  {Consider a c\`adl\`ag local martingale $M$ with square brackets $[M]$. In
this paper, we provide upper and lower bounds for expectations of the type
${\mathbb E} [M]^{q/2}_{\tau}$, for any stopping time $\tau$ and $q\ge 2$, in
terms of predictable processes. This result can be thought of as a
Burkholder-Davis-Gundy type inequality in the sense that it can be used to
relate the expectation of the running maximum $|M^*|^q$ to the expectation of
the dual previsible projections of the relevant powers of the associated jumps
of $M$. The case for a class of moderate functions is also discussed.
