On the vector-valued Fourier transform and compatibility of operators

In Sook Park

Studia Mathematica (2005)

  • Volume: 168, Issue: 2, page 95-108
  • ISSN: 0039-3223

Abstract

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Let be a locally compact abelian group and let 1 < p ≤ 2. ’ is the dual group of , and p’ the conjugate exponent of p. An operator T between Banach spaces X and Y is said to be compatible with the Fourier transform F if F ⊗ T : L p ( ) ⊗ X → L p ' ( ' ) ⊗ Y admits a continuous extension [ F , T ] : [ L p ( ) , X ] → [ L p ' ( ' ) , Y ] . Let ℱ T p denote the collection of such T’s. We show that ℱ T p ℝ × = ℱ T p ℤ × = ℱ T p ℤ ⁿ × for any and positive integer n. Moreover, if the factor group of by its identity component is a direct sum of a torsion-free group and a finite group with discrete topology then ℱ T p = ℱ T p ℤ .

How to cite

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In Sook Park. "On the vector-valued Fourier transform and compatibility of operators." Studia Mathematica 168.2 (2005): 95-108. <http://eudml.org/doc/285079>.

@article{InSookPark2005,
abstract = {Let be a locally compact abelian group and let 1 < p ≤ 2. ’ is the dual group of , and p’ the conjugate exponent of p. An operator T between Banach spaces X and Y is said to be compatible with the Fourier transform $F^\{\}$ if $F^\{\} ⊗ T: L_\{p\}() ⊗ X → L_\{p^\{\prime \}\}(^\{\prime \}) ⊗ Y $ admits a continuous extension $[F^\{\},T]:[L_\{p\}(),X] → [L_\{p^\{\prime \}\}(^\{\prime \}),Y]$. Let $ℱT_\{p\}^\{\}$ denote the collection of such T’s. We show that $ℱT_\{p\}^\{ℝ×\} = ℱT_\{p\}^\{ℤ×\} = ℱT_\{p\}^\{ℤⁿ×\}$ for any and positive integer n. Moreover, if the factor group of by its identity component is a direct sum of a torsion-free group and a finite group with discrete topology then $ℱT_\{p\}^\{\} = ℱT_\{p\}^\{ℤ\}$.},
author = {In Sook Park},
journal = {Studia Mathematica},
keywords = {Banach space; operator; Fourier transform; vector-valued function; locally compact abelian group; dual group},
language = {eng},
number = {2},
pages = {95-108},
title = {On the vector-valued Fourier transform and compatibility of operators},
url = {http://eudml.org/doc/285079},
volume = {168},
year = {2005},
}

TY - JOUR
AU - In Sook Park
TI - On the vector-valued Fourier transform and compatibility of operators
JO - Studia Mathematica
PY - 2005
VL - 168
IS - 2
SP - 95
EP - 108
AB - Let be a locally compact abelian group and let 1 < p ≤ 2. ’ is the dual group of , and p’ the conjugate exponent of p. An operator T between Banach spaces X and Y is said to be compatible with the Fourier transform $F^{}$ if $F^{} ⊗ T: L_{p}() ⊗ X → L_{p^{\prime }}(^{\prime }) ⊗ Y $ admits a continuous extension $[F^{},T]:[L_{p}(),X] → [L_{p^{\prime }}(^{\prime }),Y]$. Let $ℱT_{p}^{}$ denote the collection of such T’s. We show that $ℱT_{p}^{ℝ×} = ℱT_{p}^{ℤ×} = ℱT_{p}^{ℤⁿ×}$ for any and positive integer n. Moreover, if the factor group of by its identity component is a direct sum of a torsion-free group and a finite group with discrete topology then $ℱT_{p}^{} = ℱT_{p}^{ℤ}$.
LA - eng
KW - Banach space; operator; Fourier transform; vector-valued function; locally compact abelian group; dual group
UR - http://eudml.org/doc/285079
ER -

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