Hausdorff–Young inequality on a locally compact abelian group

Statement

Let $G$ be a locally compact group that is abelian, choose Haar measures on $G$ and its dual $\widehat G$ in Plancherel duality, and let $1\leq p\leq2$ . If $q$ is the conjugate exponent, so $p^{-1}+q^{-1}=1$ , then the Fourier transform extends uniquely to a bounded map

L^p(G)\longrightarrow L^q(\widehat G)

satisfying

\lVert\widehat f\rVert_q\leq\lVert f\rVert_p.

At $p=1$ , interpret $q=\infty$ . At $p=2$ , the extension is the unitary Plancherel transform. The statement concerns equivalence classes in the corresponding $L^p$ spaces. On the dense intersection $L^1(G)\cap L^p(G)$ , the extension agrees with the defining Fourier integral.

Proof mechanism

The $p=1$ estimate follows directly from the defining integral: $\lvert\widehat f(\gamma)\rvert\leq\lVert f\rVert_1$ . The $p=2$ estimate is equality by Plancherel's theorem. Complex interpolation between these endpoint operators gives the intermediate exponents Folland, Chapter 4. The constant $1$ is the interpolation bound for the paired Haar normalization, not necessarily the sharp interior constant in every concrete model.

Examples and boundary cases

For $G=\mathbb R^n$ with the standard dual normalization, this is the classical $L^p(\mathbb R^n)$ -to- $L^q(\mathbb R^n)$ Hausdorff–Young inequality. For a compact group such as the circle, it bounds Fourier coefficients in $\ell^q$ . The direction generally cannot be extended to $p>2$ : an arbitrary $L^p$ function then need not have an $L^q$ Fourier transform.

Normalization and scope

Rescaling the Haar measure on $G$ requires the reciprocal Plancherel normalization on $\widehat G$ , and the numerical bound changes if unrelated normalizations are chosen. The scalar theorem stated here uses commutativity. For nonabelian groups, Hausdorff–Young inequalities require operator-valued targets and Schatten norms rather than the displayed scalar $L^q$ space.

References

Walter Rudin, Fourier Analysis on Groups, Wiley-Interscience, 1962. Wiley DOI record. Relevant: Chapter 1 on Fourier transformation and the Hausdorff–Young inequality.
Gerald B. Folland, A Course in Abstract Harmonic Analysis, 2nd ed., CRC Press, 2016. DOI record. Relevant: Chapter 4 on Fourier analysis on locally compact abelian groups.