, ... , \xi_n$ in $[a,b]$ such that

$$ \sum_{i=0}^{n-1} \frac{|G_m(x_{i+1})-G_m(x_i)|^p}{(x_{i+1}-x_i)^{p-1}} = \sum_{i=0}^{n-1} \left|\frac{G_m(x_{i+1})-G_m(x_i)}{x_{i+1}-x_i}\right|^p(x_{i+1}-x_i) = \sum_{i=0}^{n-1} \big|g_m(\xi_i)|^p(x_{i+1}-x_i). $$

Now, since $|g_m|^p$is Riemann integrable on the interval, there exists a partition $\pi'$ such that

$$ S(\pi') < \int_a^b|g_m(t)|^pdt +1 = \|g_m\|_{p}^p + 1 $$

where $S(\pi')$ here denotes the upper sum. Now if $\pi \cup \pi' = \{y_j\}_{0\leq j \leq k}$ and $\delta_0 , ... , \delta_k$ are such that $\delta_j = \xi_i$ if $y_j \in [x_i,x_{i+1})$,

$$ \sum_{i=0}^{n-1} \big|g_m(\xi_i)|^p(x_{i+1}-x_i) = \sum_{j=0}^{k-1} \big|g_m(\delta_i)|^p(y_{j+1}-y_j) \leq S(\pi \cup \pi') < \|g_m\|_{p}^p + 1 $$

with this last inequality given by $S(\pi \cup \pi') \leq S(\pi')$, since $\pi \cup \pi'$ refines $\pi'$. Now,

\begin{align*} &\sum_{i=0}^{n-1} \frac{|F(x_{i+1})-F(x_i)|^p}{(x_{i+1}-x_i)^{p-1}} \leq \ ... \\ &\sum_{i=0}^{n-1} \frac{|F(x_{i+1}) - G_m(x_{i+1}) + G_m(x_i) - F(x_i)|^p}{(x_{i+1}-x_i)^{p-1}} + \sum_{i=0}^{n-1} \frac{|G_m(x_{i+1})-G_m(x_i)|^p}{(x_{i+1}-x_i)^{p-1}} \leq \\ &\sum_{i=0}^{n-1} \frac{2^p\|f-g_m\|_{1}^p}{(x_{i+1}-x_i)^{p-1}} + \|g_m\|_{p}^p + 1 \stackrel{\text{Hölder}}{\leq} \sum_{i=0}^{n-1} \frac{2^p\|f-g_m\|_{p}^p\|1\|_{p'}^p}{(x_{i+1}-x_i)^{p-1}} + \|g_m\|_{L^p}^p + 1 \end{align*}

Finally, since $\|1\|_{p'}^p < +\infty$, $\|g_m\|_{p} \to \|f\|_{p}$ and $\|f-g_m\|_{p} \to 0$,

$$ \sum_{i=0}^{n-1} \frac{|F(x_{i+1})-F(x_i)|^p}{(x_{i+1}-x_i)^{p-1}} \leq \|f\|_{p}^p + 1 $$

for each fixed partition $\pi$. Since the bound only depends on $f$, we have the desired result.

Does this argument seem sound? If so, in which ways can we optimize it?

Answer 1

A better estimate can be obtained as follows: $$ F(x_{i+1})-F(x_i)=\int_{x_i}^{x_{i+1}}f(t)dt=(x_{i+1}-x_i)\int_{0}^{1}f(x_{i}+t(x_{i+1}-x_i))dt$$ Hence, by Hölder’s inequality we have $$\eqalign{\vert F(x_{i+1})-F(x_i)\vert^p&=\vert x_{i+1}-x_i\vert^p\left\vert \int_{0}^{1}f(x_{i}+t(x_{i+1}-x_i))dt\right\vert^p\\ &\le ( x_{i+1}-x_i)^p \int_{0}^{1}\vert f(x_{i}+t(x_{i+1}-x_i))\vert^p dt \\ &\le ( x_{i+1}-x_i)^{p-1} \int_{x_{i}}^{x_{i+1}}\vert f(t)\vert^p dt }$$ Thus $$\sum_{i=0}^{n-1}\frac{\vert F(x_{i+1})-F(x_i)\vert^p}{( x_{i+1}-x_i)^{p-1} }\le \int_{x_0}^{x_n}\vert f(t)\vert^p dt=\Vert f\Vert^p_p$$ Note that equality holds when $f$ is constant, so this is the best inequality of this kind.