As written in my paper [1], the inequality $$P(f^*>r) \le 2\exp\big(-r^2/2(p-1)\big) $$ in Theorem 3 in [1] for martingales in $\mathcal{X}=L^p$ can be compared with the inequality $$ P(f^*>r) \le C (r+1)\exp(-r^2/2) $$ with an unidentified absolute constant $C>0$, given (as noted in the comment by Nate Eldredge) in Theorem 5.3 of Kallenberg and Sztencel, proved for $\mathcal{X} = L^2$.

As for the inequality $$P(f^*>r) \le C \frac{1+r}{1+rc}\exp\big(-\frac{r}{2c} \ln(1+rc)\big) $$ with an absolute constant $C>0$, which is inequality (5.2) of Kallenberg and Sztencel, it can be compared with the inequality in Theorem 2 of [1].

More general results were given in [3].

Also, you wrote: "I still cannot see the path he takes to prove his theorem." Can you specify the steps that seem unclear?

As written in my paper [1], the inequality $$P(f^*>r) \le 2\exp\big(-r^2/2(p-1)\big) $$ in Theorem 3 in [1] for martingales in $\mathcal{X}=L^p$ can be compared with the inequality $$ P(f^*>r) \le C (r+1)\exp(-r^2/2) $$ with an unidentified absolute constant $C>0$, given (as noted in the comment by Nate Eldredge) in Theorem 5.3 of [Kallenberg and Sztencel], proved for $\mathcal{X} = L^2$.

As for the inequality $$P(f^*>r) \le C \frac{1+r}{1+rc}\exp\big(-\frac{r}{2c} \ln(1+rc)\big) $$ with an absolute constant $C>0$, which is inequality (5.2) of [Kallenberg and Sztencel], it can be compared with the inequality in Theorem 2 of [1].

More general results were given in [3].

Also, you wrote: "I still cannot see the path he takes to prove his theorem." Can you specify the steps that seem unclear?