Algebras of Convolution-Type Operators with Piecewise Slowly Oscillating Data on Weighted Lebesgue Spaces

Abstract

Let ${\mathcal B}_{p,w}$ be the Banach algebra of all bounded linear operators acting on the weighted Lebesgue space $L^p(\mathbb {R},w)$, where $p\in (1,\infty )$ and w is a Muckenhoupt weight. We study the Banach subalgebra $\mathfrak {A}_{p,w}$ of ${\mathcal B}_{p,w}$ generated by all multiplication operators aI ($a\in \mathrm{PSO}^\diamond$) and all convolution operators $W^0(b)$ ($b\in \mathrm{PSO}_{p,w}^\diamond$), where $\mathrm{PSO}^\diamond \subset L^\infty (\mathbb {R})$ and $\mathrm{PSO}_{p,w}^\diamond \subset M_{p,w}$ are algebras of piecewise slowly oscillating functions that admit piecewise slowly oscillating discontinuities at arbitrary points of $\mathbb {R}\cup \{\infty \}$, and $M_{p,w}$ is the Banach algebra of Fourier multipliers on $L^p(\mathbb {R},w)$. For any Muckenhoupt weight w, we study the Fredholmness in the Banach algebra ${\mathcal Z}_{p,w}\subset \mathfrak {A}_{p,w}$ generated by the operators $aW^0(b)$ with slowly oscillating data $a\in \mathrm{SO}^\diamond$ and $b\in \mathrm{SO}^\diamond _{p,w}$. Then, under some condition on the weight w, we complete constructing a Fredholm symbol calculus for the Banach algebra $\mathfrak {A}_{p,w}$ in comparison with Karlovich and Loreto Hernández (Integr. Equations Oper. Theory 74:377–415, 2012) and Karlovich and Loreto Hernández (Integr. Equations Oper. Theory 75:49–86, 2013) and establish a Fredholm criterion for the operators $A\in \mathfrak {A}_{p,w}$ in terms of their symbols. A new approach to determine local spectra is found.