## Communications in Mathematical Analysis

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

### Commutators of Convolution Type Operators with Piecewise Quasicontinuous Data

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

#### 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