= Solution
Let $(Z_k,\mathcal F_k)_{0\leq k\leq n}$ be an <integrable> real <submartingale>, and put $Z_n^*=\max_{0\leq k\leq n}Z_k$. The refined <Doob maximal inequality> is
$$
\boxed{\lambda\,\mathbb P(Z_n^*\geq\lambda)
\leq\mathbb E[Z_n\mathbf1_{\{Z_n^*\geq\lambda\}}],\qquad\lambda>0.}
$$
In particular, for a nonnegative <submartingale> the right-hand side is at most $\mathbb EZ_n$, giving the usual weak maximal bound.
For the proof, partition $A=\{Z_n^*\geq\lambda\}$ by its first crossing time: $A_k=\{Z_j<\lambda\ (j<k),\ Z_k\geq\lambda\}\in\mathcal F_k$. The <submartingale> property gives $\mathbb E[Z_n\mid\mathcal F_k]\geq Z_k$, so
$$
\mathbb E[Z_n\mathbf1_{A_k}]
=\mathbb E[\mathbf1_{A_k}\mathbb E(Z_n\mid\mathcal F_k)]
\geq\mathbb E[Z_k\mathbf1_{A_k}]
\geq\lambda\mathbb P(A_k).
$$
Summing over the disjoint crossing events proves the refined inequality, including a possible crossing at time zero. No <independence> assumption is needed.
The strong form is the <Doob Lp maximal inequality>. For $p>1$ and a nonnegative <submartingale> with $Z_n\in L^p$,
$$
\boxed{\|Z_n^*\|_p\leq\frac p{p-1}\|Z_n\|_p.}
$$
Indeed, integrate the refined inequality against $p\lambda^{p-2}$ up to $K$. The <tail integral formula for moments> and the <Tonelli theorem> give
$$
\mathbb E[(Z_n^*\wedge K)^p]
\leq\frac p{p-1}\mathbb E[Z_n(Z_n^*\wedge K)^{p-1}].
$$
Apply the <Holder inequality>, divide by $\|Z_n^*\wedge K\|_p^{p-1}$ when it is nonzero, and let $K\to\infty$ by the <monotone convergence theorem>. For a <martingale> $M$, the process $|M_k|$ is a <submartingale> by the <conditional Jensen inequality>, giving the corresponding bound for $\max|M_k|$. On a bounded continuous-time interval the nonnegative, right-continuous version follows by applying these inequalities on refining finite time grids and taking limits.
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