Hausdorff–Young inequality

Remarks

The two endpoint cases are known: for $p = 1$, $\abs{\hat f(\xi)} \le \norm{f}_1$ for every $\xi$, and for $p = 2$, the Plancherel theorem gives $\norm{\hat f}_2 = \norm{f}_2$. The Riesz–Thorin theorem interpolates between them and gives the inequality for $1 < p < 2$.

The constant $1$ is not sharp for $1 < p < 2$; the sharp constant is given by the Babenko–Beckner inequality. The restriction $p \le 2$ is necessary: for $p > 2$ there are functions in $L^p(\R^n)$ whose Fourier transform, as a tempered distribution, is not a function.

There is a version for Fourier series on the circle $\mathbb{T} = \R / \Z$ with normalized measure: for $1 \le p \le 2$ and $f \in L^p(\mathbb{T})$, $\left(\sum_{k \in \Z} \abs{\hat f(k)}^q\right)^{1/q} \le \norm{f}_{L^p(\mathbb{T})}$.

Example: for $f(x) = e^{-\pi x^2}$ on $\R$, $\hat f = f$, and $\norm{f}_r = r^{-1/(2r)}$ for every $r > 0$. So $\norm{\hat f}_q / \norm{f}_p = \left( p^{1/p} / q^{1/q} \right)^{1/2}$, and the theorem says that this number is at most $1$.

Going deeper

For a student

The inequality is the first example of an $L^p$ estimate obtained by interpolation, and it is the model for the Riesz–Thorin method. It shows that the Fourier transform is defined as a function on $L^p$ for $1 \le p \le 2$. It is used in restriction theory, in the theory of Fourier multipliers and in the entropic uncertainty principle. On locally compact abelian groups the same statement holds, with the dual group in place of the frequency space.