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

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-infinity(R, V(R)), PC((R) over bar, V(R)) and Lambda r(R, V-d(R)) on the weighted Lebesgue spaces L-p(R, w), with 1 < p < infinity and w is an element of A(p)(R). The Banach algebras L-infinity(R, V(R)) and PC((R) over bar, V(R)) consist, respectively, of all bounded measurable or piecewise continuous-valued functions on where is the Banach algebra of all functions on of bounded total variation, and the Banach algebra consists Lambda(r)(R, V-d(R)) of all Lipschitz V-d(R)-valued functions of exponent r is an element of (0,1] on where V-d(R) is the Banach algebra of all functions on of bounded variation on dyadic shells. Finally, for the Banach algebra u(p,w) generated by all pseudodifferential operators a(x, D) with symbols a(x, lambda) is an element of PC((R) over bar, V(R)) on the space L-p(R, w), we construct a non-commutative Fredholm symbol calculus and give a Fredholm criterion for the operators A is an element of U-p,U-w.