Boundedness and Compactness of Pseudodifferential Operators with Non-Regular Symbols on Weighted Lebesgue Spaces

Abstract

Applying the boundedness on weighted Lebesgue spaces of the maximal singular integral operator S * related to the Carleson–Hunt theorem on almost everywhere convergence, we study the boundedness and compactness of pseudodifferential operators a(x, D) with non-regular symbols in \({L^\infty(\mathbb{R}, V(\mathbb{R})), PC(\overline{\mathbb{R}}, V(\mathbb{R}))}\) and \({\Lambda_\gamma(\mathbb{R}, V_d(\mathbb{R}))}\) on the weighted Lebesgue spaces \({L^p(\mathbb{R},w)}\) , with 1 < p < ∞ and \({w\in A_p(\mathbb{R})}\) . The Banach algebras \({L^\infty(\mathbb{R}, V(\mathbb{R}))}\) and \({PC(\overline{\mathbb{R}}, V(\mathbb{R}))}\) consist, respectively, of all bounded measurable or piecewise continuous \({V(\mathbb{R})}\) -valued functions on \({\mathbb{R}}\) where \({V(\mathbb{R})}\) is the Banach algebra of all functions on \({\mathbb{R}}\) of bounded total variation, and the Banach algebra \({\Lambda_\gamma(\mathbb{R}, V_d(\mathbb{R}))}\) consists of all Lipschitz \({V_d(\mathbb{R})}\) -valued functions of exponent \({\gamma \in (0,1]}\) on \({\mathbb{R}}\) where \({V_d(\mathbb{R})}\) is the Banach algebra of all functions on \({\mathbb{R}}\) of bounded variation on dyadic shells. Finally, for the Banach algebra \({\mathfrak{A}_{p,w}}\) generated by all pseudodifferential operators a(x, D) with symbols \({a(x, \lambda) \in PC(\overline{\mathbb{R}}, V(\mathbb{R}))}\) on the space \({L^p(\mathbb{R}, w)}\) , we construct a non-commutative Fredholm symbol calculus and give a Fredholm criterion for the operators \({A \in \mathfrak{A}_{p,w}}\) .