Alexei Yu. Karlovich

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

AbstractThe paper is devoted to some only recently uncovered phenomena emerging in the study of singular integral operators (SIOs) with piecewise continuous (PC) coefficients in reflexive rearrangement-invariant spaces over Carleson curves. We deal with several kinds of indices of submultiplicative functions which describe properties of spaces (Boyd and Zippin indices) and curves (spirality indices). We consider some “disintegration condition” which combines properties of spaces and curves, the Boyd and spirality indices.We show that the essential spectrum of SIO associated with the Riemann boundary value problem with PC coefficient arises from the essential range of the coefficient by filling in certain massive connected sets (so-called logarithmic 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

On the interpolation constant for Orlicz spaces

\mathfrak{A}

Singular Integral Operators on Variable Lebesgue Spaces with Radial Oscillating Weights

generated by SIOs with matrix-valued piecewise continuous coefficients. We construct a symbol calculus for this Banach algebra which provides a Fredholm criterion and gives a basis for an index formula for arbitrary SIOs from

Criteria for One-Sided Invertibility of a Functional Operator in Rearrangement-Invariant Spaces of Fundamental Type

\mathfrak{A}

Higher-order Asymptotic Formulas for Toeplitz Matrices with Symbols in Generalized Holder Spaces

in terms of their symbols.

Some Algebras of Functions with Fourier Coefficients in Weighted Orlicz Sequence Spaces

AbstractIn this paper we extend necessary conditions for Fredholmness of singular integral operators with piecewise continuous coefficients in rearrangement-invariant spaces [19] to the weighted caseX(Γ,w). These conditions are formulated in terms of indices α(Qtw) and β(Qtw) of a submultiplicative functionQtw, which is associated with local properties of the space, of the curve, and of the weight at the pointt∈Γ. Using these results we obtain a lower estimate for the essential norm |S| of the Cauchy singular integral operatorS in reflexive weighted rearrangement-invariant spacesX(Γ,w) over arbitrary Carleson curves Γ: