with the operation of the nonincreasing rearrangement of a measurable function.

Throughout this section, let $(R,\mu )$ be a $\sigma $-finite nonatomic measure space. We set $$\begin{aligned}
\mathcal{M}(R,\mu )= \{f: f \mbox{ is a } \mu {-}\mbox{measurable function on }R \mbox{ with values in }[-\infty ,\infty ]\},\end{aligned}$$ $$\begin{aligned}
\mathcal{M}_{0}(R,\mu )= \{f \in \mathcal{M}(R,\mu ): f  \mbox{ is
finite } \mu \mbox{-a.e. on } R\}\end{aligned}$$ and $$\begin{aligned}
\mathcal{M}_{+}(R,\mu )= \{f \in \mathcal{M}(R,\mu ): f \geq 0\}.\end{aligned}$$ The *nonincreasing rearrangement* $f^{*} \colon [0,\infty )
\to [0, \infty ]$ of a function $f\in \mathcal{M}(R,\mu )$ is defined as $$\begin{aligned}
f^{*}(t)=\inf \{\lambda \in (0,\infty ): \mu(\{s\in R: |f(s)|>
\lambda \})\leq t\},\ t\in [0,\infty ).\end{aligned}$$ The *maximal nonincreasing rearrangement* $f^{**} \colon (0,
\infty ) \to [0, \infty ]$ of a function $f\in \mathcal{M}(R,\mu )$ is defined as $$\begin{aligned}
f^{**}(t)=\frac{1}{t}\int _{0}^{t} f^{*}(s)\,\mathrm{d}s,\quad t\in (0,\infty ).\end{aligned}$$ If $|f|\leq |g|$ $\mu $-a.e. in $R$, then $f^{*}\leq g^{*}$. The operation $f\mapsto f^{*}$ does not preserve sums or products of functions, and is known not to be subadditive. The lack of subadditivity of the operation of taking the nonincreasing rearrangement is, up to some extent, compensated by the following fact [@BS Chapter 2, (3.10)]: for every $t\in (0,\infty )$ and 