Martingale inequalities in symmetric spaces with semimultiplicative weight

Let $(\Omega,\Sigma,P)$ be a complete probability space, $\mathcal F=\{\mathcal F_n\}^\infty_{n=0}$ be an increasing sequence of $\sigma$-algebras such that $\cup^\infty_{n=0}\mathcal F_n$ generates $\Sigma$. If $f=\{f_n\}^\infty_{n=0}$ is a martingale with respect to $\mathcal F$ and $\mathbb E_n$ is the conditional expectation with respect to $\mathcal F_n$, then one can introduce a maximal function $M(f)=\sup_{n\geq 0}|f_n|$ and a square function $S(f)=\left(\sum\limits^\infty_{i=0}|f_i-f_{i-1}|^2\right)^{1/2}$, $f_{-1}=0$. In the case of uniformly integrable martingales there exists $g\in L^1(\Omega)$ such that $\mathbb E_ng=f_n$ and we consider a sharp maximal function $f^\sharp=\sup_{n\geq 0}\mathbb E_n|g-f_{n-1}|$. The result of Burkholder – Davis – Gundy is that $C_1\|M(f)\|_p\leq \|S(f)\|_p\leq C_2\|M(f)\|$ for $1<p<\infty$, where $\|\cdot\|_p$ is the norm in $L^p(\Omega)$ and $C_2>C_1>0$. We call the inequality of type $\|M(f)\|_p\leq C\|f^\sharp\|_p$, $1<p<\infty$ Fefferman – Stein inequality. It is known that Burkholder – Davis – Gundy martingale inequality is valid in rearrangement invariant Banach function spaces with non-trivial Boyd indices. We prove this inequality in a more wide class of symmetric spaces (the last notion is defined as in the famous monograph by S. G. Krein, Yu. I. Petunin and E. M. Semenov) with semimultiplicative weight. Also, the Fefferman – Stein type inequalities of sharp maximal function and sharp square functions are obtained in this class of symmetric spaces.