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On the vector-valued Fourier transform and compatibility of operators

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Park I.

Studia Mathematica

2005

tom 168

nr 2

95-108

Let 𝔾 be a locally compact abelian group and let 1 < p ≤ 2. 𝔾' is the dual group of 𝔾, 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 $F^{𝔾}$ if $F^{𝔾} ⊗ T: L_{p}(𝔾) ⊗ X → L_{p'}(𝔾') ⊗ Y $ admits a continuous extension $[F^{𝔾},T]:[L_{p}(𝔾),X] → [L_{p'}(𝔾'),Y]$. Let $ℱT_{p}^{𝔾}$ denote the collection of such T's. We show that $ℱT_{p}^{ℝ×𝔾} = ℱT_{p}^{ℤ×𝔾} = ℱT_{p}^{ℤⁿ×𝔾}$ for any 𝔾 and positive integer n. Moreover, if the factor group of 𝔾 by its identity component is a direct sum of a torsion-free group and a finite group with discrete topology then $ℱT_{p}^{𝔾} = ℱT_{p}^{ℤ}$.

Ograniczanie wyników

1 Studia Mathematica

1 Park I.

1 2005

Wyniki wyszukiwania

On the vector-valued Fourier transform and compatibility of operators