AN ALGEBRA OF PSEUDODIFFERENTIAL OPERATORS WITH SLOWLY OSCILLATING SYMBOLS

Abstract

Let $V(\mathbb{R})$ denote the Banach algebra of absolutely continuous functions of bounded total variation on $\mathbb{R}$, and let $\mathcal{B}_p$ be the Banach algebra of bounded linear operators acting on the Lebesgue space $L^p(\mathbb{R})$ for $1 < p < \infty$. We study the Banach algebra $\mathfrak{A}\subset\mathcal{B}_p$ generated by the pseudodifferential operators of zero order with slowly oscillating $V(\mathbb{R})$-valued symbols on $\mathbb{R}$. Boundedness and compactness conditions for pseudodifferential operators with symbols in $L^\infty (\mathbb{R}, V(\mathbb{R}))$ are obtained. A symbol calculus for the non-closed algebra of pseudodifferential operators with slowly oscillating $V(\mathbb{R})$-valued symbols is constructed on the basis of an appropriate approximation of symbols by infinitely differentiable ones and by use of the techniques of oscillatory integrals. As a result, the quotient Banach algebra $\mathfrak{A}^\pi = {\mathfrak A} / \mathcal{K}$, where $\mathcal{K}$ is the ideal of compact operators in $\mathcal{B}_p$, is commutative and involutive. An isomorphism between the quotient Banach algebra $\mathfrak{A}^\pi$ of pseudodifferential operators and the Banach algebra $\widehat{\mathfrak{A}}$ of their Fredholm symbols is established. A Fredholm criterion and an index formula for the pseudodifferential operators $A \in \mathfrak{A}$ are obtained in terms of their Fredholm symbols.