Communications in Mathematical Analysis

Commutators of Convolution Type Operators with Piecewise Quasicontinuous Data

I. De la Cruz-Rodriguez, Yu. I. Karlovich, and I. Loreto-Hernandez

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

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 $A\in Z_i$.

Article information

Source
Commun. Math. Anal., Volume 17, Number 2 (2014), 131-150.

Dates
First available in Project Euclid: 18 December 2014

Permanent link to this document
https://projecteuclid.org/euclid.cma/1418919760

Mathematical Reviews number (MathSciNet)
MR3292964

Zentralblatt MATH identifier
1319.47033

Subjects
Primary: 47B47: Commutators, derivations, elementary operators, etc.

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

Citation

De la Cruz-Rodriguez, I.; Karlovich, Yu. I.; Loreto-Hernandez, I. Commutators of Convolution Type Operators with Piecewise Quasicontinuous Data. Commun. Math. Anal. 17 (2014), no. 2, 131--150. https://projecteuclid.org/euclid.cma/1418919760


Export citation