Let \(1\le p\le 2\) and \(\frac{1}{p}+\frac{1}{q}=1\). For \(f\in \mathcal{S}(\mathbb{R}^n)\) we have \(\mathcal{F}f\in L^q(\mathbb{R}^n)\). To be more precise the Hausdorff-Young inequality holds
\begin{equation*} \lVert \mathcal{F}f\rVert_{L^q(\mathbb{R}^n)}\le \frac{1}{(2\pi)^{n/p-n/2}}\lVert f\rVert_{L^p(\mathbb{R}^n)}. \end{equation*}
Remarks
- This implies \(\mathcal{F}\colon L^p(\mathbb{R}^n)\to L^q(\mathbb{R}^n)\) is a bounded operator.