# The rearrangement-invariant space $$\Gamma _{p,\phi }$$ Γ p , ϕ

The rearrangement-invariant space $$\Gamma _{p,\phi }$$ Γ p , ϕ Fix $$b\in \mathbb R _+$$ b ∈ R + and $$p\in (1,\infty )$$ p ∈ ( 1 , ∞ ) . Let $$\phi$$ ϕ be a positive measurable function on $$I_b:=(0,b)$$ I b : = ( 0 , b ) . Define the Lorentz Gamma norm, $$\rho _{p,\phi }$$ ρ p , ϕ , at the measurable function $$f:\mathbb R _+\rightarrow \mathbb R _+$$ f : R + → R + by $$\rho _{{}_{p,\phi }}(f):=\left[ \int _0^bf^{**}(t)^p\phi (t)\,dt\right] ^{\frac{1}{p}}$$ ρ p , ϕ ( f ) : = ∫ 0 b f ∗ ∗ ( t ) p ϕ ( t ) d t 1 p , in which $$f^{**}(t):=t^{-1}\int _0^tf^{*}(s)\,ds$$ f ∗ ∗ ( t ) : = t - 1 ∫ 0 t f ∗ ( s ) d s , where $$f^*(t):=\mu _f^{-1}(t)$$ f ∗ ( t ) : = μ f - 1 ( t ) , with $$\mu _f(s):=|\{ x\in I_b: |f(x)|>s\}|$$ μ f ( s ) : = | { x ∈ I b : | f ( x ) | > s } | . Our aim in this paper is to study the rearrangement-invariant space determined by $$\rho _{{}_{p,\phi }}$$ ρ p , ϕ . 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# The rearrangement-invariant space $$\Gamma _{p,\phi }$$ Γ p , ϕ

, Volume 18 (2) – Jun 18, 2013
27 pages

/lp/springer_journal/the-rearrangement-invariant-space-gamma-p-phi-p-PiYUrFQCvH
Publisher
Springer Journals
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-013-0246-4
Publisher site
See Article on Publisher Site

### Abstract

Fix $$b\in \mathbb R _+$$ b ∈ R + and $$p\in (1,\infty )$$ p ∈ ( 1 , ∞ ) . Let $$\phi$$ ϕ be a positive measurable function on $$I_b:=(0,b)$$ I b : = ( 0 , b ) . Define the Lorentz Gamma norm, $$\rho _{p,\phi }$$ ρ p , ϕ , at the measurable function $$f:\mathbb R _+\rightarrow \mathbb R _+$$ f : R + → R + by $$\rho _{{}_{p,\phi }}(f):=\left[ \int _0^bf^{**}(t)^p\phi (t)\,dt\right] ^{\frac{1}{p}}$$ ρ p , ϕ ( f ) : = ∫ 0 b f ∗ ∗ ( t ) p ϕ ( t ) d t 1 p , in which $$f^{**}(t):=t^{-1}\int _0^tf^{*}(s)\,ds$$ f ∗ ∗ ( t ) : = t - 1 ∫ 0 t f ∗ ( s ) d s , where $$f^*(t):=\mu _f^{-1}(t)$$ f ∗ ( t ) : = μ f - 1 ( t ) , with $$\mu _f(s):=|\{ x\in I_b: |f(x)|>s\}|$$ μ f ( s ) : = | { x ∈ I b : | f ( x ) | > s } | . Our aim in this paper is to study the rearrangement-invariant space determined by $$\rho _{{}_{p,\phi }}$$ ρ p , ϕ . 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.

### Journal

PositivitySpringer Journals

Published: Jun 18, 2013

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