I am studying the local time of Brownian motion from Karatzas and Shreve book 'Brownian motion and stochastic calculus'.
I've found that $\frac{1}{4 \epsilon}\int_0^T \mathbf{1}_{W_t \in [a-\epsilon,a+ \epsilon ]} dt$ converges in $L^2$ sense to $|W_T-a|-|a|-\int_0^T sign(W_s-a)dW_s$.
Is it true that the convergence holds in any $L^p$ with $p<\infty$? Can it be proved using Burkholder-Davis-Gundy inequality?
