Open Access
2014 Commutators of Convolution Type Operators with Piecewise Quasicontinuous Data
I. De la Cruz-Rodriguez, Yu. I. Karlovich, I. Loreto-Hernandez
Commun. Math. Anal. 17(2): 131-150 (2014).


Applying the theory of Calderon-Zygmund operators, we study the compactness of the commutators $[aI,W^0(b)]$ of multiplication operators $aI$ and convolution operators $W^0(b)$ on weighted Lebesgue spaces $L^p({\mathbb R},w)$ with $p\in(1,\infty)$ and Muckenhoupt weights $w$ for some classes of piecewise quasicontinuous functions $a\in PQC$ and $b\in PQC_{p,w}$ on the real line ${\mathbb R}$. Then we study two $C^*$-algebras $Z_1$ and $Z_2$ generated by the operators $aW^0(b)$, where $a,b$ are piecewise quasicontinuous functions admitting slowly oscillating discontinuities at $\infty$ or, respectively, quasicontinuous functions on ${\mathbb R}$ admitting piecewise slowly oscillating discontinuities at $\infty$. We describe the maximal ideal spaces and the Gelfand transforms for the commutative quotient $C^*$-algebras $Z_i^\pi=Z_i/{\mathcal K}$ $(i=1,2)$ where ${\mathcal K}$ is the ideal of compact operators on the space $L^2({\mathbb R})$, and establish the Fredholm criteria for the operators $A\in Z_i$.


Download Citation

I. De la Cruz-Rodriguez. Yu. I. Karlovich. I. Loreto-Hernandez. "Commutators of Convolution Type Operators with Piecewise Quasicontinuous Data." Commun. Math. Anal. 17 (2) 131 - 150, 2014.


Published: 2014
First available in Project Euclid: 18 December 2014

zbMATH: 1319.47033
MathSciNet: MR3292964

Primary: 47B47

Keywords: BMO and VMO functions , commutator , Convolution type operator , Fredholmness , Gelfand transform , maximal ideal space , piecewise quasicontinuous function , slowly oscillating function

Rights: Copyright © 2014 Mathematical Research Publishers

Vol.17 • No. 2 • 2014
Back to Top