Hausdorff-Young inequality on T space

Question

I know that this inequality work well on space $L^p(\mathbb{R})$. But is it possible to generalize this inequality to the $\mathbb{T}$ space?

I think that on this space I can write this inequality in this way:

H-Y inequality: If $1<p<2$ and let $f\in L^p(\mathbb{T})\Rightarrow(\sum_{n\in \mathbb{Z}}|\hat{f}(n)|^q)^\frac{1}{q}\le |f|_{p}$

So is it possible to prove that? And if it's possible, how can this be prove?

Answer 1

From Folland, Real Analysis, Theorem 8.21.

Suppose $1 \leq p \leq 2$ and $q$ s.t. $q^{-1} + p^{-1} = 1$. If $f \in L^p(\mathbb{T}^n)$, then $\hat{f} \in \ell^q(\mathbb{Z}^n)$ and $\|\hat{f}\|_q \leq \|f\|_p$.

Proof. Since $\|\hat{f}\|_\infty \leq \|f\|_1$ and $\|\hat{f}\|_2 = \|f\|_2$ for $f \in L^1$ of $f \in L^2$, the assertion follows from the Riesz-Thorin interpolation theorem.