Let $(u_t,{\cal G}_t)_{t\geq 0}$ be Azéma martingale and its filtration, and let $(\lambda^x_t; x \in \R, t\geq 0)$ be the local times of the Azéma martingale defined by the following Tanaka formula: $$ u_t1_{\{u_t>x\}}=\int_0^t 1_{\{u_{s^-}>x\}}\d u_s +\half \lambda_t^x . $$ Then, for every $({\cal G}_t)_{t\geq 0}$ stopping time T and every p>0, there exist two universal constants cp, Cp >0 depending only on p, such that $$ c_p\|T^{1/2}\|_p \leq \|\lambda^*_T\|_p \leq C_p \|T^{1/2}\|_{p}, $$ where $\lambda^*_t=\sup_{x\in \R} \lambda^x_t$ .