Banach algebra of the Fourier multipliers on weighted Banach function spaces

Let MX,w(ℝ) denote the algebra of the Fourier multipliers on a separable weighted Banach function space X(ℝ,w).We prove that if the Cauchy singular integral operator S is bounded on X(ℝ, w), thenMX,w(ℝ) is continuously embedded into L∞(ℝ). An important consequence of the continuous embedding MX,w(ℝ) ⊂ L∞(ℝ) is that MX,w(ℝ) is a Banach algebra.

Remark on the boundedness of the Cauchy singular integral operator on variable Lebesgue spaces with radial oscillating weights

Recently V. Kokilashvili, N. Samko, and S. Samko have proved a sufficient condition for the boundedness of the Cauchy singular integral operator on variable Lebesgue spaces with radial oscillating weights over Carleson curves. This condition is formulated in terms of Matuszewska-Orlicz indices of weights. We prove a partial converse of their result.

Singular Integral Equations in Special Weighted Spaces

We prove the boundedness of the Cauchy singular integral operator in special weighted Sobolev and Hölder-Zygmund spaces for large values of the smoothness parameter, which is an integer m ≥ 0, when the underlying contour is piecewise-smooth with angular points and even with cusps. We obtain Fredholm criteria and an index formula for singular integral equations with piecewise-continuous coefficients and complex conjugation in the spaces and provided that the underlying contour has only angular points but no cusps. The Fredholm property and the index turn out to be independent of the smoothness parameter m.