A Estimation about Hölder condition

Question

Let $p:[0,\inf) \to \mathbb{R}$ be a contionous function such that $p(0)=0$
Fix $a>1/2 , k$ is a positive integer $>\frac{1}{a-\frac{1}{2}}$.
Suppose for all $n \in \mathbb{N}$ and $\lambda >0$
If there is a $t_0 \in [0,1]$ so that $$\sup_{0<h\leq \frac{k+1}{n}} \frac{|p(t_0+h)-p(t_0)|}{h^a} \leq \lambda$$ Show that there is a $j_0 \in \mathbb{N}, 0\leq j_0 \leq n-1$, so that $$\max_{1\leq l \leq k} |p\bigg(\frac{j_0+l+1}{n}\bigg)-p\bigg(\frac{j_0+l}{n}\bigg)| \leq C_k \lambda n^{-a}$$ where $C_k = 2(k+1)^{a}$

Answer 1

Fix $1 \leq l \leq k$, and $t_0 \in[0, 1]$ as in the assumption.

Now, choose $0 \leq j_0 \leq n - 1$ such that $\left|\frac{j_0 + 1}{n} - t_0\right| \leq \frac{1}{n}$, then we have

\begin{align*} \left|p(\frac{j_0 + l + 1}{n}) - p(\frac{j_0 + l}{n})\right| &\leq \left|p(\frac{j_0 + l + 1}{n}) - p(t_0)\right| + \left|p(\frac{j_0 + l}{n}) - p(t_0)\right| \\ &\leq \lambda \left(\left(\frac{l + 1}{n}\right)^a + \left(\frac{l}{n}\right)^a\right) \\ &\leq 2\left(k + 1\right)^a\lambda n^{-a}. \end{align*}

Since the estimate is independent on $1 \leq l \leq k$, we have $$\max_{1 \leq l \leq k}\left|p(\frac{j_0 + l + 1}{n}) - p(\frac{j_0 + l}{n})\right| \leq 2\left(k + 1\right)^a\lambda n^{-a}.$$

Remark: Note that the estimate depends only on the assumption given $n, \lambda$, there exists $t_0 \in [0, 1]$ such that $$\sup_{0 < h \leq \frac{k + 1}{n}} \frac{\left|p(t_0 + h) - p(t_0)\right|}{h^a} \leq \lambda.$$ Whether $p$ being continuous or not or $p(0) = 0$ or not do not matter. Besides, there has nothing to do with $a > \frac{1}{2}$ or $k > \frac{1}{a - \frac{1}{2}}$.