On the vector-valued Fourier transform and compatibility of operators

In Sook Park

In Sook Park

Studia Mathematica, Tome 166 (2005), p. 95-108 / Harvested from The Polish Digital Mathematics Library

Résumé

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 $F^{} \otimes T : L_{p} () \otimes X \to L_{p^{'}} (^{'}) \otimes Y$ admits a continuous extension $[F^{}, T] : [L_{p} (), X] \to [L_{p^{'}} (^{'}), Y]$ . Let $ℱ T_{p}^{}$ denote the collection of such T’s. We show that $ℱ T_{p}^{ℝ \times} = ℱ T_{p}^{ℤ \times} = ℱ T_{p}^{ℤ ⁿ \times}$ 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 $ℱ T_{p}^{} = ℱ T_{p}^{ℤ}$ .

Publié le : 2005-01-01

Zbl 1062.47071

EUDML-ID : urn:eudml:doc:285079

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In Sook Park. On the vector-valued Fourier transform and compatibility of operators. Studia Mathematica, Tome 166 (2005) pp. 95-108. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm168-2-1/