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In a few weeks, Princeton University will host a conference in Analysis and Applications in honour of the 80th birthday of Elias Stein (though, technically, Eli’s 80th birthday was actually in January). As one of Eli’s students, I was originally scheduled to be one of the speakers at this conference; but unfortunately, for family reasons I will be unable to attend. In lieu of speaking at this conference, I have decided to devote some space on this blog for this month to present some classic results of Eli from his many decades of work in harmonic analysis, ergodic theory, several complex variables, and related topics. My choice of selections here will be a personal and idiosyncratic one; the results I present are not necessarily the “best” or “deepest” of his results, but are ones that I find particularly elegant and appealing. (There will also inevitably be some overlap here with Charlie Fefferman’s article “Selected theorems by Eli Stein“, which not coincidentally was written for Stein’s 60th birthday conference in 1991.)

In this post I would like to describe one of Eli Stein’s very first results that is still used extremely widely today, namely his interpolation theorem from 1956 (and its refinement, the Fefferman-Stein interpolation theorem from 1972). This is a deceptively innocuous, yet remarkably powerful, generalisation of the classic Riesz-Thorin interpolation theorem which uses methods from complex analysis (and in particular, the Lindelöf theorem or the Phragmén-Lindelöf principle) to show that if a linear operator ${T: L^{p_0}(X) + L^{p_1}(X) \rightarrow L^{q_0}(Y) + L^{q_1}(Y)}$ from one (${\sigma}$-finite) measure space ${X = (X,{\mathcal X},\mu)}$ to another ${Y = (Y, {\mathcal Y}, \nu)}$ obeyed the estimates

$\displaystyle \| Tf \|_{L^{q_0}(Y)} \leq B_0 \|f\|_{L^{p_0}(X)} \ \ \ \ \ (1)$

for all ${f \in L^{p_0}(X)}$ and

$\displaystyle \| Tf \|_{L^{q_1}(Y)} \leq B_1 \|f\|_{L^{p_1}(X)} \ \ \ \ \ (2)$

for all ${f \in L^{p_1}(X)}$, where ${1 \leq p_0,p_1,q_0,q_1 \leq \infty}$ and ${B_0,B_1 > 0}$, then one automatically also has the interpolated estimates

$\displaystyle \| Tf \|_{L^{q_\theta}(Y)} \leq B_\theta \|f\|_{L^{p_\theta}(X)} \ \ \ \ \ (3)$

for all ${f \in L^{p_\theta}(X)}$ and ${0 \leq \theta \leq 1}$, where the quantities ${p_\theta, q_\theta, B_\theta}$ are defined by the formulae

$\displaystyle \frac{1}{p_\theta} = \frac{1-\theta}{p_0} + \frac{\theta}{p_1}$

$\displaystyle \frac{1}{q_\theta} = \frac{1-\theta}{q_0} + \frac{\theta}{q_1}$

$\displaystyle B_\theta = B_0^{1-\theta} B_1^\theta.$

The Riesz-Thorin theorem is already quite useful (it gives, for instance, by far the quickest proof of the Hausdorff-Young inequality for the Fourier transform, to name just one application), but it requires the same linear operator ${T}$ to appear in (1), (2), and (3). Eli Stein realised, though, that due to the complex-analytic nature of the proof of the Riesz-Thorin theorem, it was possible to allow different linear operators to appear in (1), (2), (3), so long as the dependence was analytic. A bit more precisely: if one had a family ${T_z}$ of operators which depended in an analytic manner on a complex variable ${z}$ in the strip ${\{ z \in {\bf C}: 0 \leq \hbox{Re}(z) \leq 1 \}}$ (thus, for any test functions ${f, g}$, the inner product ${\langle T_z f, g \rangle}$ would be analytic in ${z}$) which obeyed some mild regularity assumptions (which are slightly technical and are omitted here), and one had the estimates

$\displaystyle \| T_{0+it} f \|_{L^{q_0}(Y)} \leq C_t \|f\|_{L^{p_0}(X)}$

and

$\displaystyle \| T_{1+it} f \|_{L^{q_1}(Y)} \leq C_t\|f\|_{L^{p_1}(X)}$

for all ${t \in {\bf R}}$ and some quantities ${C_t}$ that grew at most exponentially in ${t}$ (actually, any growth rate significantly slower than the double-exponential ${e^{\exp(\pi |t|)}}$ would suffice here), then one also has the interpolated estimates

$\displaystyle \| T_\theta f \|_{L^{q_\theta}(Y)} \leq C' \|f\|_{L^{p_\theta}(X)}$

for all ${0 \leq \theta \leq 1}$ and a constant ${C'}$ depending only on ${C, p_0, p_1, q_0, q_1}$.