The interpolation input belongs to measure theory

The bound $\mathrm{\parallel }\widehat{f}{\mathrm{\parallel }}_{\mathrm{\infty }}\le \mathrm{\parallel }f{\mathrm{\parallel }}_1$ in The L1 transform is bounded and uniformly continuous supplies the Fourier $L^1\to L^{\mathrm{\infty }}$ endpoint. Interpolation requires a theorem that actually permits an infinite target exponent. A theorem stated only with finite target endpoint exponents cannot supply that step.

The published page complex-riesz-thorin-endpoint-interpolation is the designated endpoint-capable supplier; it lies outside this pair's current prerequisite closure. Its role here is orientation. Intermediate-exponent Fourier bounds are not asserted or used in this page, and no second Riesz–Thorin theorem is introduced. The later Plancherel pair supplies the other Fourier endpoint and common-domain agreement needed for that route.