Classification and geometric aspects of vector valued Fourier transforms

✍ Scribed by In Sook Park

• Publisher: John Wiley and Sons
• Year: 2008
• Tongue: English
• Weight: 208 KB
• Volume: 281
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200310625

✦ Synopsis

Abstract

It is shown that for any locally compact abelian group 𝔾 and 1 ≤ p ≤ 2, the Fourier type p norm with respect to 𝔾 of a bounded linear operator T between Banach spaces, denoted by ‖T |ℱ𝒯^𝔾^~p~‖, satisfies ‖T |ℱ𝒯^𝔾^~p~‖ ≤ ‖T |ℱ𝒯^𝔸^~p~‖, where 𝔸 is the direct product of ℤ~2~, ℤ~3~, ℤ~4~, … It is also shown that if 𝔾 is not of bounded order then C^n^~p~ ‖T |ℱ𝒯^𝕋^~p~‖ ≤ ‖T |ℱ𝒯^𝔾^~p~‖, where 𝕋 is the circle group, n is a onnegative integer and C^p^ = . From these inequalities, for any locally compact abelian group 𝔾 ‖T |ℱ𝒯^𝔾^~2~‖ ≤ ‖T |ℱ𝒯^𝕋^~2~‖, and moreover if 𝔾 is not of bounded order then ‖T |ℱ𝒯^𝔾^~2~‖ = ‖T |ℱ𝒯^𝕋^~2~‖. The Hilbertian property and B‐convexity are discussed in the framework of Fourier type p norms. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)