On a class of convolution algebras of functions

Hans G. Feichtinger

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Convolution algebras with weighted rearrangement-invariant norm

R. Kerman, E. Sawyer (1994)

Studia Mathematica

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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 $X (w) = F : ℝ^{n} \to ℂ : \infty > ∥ F ∥_{X (w)} : = ∥ F w ∥_{X}$ . We investigate conditions on such a weight w that guarantee X(w) is an algebra under the convolution product F∗G defined at $x \in ℝ^{n}$ by $(F * G) (x) = ʃ_{ℝ^{n}} F (x - y) G (y) d y$ ; more precisely, when $∥ F * G ∥_{X (w)} \leq ∥ F ∥_{X (w)} ∥ G ∥_{X (w)}$ for all F,G ∈ X(w).