Let $f,g\in BV([0,T],\mathbb{R}^d)$ and $p,q\geq 1$ with $$ \theta= \frac{1}{p} + \frac{1}{q} >1.$$ Show that for $0\leq s\leq t\leq T, $ $$\left\| \int_s^t f dg- f(s)\cdot\left(g(t)-g(s)\right) \right\| \leq \frac{ V_p\left(f,[0,T]\right) V_q\left(g,[0,T]\right)}{1-2^{1-\theta}}.$$
Definitions:
$BV([0,T],\mathbb{R}^d)$ is the space of functions from [0,T] to $\mathbb{R}^d$ with finite variation.
A function $g:[0,T]\to \mathbb{R}^d$ is said to be of bounded variation if $$ V_1(g,[0,T])= \sup_{\tau\in \Pi([0,T])} \sum_\tau \|g(\tau_{k+1})-g(\tau_k)\| < \infty$$ where we denote by $\Pi([0,T])$ the set of all partitions of $[0,T]$. A partition of $[0,T]$ is a finite sequence $0=t_0<t_1...<t_{m}=T$. Denoting $\tau$ such a partition.
The p-variation of a function $g:[0,T]\to \mathbb{R}^d$ along a partition $\tau=(0=\tau_0< \tau_1<...\tau_n=T)$ as $$V_p(g,\tau,[0,T])= \sum_\tau \|g(\tau_{k+1})-g(\tau_k)\|^p.$$ Replace p with q for the definition of the q-variation of a function.
I have shown that $V_p(f,[0,T])<\infty$, but stuck without idea after proving that. Any hint or idea of how to use the scaling factor $\frac{1}{1-2^{1-\theta}}$ will be appreciated.
Thank you!