Convolution algebras with weighted rearrangement-invariant norm

R. Kerman; E. Sawyer

Studia Mathematica (1994)

  • Volume: 108, Issue: 2, page 103-126
  • ISSN: 0039-3223

Abstract

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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 → ℂ : ∞ > ∥ 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 ∈ ℝ n by ( F ∗ G ) ( x ) = ʃ ℝ n F ( x - y ) G ( y ) d y ; more precisely, when ∥ F ∗ G ∥ X ( w ) ≤ ∥ F ∥ X ( w ) ∥ G ∥ X ( w ) for all F,G ∈ X(w).

How to cite

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Kerman, R., and Sawyer, E.. "Convolution algebras with weighted rearrangement-invariant norm." Studia Mathematica 108.2 (1994): 103-126. <http://eudml.org/doc/216044>.

@article{Kerman1994,
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).},
author = {Kerman, R., Sawyer, E.},
journal = {Studia Mathematica},
keywords = {rearrangement-invariant space of Lebesgue-measurable functions; Lebesgue, Lorentz or Orlicz spaces; algebra; convolution product},
language = {eng},
number = {2},
pages = {103-126},
title = {Convolution algebras with weighted rearrangement-invariant norm},
url = {http://eudml.org/doc/216044},
volume = {108},
year = {1994},
}

TY - JOUR
AU - Kerman, R.
AU - Sawyer, E.
TI - Convolution algebras with weighted rearrangement-invariant norm
JO - Studia Mathematica
PY - 1994
VL - 108
IS - 2
SP - 103
EP - 126
AB - 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).
LA - eng
KW - rearrangement-invariant space of Lebesgue-measurable functions; Lebesgue, Lorentz or Orlicz spaces; algebra; convolution product
UR - http://eudml.org/doc/216044
ER -

References

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  11. [11] G. Ya. Lozanovskiĭ, On some Banach lattices, Sibirsk. Mat. Zh. 10 (1969), 419-431 (in Russian). Zbl0194.43302
  12. [12] N. K. Nikol'skiĭ, Invariant subspaces of the shift operator in some sequence spaces, Candidate's Dissertation, Leningrad, 1966 (in Russian). 
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