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

Abstract

Fix $b\in (0,\infty )$ and $p\in (1,\infty )$. Let $\varphi$ be a positive measurable function on $I_b:=(0,b)$. Define the Lorentz Gamma norm, $\r_{p,ϕ}$, at the measurable function $f:\mathbb{R}+\to \mathbb{R}+$ by $\rph(f):=[\int_0^bf^{**}(t)^pϕ(t)dt]^{\frac1p}$, in which $f^{\ast \ast }(t):=t^{-1}{\int }_0^t{f}^{\ast }(s)ds$, where $f^{\ast }(t):={\mu }_f^{-1}(t)$, with ${\mu }_f(s):=|\{x\in I_b:|f(x)|>s\}|$. Our aim in this paper is to study the rearrangement-invariant space determined by $\rph$. In particular, we determine its Köthe dual and its Boyd indices. Using the latter a sufficient condition is given for a Caldéron-Zygmund operator to map such a space into itself.