The Question: Let $X$ be a continuous process, and suppose $0 < p < q$.
Prove the case $V_t^p(X) < \infty \implies V_t^q(X) = 0$.
Definitions:
The standard setup.
$\Pi := \{t_0,t_1,...,t_N\}$ where $0 = t_0 \leq t_1 \leq ... \leq t_N = t$.
$||\Pi|| := \text{max}_{1 \leq i \leq N}|t_i - t_{i-1}|$
$V_t^a(X) := \lim_{||\Pi||\to 0}\sum_{i=1}^N |X_{t_i} - X_{t_{i-1}}|^a$
My Current Progress:
$V_t^q(X) = \lim_{||\Pi||\to 0}\sum_{i=1}^N |X_{t_i} - X_{t_{i-1}}|^p |X_{t_i} - X_{t_{i-1}}|^{q-p}$.
By the Holder Inequality, for $\frac1a + \frac1b = 1$:
$V_t^q(X) = \lim_{||\Pi|| \to 0}\sum_{i=1}^N|X_{t_i}-X_{t_{i-1}}|^p|X_{t_i}-X_{t_{i-1}}|^{q-p} $ $ V_t^q(X) \leq \lim_{||\Pi|| \to 0}\Big[\sum_{i=1}^N(|X_{t_i}-X_{t_{i-1}}|^p)^a\Big]^\frac1b \Big[\sum_{i=1}^N(|X_{t_i}-X_{t_{i-1}}|^{p-q})^b\Big]^\frac1a $ $ = \lim_{||\Pi|| \to 0}\Big[\sum_{i=1}^N(|X_{t_i}-X_{t_{i-1}}|^p)^a\Big]^\frac1b \lim_{||\Pi|| \to 0}\Big[\sum_{i=1}^N(|X_{t_i}-X_{t_{i-1}}|^{p-q})^b\Big]^\frac1a $ My Request:
Please help me to progress further on this interesting problem.