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

TL;DR

This paper investigates the properties of a specific rearrangement-invariant space defined via Lorentz Gamma norms, including its dual space, Boyd indices, and conditions for boundedness of certain operators.

Contribution

It characterizes the K"othe dual and Boyd indices of the space $ _{p,}$, and provides a sufficient condition for Caldéron-Zygmund operators to act boundedly on it.

Findings

Determined the K"othe dual of the space.

Computed Boyd indices for the space.

Established a sufficient condition for operator boundedness.

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.