the functional $\varrho _{p,q}$ by $$\begin{aligned}
\varrho _{p,q}(f)=\|f\|_{p,q}=
\left \|  s^{\frac{1}{p}-\frac{1}{q}}f
^{*}(s)\right \|  _{q}\end{aligned}$$ for $f \in \mathcal{M}_{+}(R,\mu )$. The set $L^{p,q}$, defined as the collection of all $f\in \mathcal{M}(R,\mu )$ satisfying $\varrho _{p,q}(|f|)<
\infty $, is called a *Lorentz space*. If either $1<p<\infty $ and $1\leq q\leq \infty $ or $p=q=1$ or $p=q=\infty $, then $\varrho _{p,q}$ is equivalent to an r.i. norm in the sense that there exists an r.i. norm $\sigma $ and a constant $C$, $0<C<\infty $, depending on $p,q$ but independent of $f$, such that $$\begin{aligned}
C^{-1}\sigma (f)\leq \varrho _{p,q}(f)\leq C\sigma (f).\end{aligned}$$ As a consequence, $L^{p,q}$ is considered to be an r.i. space for these cases of $p,q$, see [@BS Chapter 4]. If either $0<p<1$ or $p=1$ and $q>1$, then $L^{p,q}$ is a quasi-normed space. If $p=\infty $ and $q<\infty $, then $L^{p,q}=\{0\}$. For every $p\in [1,\infty ]$, we have $L^{p,p}=L^{p}$. Furthermore, if $p,q,r\in (0,\infty ]$ and $q\leq r$, then the inclusion $L^{p,q}\subset L^{p,r}$ holds.

If ${\mathbb{A}}=[\alpha _{0},\alpha _{\infty }]\in \mathbb{R}^{2}$ and $t\in \mathbb{R}$, then we shall use the notation ${\mathbb{A}}+t=[
\alpha _{0}+t,\alpha _{\infty }+t]$.

Let $0<p,q\le \infty $, ${\mathbb{A}}=[\alpha _{0},\alpha _{\infty }]
\in \mathbb{R}^{2}$ and ${\mathbb{B}}=[\beta _{0},\beta _{\infty }]
\in \mathbb{R}^{2}$. Then we define 