Hausdorff–Young and interpolation orientation

Assume countable choice. In the fixed negative-sign, $2\pi$ normalization, the Fourier operator has norm one at both endpoints $L^1\to L^{\mathrm{\infty }}$ and $L^2\to L^2$.

For the first endpoint, The L1 transform is bounded and uniformly continuous gives the upper bound one. The nonnegative Gaussian $g(x)=e^{-\pi \mathrm{\mid }x{\mathrm{\mid }}^2}$ has $\mathrm{\parallel }g{\mathrm{\parallel }}_1=1$ and $\widehat{g}(0)=1$ by Euclidean Gaussian transform with the 2π normalization. Continuity means its essential supremum is also one: every smaller positive bound is exceeded on a neighbourhood of zero of positive measure. Thus the operator norm is at least one. Plancherel theorem supplies the second norm-one endpoint, and Agreement of the integral and L2 transforms verifies agreement on their common domain.

These are the inputs to the Hausdorff–Young interpolation route. Teschl's endpoint-capable interpolation theorem, Theorem 15.2 and its extension Corollary 15.3, permits infinite endpoint exponents. A theorem restricting both target endpoint exponents to finite values cannot supply the $L^1\to L^{\mathrm{\infty }}$ endpoint. The published page complex-riesz-thorin-endpoint-interpolation develops this separate subject outside this pair's authorized prerequisite closure. It is orientation here, not an input to any proof on this pair; no intermediate-exponent Hausdorff–Young theorem is asserted by this remark. Countable choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) is inherited from the Gaussian and Plancherel suppliers.