Riesz–Thorin theorem

Remarks

The proof is Thorin's complex-variable argument. By density of simple functions and duality, it suffices to bound $\int (T f) g \, d\nu$ for simple $f$ and $g$. One embeds $f$ and $g$ in analytic families $f_z$, $g_z$ for $z$ in the strip $0 \le \operatorname{Re} z \le 1$, with $f_\theta = f$ and $g_\theta = g$. The function $z \mapsto \int (T f_z) g_z \, d\nu$ is bounded and analytic, and the Hadamard three-lines theorem interpolates the bounds on the two edges.

Example: the Fourier transform satisfies $\norm{\hat f}_\infty \le \norm{f}_1$ and, by Plancherel, $\norm{\hat f}_2 = \norm{f}_2$ on $\R^n$ with suitable normalisation. Riesz–Thorin with $p_0 = 1$, $q_0 = \infty$, $p_1 = q_1 = 2$ gives the Hausdorff–Young inequality $\norm{\hat f}_{p'} \le \norm{f}_p$ for $1 \le p \le 2$, where $1/p + 1/p' = 1$.

Complex scalars matter for the constant. For spaces of real-valued functions the conclusion still holds, but the constant $M_0^{1-\theta} M_1^\theta$ may need an extra factor of at most $2$.

Going deeper

For a student

Riesz–Thorin is the basic tool for proving $L^p$ bounds from two easy endpoint bounds. Typical uses are the Hausdorff–Young inequality, Young's convolution inequality and bounds for integral operators. It needs linear operators and strong-type bounds at the endpoints. The Marcinkiewicz interpolation theorem allows sublinear operators and weak-type endpoint bounds, at the cost of a worse constant and, in its basic form, only on the diagonal $p = q$. Both are the starting points of abstract interpolation space theory, the complex and the real method.