Let $u: [a,b] \to \mathbb{R}$ and define its $p$-Wiener variation as $$V_p(u) := \sup \left\{ \left(\sum_i |u(x_i)-u(x_{i-1})|^p\right)^{1/p}\right\}, $$ where the supremum ranges over finite partitions $P=\{x_1,\ldots, x_n\}$ of $[a,b]$.
For $p = 1$, it is clear (triangle inequality) that the sum over some partition $P$ will only increase if an extra point is added to $P$. Thus the supremum can always be achieved approximating with partitions for which the mesh size $\Delta(P) := \max_i (x_i-x_{i-1})$ tends to $0$.
However, for $p > 1$, adding a new point in to a given sum may actually decrease it, so one can construct a function (take a step function with 3 values) for which $V_p(u)$ is attained without $\Delta(P)$ tending to $0$. Intuitively this feels a little funny, as the $p$-variation is "missing" some parts of the function.
Question Is there any characterization of $u$ for when it is guaranteed that $\Delta(P) \to 0$ as $\left(\sum_i |u(x_i)-u(x_{i-1})|^p\right)^{1/p} \to V_p(u)$? I.e. when is the $p$-variation forced to "look at" the whole function?
