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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 ${C, p_0, p_1, q_0, q_1}$.

In the previous two quarters, we have been focusing largely on the “soft” side of real analysis, which is primarily concerned with “qualitative” properties such as convergence, compactness, measurability, and so forth. In contrast, we will begin this quarter with more of an emphasis on the “hard” side of real analysis, in which we study estimates and upper and lower bounds of various quantities, such as norms of functions or operators. (Of course, the two sides of analysis are closely connected to each other; an understanding of both sides and their interrelationships, are needed in order to get the broadest and most complete perspective for this subject.)

One basic tool in hard analysis is that of interpolation, which allows one to start with a hypothesis of two (or more) “upper bound” estimates, e.g. ${A_0 \leq B_0}$ and ${A_1 \leq B_1}$, and conclude a family of intermediate estimates ${A_\theta \leq B_\theta}$ (or maybe ${A_\theta \leq C_\theta B_\theta}$, where ${C_\theta}$ is a constant) for any choice of parameter ${0 < \theta < 1}$. Of course, interpolation is not a magic wand; one needs various hypotheses (e.g. linearity, sublinearity, convexity, or complexifiability) on ${A_i, B_i}$ in order for interpolation methods to be applicable. Nevertheless, these techniques are available for many important classes of problems, most notably that of establishing boundedness estimates such as ${\| T f \|_{L^q(Y, \nu)} \leq C \| f \|_{L^p(X, \mu)}}$ for linear (or “linear-like”) operators ${T}$ from one Lebesgue space ${L^p(X,\mu)}$ to another ${L^q(Y,\nu)}$. (Interpolation can also be performed for many other normed vector spaces than the Lebesgue spaces, but we will just focus on Lebesgue spaces in these notes to focus the discussion.) Using interpolation, it is possible to reduce the task of proving such estimates to that of proving various “endpoint” versions of these estimates. In some cases, each endpoint only faces a portion of the difficulty that the interpolated estimate did, and so by using interpolation one has split the task of proving the original estimate into two or more simpler subtasks. In other cases, one of the endpoint estimates is very easy, and the other one is significantly more difficult than the original estimate; thus interpolation does not really simplify the task of proving estimates in this case, but at least clarifies the relative difficulty between various estimates in a given family.

As is the case with many other tools in analysis, interpolation is not captured by a single “interpolation theorem”; instead, there are a family of such theorems, which can be broadly divided into two major categories, reflecting the two basic methods that underlie the principle of interpolation. The real interpolation method is based on a divide and conquer strategy: to understand how to obtain control on some expression such as ${\| T f \|_{L^q(Y, \nu)}}$ for some operator ${T}$ and some function ${f}$, one would divide ${f}$ into two or more components, e.g. into components where ${f}$ is large and where ${f}$ is small, or where ${f}$ is oscillating with high frequency or only varying with low frequency. Each component would be estimated using a carefully chosen combination of the extreme estimates available; optimising over these choices and summing up (using whatever linearity-type properties on ${T}$ are available), one would hope to get a good estimate on the original expression. The strengths of the real interpolation method are that the linearity hypotheses on ${T}$ can be relaxed to weaker hypotheses, such as sublinearity or quasilinearity; also, the endpoint estimates are allowed to be of a weaker “type” than the interpolated estimates. On the other hand, the real interpolation often concedes a multiplicative constant in the final estimates obtained, and one is usually obligated to keep the operator ${T}$ fixed throughout the interpolation process. The proofs of real interpolation theorems are also a little bit messy, though in many cases one can simply invoke a standard instance of such theorems (e.g. the Marcinkiewicz interpolation theorem) as a black box in applications.

The complex interpolation method instead proceeds by exploiting the powerful tools of complex analysis, in particular the maximum modulus principle and its relatives (such as the Phragmén-Lindelöf principle). The idea is to rewrite the estimate to be proven (e.g. ${\| T f \|_{L^q(Y, \nu)} \leq C \| f \|_{L^p(X, \mu)}}$) in such a way that it can be embedded into a family of such estimates which depend holomorphically on a complex parameter ${s}$ in some domain (e.g. the strip ${\{ \sigma+it: t \in {\bf R}, \sigma \in [0,1]\}}$. One then exploits things like the maximum modulus principle to bound an estimate corresponding to an interior point of this domain by the estimates on the boundary of this domain. The strengths of the complex interpolation method are that it typically gives cleaner constants than the real interpolation method, and also allows the underlying operator ${T}$ to vary holomorphically with respect to the parameter ${s}$, which can significantly increase the flexibility of the interpolation technique. The proofs of these methods are also very short (if one takes the maximum modulus principle and its relatives as a black box), which make the method particularly amenable for generalisation to more intricate settings (e.g. multilinear operators, mixed Lebesgue norms, etc.). On the other hand, the somewhat rigid requirement of holomorphicity makes it much more difficult to apply this method to non-linear operators, such as sublinear or quasilinear operators; also, the interpolated estimate tends to be of the same “type” as the extreme ones, so that one does not enjoy the upgrading of weak type estimates to strong type estimates that the real interpolation method typically produces. Also, the complex method runs into some minor technical problems when target space ${L^q(Y,\nu)}$ ceases to be a Banach space (i.e. when ${q<1}$) as this makes it more difficult to exploit duality.

Despite these differences, the real and complex methods tend to give broadly similar results in practice, especially if one is willing to ignore constant losses in the estimates or epsilon losses in the exponents.

The theory of both real and complex interpolation can be studied abstractly, in general normed or quasi-normed spaces; see e.g. this book for a detailed treatment. However in these notes we shall focus exclusively on interpolation for Lebesgue spaces ${L^p}$ (and their cousins, such as the weak Lebesgue spaces ${L^{p,\infty}}$ and the Lorentz spaces ${L^{p,r}}$).