The Rearrangement-Invariant space $\Gamma_{p,\phi}$

Abstract

Fix $b\in (0,\infty)$ and $p\in (1,\infty)$. Let $\phi$ be a positive measurable function on $I_b:=(0,b)$. Define the Lorentz Gamma norm, $\r_{p,\phi}$, at the measurable function $f:\R+\to\R+$ by $\rph(f):=[\int_0^bf^{**}(t)^p\phi(t)dt]^{\frac1p}$, in which $f^{**}(t):=t^{-1}\int_0^tf^{*}(s)ds$, where $f^*(t):=\mu_f^{-1}(t)$, 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 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.