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For martingales of the form $X_{t} = \int_{0}^{t}f(s,\omega)dB_{s}$ where $B_{t}$ is the standard Brownian motion, $f(s, \omega)$ is non-anticipating function such that $\int_{0}^{\infty}f^{2}(s, \omega)ds <\infty$ a.s. (I guess they are called $L^{2}$ integrable continuous martingales) one can obtain the bound of order $\sqrt{p}$ for large $p$. This is due to Burgess Davis, (see Section 3) of the reference. The result essentially follows from the Brownian case with arbitrary $L^p$ integrable stopping times.

However, for discrete time martingales, this is not true, and the best possible bound is of order $p$ for large $p$, for example, see paper of G. Wang, Remark 2. On the other hand, there are very special ``conditionally symmetric'' discrete martingales for which one can obtain the bound of order $\sqrt{p}$. And it looks like continuous $L^{2}$ integrable martingales in this sense are similar to conditionally symmetric ones.

*Wang, Gang*, **Sharp inequalities for the conditional square function of a martingale**, Ann. Probab. 19, No.4, 1679-1688 (1991). ZBL0744.60046.

*Davis, Burgess*, **On the $L^p$ norms of stochastic integrals and other martingales**, Duke Math. J. 43, 697-704 (1976). ZBL0349.60061.