Banach algebra of the Fourier multipliers on weighted Banach function spaces

Abstract

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.

Keywords

• Fourier convolution operator
• Fourier multiplier
• Banach function space
• Cauchy singular integral operator
• rearrangement-invariant space
• variable Lebesgue space
• Muckenhoupt-type weight