The Rearrangement-Invariant space ${\mathrm{\Gamma }}_{p,\varphi }$

Fix $b\in (0,\mathrm{\infty })$ and $p\in (1,\mathrm{\infty })$. Let $\varphi$ be a positive measurable function on $I_b:=(0,b)$. Define the Lorentz Gamma norm, \r_{p,\phi}, at the measurable function $f:\mathbb{R}+\to \mathbb{R}+$ by $\text{rph}(f):=[{\int }_0^b{f}^{\ast \ast }(t{)}^p\varphi (t)dt{]}^{\frac{1}{p}}$, in which $f^{\ast \ast }(t):=t^{-1}{\int }_0^t{f}^{\ast }(s)ds$, where $f^{\ast }(t):={\mu }_f^{-1}(t)$, with ${\mu }_f(s):=\mathrm{\mid }\{x\in I_b:\mathrm{\mid }f(x)\mathrm{\mid }>s\}\mathrm{\mid }$. Our aim in this paper is to study the rearrangement-invariant space determined by $\text{rph}$. In particular, we determine its K"othe dual and its Boyd indices. Using the latter a sufficient condition is given for a Cald'eron-Zygmund operator to map such a space into itself.