Singular integral operators with piecewise continuous coefficients in reflexive rearrangement-invariant spaces

The paper is devoted to some only recently uncovered phenomena emerging in the study of singular integral operators (SIO's) with piecewise continuous (PC) coefficients in reflexive rearrangement-invariant spaces over Carleson curves. We deal with several kinds of indices of submultiplicative functio ns which describe properties of spaces (Boyd and Zippin indices) and curves (spirality indices). We consider some "disintegration condition" which com bines properties of spaces and curves, the Boyd and spirality indices. We show that the essential spectrum of SIO associated with the Riemann boun dary value problem with PC coefficient arises from the essential range of t he coefficient by filling in certain massive connected sets (so-called loga rithmic leaves) between the endpoints of jumps. These results combined with the Allan-Douglas local principle and with the two projections theorem enable us to study the Banach algebra U generated b y SIO's with matrix-valued piecewise continuous coefficients. We construct a symbol calculus for this Banach algebra which provides a Fredholm criteri on and gives a basis for an index formula for arbitrary SIO's from U in ter ms of their symbols.