Is Fourier transform defined on $L^p(\mathbb{R})$ only for $p \in [1, 2]$?
From Lieb and Loss's Analysis, they extend the definition of Fourier transform from $L^1(\mathbb{R})$ to $L^p(\mathbb{R}), p \in (1, \infty)$, using $ \| FT(f) \|_q \leq C_{p,q} \|f\|_p $ which they said only holds when $p \in (1,2]$. Does that mean that FT cannot be defined on $L^p(\mathbb{R})$ with $p \in (2, \infty)$, possibly via other means?
Thanks and regards!