Singular integral operators with flip and unbounded coefficients on rearrangement-invariant spaces

Citation:
Karlovich, Alexei Yu. "Singular integral operators with flip and unbounded coefficients on rearrangement-invariant spaces." Functional Analysis and its Applications. Proceedings of the international conference, dedicated to the 110th anniversary of Stefan Banach, Lviv National University, Lviv, Ukraine, May 28--31, 2002. Eds. V. Kadets, and W. Zelazko. Amsterdam: Elsevier, 2004. 123-131.

Abstract:

We prove Fredholm criteria for singular integral operators of the form
$P_++M_bP_-+M_uUP_-,$
where $$P_\pm$$ are the Riesz projections, $$U$$ is the flip operator, and $$M_b,M_u$$ are operators of multiplication by functions $$b,u$$, respectively, on a reflexive rearrangement-invariant space with nontrivial Boyd indices over the unit circle. We assume a priori that $$M_b$$ is bounded, but $$M_u$$ may be unbounded. The function $$u$$ belongs to a class of, in general, unbounded functions that relates to the Douglas algebra $$H^\infty+C$$.