Banach algebra of the Fourier multipliers on weighted Banach function spaces

Abstract & Overview

Abstract Let M X,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), thenM X,w (ℝ) is continuously embedded into L ∞ (ℝ). An important consequence of the continuous embedding M X,w (ℝ) ⊂ L ∞ (ℝ) is that M X,w (ℝ) is a Banach algebra.