(The) vector valued Fourier transform with respect to locally compact abelian groups and Hausdorff-Young

Abstract

This thesis is devoted to a study on the Fourier transform of Banach space valued functions defined on a locally compact abelian group. For a locally compact abelian(LCA) group $\mathbb{G}$ and 1 ≤ p ≤ 2, the Fourier type p norm with respect to \mathbb{G}$ of a bounded linear operator T from a Banach space to a Banach space is denoted by $\|T|\mathcal{FT}_p^{\mathbb{G}}\|$ and the class of T$satisfying $\|T|\mathcal{FT}_p^{\mathbb{G}}\|< ∞$ is denoted by $\mathcal{FT}_p^{\mathbb{G}}$. For 1<p≤ 2, we find a regular form of groups $\mathbb{G}$ which satisfies $\mathcal{FT}_p^{\mathbb{G}}=\mathcal{FT}_p^{\mathbb{T}}$, where $\mathbb{T}$ is the multiplicative group of all complex numbers of absolute value 1. Secondly, for 1＜ p≤2 and any infinite LCA group $\mathbb{G}$, we prove that $\|T|\mathcal{FT}_p^{\mathbb{G}}\|≤\|T|\mathcal{FT}_p^{\mathbb{A}}\|$ where $\mathbb{A}$ is the direct product of $\mathbb{Z}_2, \mathbb{Z}_3, \mathbb{Z}_4, …. If $\mathbb{G}$ is not of bounded order then $C\|T|\mathcal{FT}_p^{\mathbb{T}}\|≤\|T|\mathcal{FT}_p^{\mathbb{G}}\|$ for a positive constant $C$ dependent on $\mathbb{G}$ and $p$. And moreover $\|T|\mathcal{FT}_2^{\mathbb{G}}\|=\|T|\mathcal{FT}_2^{\mathbb{T}}\|$. If $\mathbb{G}$ is of bounded order then $\|T|\mathcal{FT}_p^{\mathbb{Z}_b^{∞}}\|≤ \|T|\mathcal{FT}_p^{\mathbb{G}}\|≤\|T|\mathcal{FT}_p^{\mathbb{Z}_d^{∞}}\|$ for some positive integers 2 ≤ b≤ d. From these results we obtain that if a Banach space has Fourier type p with respect to an infinite LCA group $\mathbb{G}$ for some 1<p≤ 2 then it is B-convex. In addition we obtain that, for any fixed infinite LCA group $\mathbb{G}$, a Banach space X has Fourier type 2 with respect to $\mathbb{G}$ if and only if X is isomorphic to a Hilbert space.