Fredholm theory for singular integral operators with discontinuous coefficients in spaces of integrable functions.
Main publications:
A. Yu. Karlovich, A. K. Lerner, “Commutators of singular integrals on generalized $L^p$ spaces with variable exponent”, Publ. Mat., 49:1 (2005), 111–125
A. Yu. Karlovich, “The index of singular integral operators in reflexive Orlicz spaces”, Mat. Zametki, 64:3 (1998), 383–396; Math. Notes, 64:3 (1998), 330–341
A. Yu. Karlovich, “Singular integral operators with piecewise-continuous coefficients
in reflexive Orlicz spaces”, Dokl. Akad. Nauk, 349:1 (1996), 10–12