A. Yu, ALGEBRAS OF SINGULAR INTEGRAL-OPERATORS WITH PIECEWISE CONTINUOUS COEFFICIENTS ON REFLEXIVE ORLICZ SPACES, Mathematische Nachrichten, 179, 1996, pp. 187-222
We consider singular integral operators with piecewise continuous coef
ficients on reflexive Orlicz spaces L(M)(Gamma), which are generalizat
ions of the Lebesgue spaces L(p)(Gamma), 1 < p < infinity. We suppose
that Gamma belongs to a large class of Carleson curves, including curv
es with corners and cusps as well as curves that look locally like two
logarithmic spirals scrolling up at the same point. For the singular
integral operator associated with the Riemann boundary value problem w
ith a piecewise continuous coefficient G, we establish a Fredholm crit
erion and an index formula in terms of the essential range of G comple
mented by spiralic horns depending on the Boyd indices of L(M)(Gamma)
and contour properties. Our main result is a symbol calculus for the c
losed algebra of singular integral operators with piecewise continuous
matrix-valued coefficients on L(M)(n)(Gamma).