Stein interpolation theorem

Remarks

The proof follows the proof of the Riesz–Thorin theorem. For fixed simple $f$ and $g$ one builds families $f_z$, $g_z$ of simple functions whose absolute values are powers of $\abs{f}$ and $\abs{g}$ depending on $z$. Then one studies the holomorphic function $z \mapsto \int_Y T_z(f_z) g_z \, d\nu$ on the strip. The hypotheses bound it on the two boundary lines. A form of the Hadamard three-lines theorem for functions of admissible growth, proved with the Poisson kernel of the strip, then bounds it at $z = \theta$. The two kernels in the formula for $M_\theta$ are the Poisson kernels of the strip at the point $\theta$.

If $T_z = T$ does not depend on $z$ and $M_0$, $M_1$ are constants, the formula gives $M_\theta = M_0^{1 - \theta} M_1^{\theta}$, which is the Riesz–Thorin theorem. A typical use is the family of fractional integration or Bochner–Riesz type operators $T_z$, where the exponent of a kernel depends analytically on $z$. The bound $a < \pi$ in the growth condition cannot be relaxed to $a = \pi$. The function $h(z) = \exp(i e^{-i\pi z})$ has modulus $1$ on both boundary lines of $S$, but $\abs{h(\tfrac12 + iy)} = \exp(e^{\pi y})$ is unbounded.

Going deeper

For a student

The Riesz–Thorin theorem interpolates one fixed operator. Many problems in harmonic analysis come instead with a natural family, for example operators whose kernel contains a power $\abs{x}^{z}$ or a factor $(1 - \abs{\xi}^2)^{z}_+$. At one end of the strip the family is easy to bound in $L^2$ by Plancherel's theorem. At the other end it is bounded in $L^\infty$ by a pointwise estimate of the kernel. Stein's theorem gives the bounds in between, as in results on Bochner–Riesz means and on Fourier restriction. It belongs to the complex method of interpolation, while the Marcinkiewicz interpolation theorem belongs to the real method.