On Convolution Type Operators with Piecewise Slowly Oscillating Data

Abstract

Applying the theory of pseudodifferential and Calderón-Zygmund operators, we study the compactness of commutators of multiplication operators aI and convolution operators \( W^{0}(b) \) on weighted Lebesgue spaces \( L^{p}(\mathbb{R},w)\;{\rm with}\; p \in (1,\infty)\) and Muckenhoupt weights ?? for some classes of piecewise slowly oscillating functions \( a \in PSO^{\diamond} \; and b \in PSO^{\diamond}_{p,w}\) on the real line ℝ. Then we study the Banach algebra \( Z_{p,w} \) generated by the operators \( aW^{0}(b)\) with functions \( a \in SO^{\diamond} \; and b \in SO^{\diamond}_{p,w}\) admitting slowly oscillating discontinuities at every point \( lambda \in \mathbb{R} \cup \left\{\infty \right\} \). Applying the method of limit operators under some condition on Muckenhoupt weights w, we describe the maximal ideal space of the commutative quotient Banach algebra \( Z^{\pi}_{p,w}\;=\;Z_{p,w}/K_{p,w}{\rm where}\;K_{p,w}\) is the ideal of compact operators on \( L^{p}(\mathbb{R,w}) \) define the Gelfand transform for \( Z^{\pi}_{p,w}) \) and establish the Fredholmness for the operators \( A\in Z_{p,w}\)