A theorem on the discrete groups and algebras $L_p$

Wiesław Żelazko

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The Banach spaces $Λ (A, B, X, G)$ defined in this paper consist essentially of those elements of $L^{1} (G)$ ( $G$ being a locally compact group) which can in a certain sense be well approximated by functions with compact support. The main result of this paper is the fact that in many cases $Λ (A, B, X, G)$ becomes a Banach convolution algebra. There exist many natural examples. Furthermore some theorems concerning inclusion results and the structure of these spaces are given. In particular we prove that simple conditions imply...

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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).

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We study continuous measures on a compact semisimple Lie group $G$ using representation theory. In Section 2 we prove a Wiener type characterization of a continuous measure. Next we construct central measures on $G$ which are related to the well known Riesz products on locally compact abelian groups. Using these measures we show in Section 3 that if $C$ is a compact set of continuous measures on $G$ there exists a singular measure $ν$ such that $ν * μ$ is absolutely continuous with respect to the Haar...

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Hermann Render, Andreas Sauer (1996)

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Let $H (G_{1})$ be the set of all holomorphic functions on the domain $G_{1} .$ Two domains $G_{1}$ and $G_{2}$ are called Hadamard-isomorphic if $H (G_{1})$ and $H (G_{2})$ are isomorphic algebras with respect to the Hadamard product. Our main result states that two admissible domains are Hadamard-isomorphic if and only if they are equal.

Displaying similar documents to “A theorem on the discrete groups and algebras L p ”

Displaying similar documents to “A theorem on the discrete groups and algebras $L_{p}$ ”