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

摘要

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 ${win A_p(mathbb{R})}$ . 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 ${A in mathfrak{A}_{p,w}}$ .