A weighted analogue of the Carleson-Hunt theorem and new classes of
pseudodifferential operators
Applying a weighted analogue of the Carleson-Hunt theorem on almost
everywhere convergence, we study the boundedness and compactness of
pseudodifferential operators with symbols that are bounded
measurable functions with respect to the spatial variable and
functions of bounded variation with respect to the dual variable.
Replacement of absolutely continuous functions of bounded variation
by arbitrary functions of bounded variation allows us to study
essentially more general classes of pseudodifferential operators. A
symbol calculus and a Fredholm theory for new classes of
pseudodifferential operators with non-regular symbols are
constructed. In particular, we study pseudodifferential operators
with symbols admitting discontinuities of first kind with respect
to spatial and dual variables that generate non-commutative
algebras of Fredholm symbols.
