Integral with respect to the $G$-Brownian local time

Let ${\mathscr L}$ be the local time of $G$-Brownian motion $B$. In this paper, we prove the existence of the quadratic covariation $<f(B),B>_{t}$ and the integral $\int_{\mathbb R}f(x){\mathscr L}(dx,t)$. Moreover, a sublinear version of the Bouleau-Yor identity $$\int_{\mathbb R}f(x){\mathscr L}(dx,t)=-<f(B),B>_{t}$$ is showed to hold under some suitable conditions. These allow us to write the It\^o's formula for $C^1$-functions.