Let $p$ be a real number greater than $1$. It is well known (see Hall and Heyde's *Martingale limit theory and its applications*, Theorem 2.10) that there exists a constant $C\_p$ such that if $(X\_i)\_{i=1}^n$ is a real valued martingale difference with respect to the filtration $(\mathcal F\_i)\_{i=1}^n$ (that is, $(S\_j:=\sum\_{i=1}^jX\_i)\_{j=1}^n$ is a martingale with respect to this filtration), then 
$$\frac 1{C\_p}\mathbb E\left(\sum\_{i=1}^nX\_i^2\right)^{p/2}\leqslant \mathbb E\left|\sum\_{i=1}^nX\_i\right|^p\leqslant C\_p\mathbb E\left(\sum\_{i=1}^nX\_i^2\right)^{p/2}.$$
Hence the $\mathbb L^p$ norm of the partial sum is controlled by those of the quadratic variation. 


Now define for a real valued random variable $X$:
$$\lVert X\rVert\_{p,\infty}:=\left(\sup\_{t\geqslant 0}t^p\mu\{|X|\geqslant t\}\right)^{1/p}.$$
This is equivalent to a norm (namely $N(X):=\sup\_{\mu(A)>0}\mu(A)^{-1+1/p}\int\_A|X|\mathrm d\mu$).


I would like to know whether there is a similar inequality to Rosenthal's one, that is, a control of $N(S\_n)$ in terms of those of $N\left(\sqrt{\sum\_{i=1}^nX\_i^2}\right)$ plus maybe an other term. This seems to be a natural question which has probably been investigated, but I didn't manage to find a reference. 


There are weak-$\mathbb L^p$ versions of Rosenthal's inequality for independent random variables, but I would like to see a reference dealing with an extension to martingale differences, namely:


**Let $(\Omega,\mathcal F,\mu)$ be a probability space $p\gt 2$. Is there a constant $C\_p$ such that if $n$ is an integer and $(X\_j)\_{1\leqslant j\leqslant n}$ is a martingale difference with respect to the filtration $(\mathcal F\_j)\_{1\leqslant j\leqslant n}$ with $\lVert X\_j\rVert\_{p,\infty}\lt\infty$, then 
$$C\_p^{-1}\left\lVert \sqrt{\sum\_{j=1}^nX\_j^2}\right\rVert\_{p,\infty}\leqslant \left\lVert \sum\_{j=1}^nX\_j\right\rVert\_{p,\infty}\leqslant 
C\_p\left\lVert \sqrt{\sum\_{j=1}^nX\_j^2}\right\rVert\_{p,\infty}~? $$** 



Rosenthal like inequality for weak $\mathbb L^p$-norms
The result you want is mentioned in Remark 6 in


Johnson, W. B.(1-TXAM); Schechtman, G.(IL-WEIZ)
Martingale inequalities in rearrangement invariant function spaces. 
Israel J. Math. 64 (1988), no. 3, 267–275 (1989).


