Young's theorem integrates a Hölder function of unbounded variation against itself

By [L1], $0<S<\infty$ and $\sum_nw_n=1$, so the intervals tile $(0,1]$. On one tent, the linear slope estimate and [L2] give $|f(x)-f(y)|\le2|x-y|^{3/4}$. If $x,y$ lie in different tents, let $z_x$ be the endpoint of $x$'s tent toward $y$ and $z_y$ the endpoint of $y$'s tent toward $x$. Both have value zero and $|x-z_x|,|y-z_y|\le|x-y|$, so the two one-tent estimates and the triangle inequality give $|f(x)-f(y)|\le4|x-y|^{3/4}$. Taking the endpoints of successively smaller tents gives the same estimate at zero. Thus $f$ is $3/4$-Hölder.

A partition through the endpoints and peaks of the first $N$ tents has variation at least [given] $2\sum_{n=1}^N w_n^{3/4}=2S^{-3/4}\sum_{n=1}^N\frac1n.$ This is unbounded by [L1], so $f$ is not BV.

Since $3/4+3/4>1$, [L3] nonetheless gives existence of $\int_0^1 f\,df$. This is genuinely outside the BV existence theorem.