Convolution algebras with weighted rearrangement-invariant norm
Volume 108 / 1994
Studia Mathematica 108 (1994), 103-126
DOI: 10.4064/sm-108-2-103-126
Abstract
Let X be a rearrangement-invariant space of Lebesgue-measurable functions on $ℝ^n$, such as the classical Lebesgue, Lorentz or Orlicz spaces. Given a nonnegative, measurable (weight) function on $ℝ^n$, define $X(w) = {F: ℝ^n → ℂ: ∞ > ∥F∥_{X(w)} := ∥Fw∥_X}$. We investigate conditions on such a weight w that guarantee X(w) is an algebra under the convolution product F∗G defined at $x ∈ ℝ^n$ by $(F∗G)(x) = ʃ_{ℝ^n} F(x-y)G(y)dy$; more precisely, when $∥F∗G∥_{X(w)} ≤ ∥F∥_{X(w)} ∥G∥_{X(w)}$ for all F,G ∈ X(w).
