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I am currently at Princeton for the conference “The power of Analysis” honouring Charlie Fefferman‘s 60th birthday. I myself gave a talk at this conference entitled “Recent progress on the Kakeya conjecture”; I plan to post a version of this talk on this blog shortly.

But one nice thing about attending these sorts of conferences is that one can also learn some neat mathematical facts, and I wanted to show two such small gems here; neither is particularly deep, but I found both of them cute. The first one, which I learned from my former student Soonsik Kwon, is a unified way to view the mean, median, and mode of a probability distribution ${\mu}$ on the real line. If one assumes that this is a continuous distribution ${\mu = f(x)\ dx}$ for some smooth, rapidly decreasing function ${f: {\mathbb R} \rightarrow {\mathbb R}^+}$ with ${\int_{\mathbb R} f(x)\ dx = 1}$, then the mean is the value of ${x_0}$ that minimises the second moment

$\displaystyle \int_{\mathbb R} |x-x_0|^2 f(x)\ dx,$

the median is the value of ${x_0}$ that minimises the first moment

$\displaystyle \int_{\mathbb R} |x-x_0| f(x)\ dx,$

and the mode is the value of ${x_0}$ that maximises the “pseudo-negative first moment”

$\displaystyle \int_{\mathbb R} \delta(x-x_0) f(x)\ dx.$

(Note that the Dirac delta function ${\delta(x-x_0)}$ has the same scaling as ${|x-x_0|^{-1}}$, hence my terminology “pseudo-negative first moment”.)

The other fact, which I learned from my former classmate Diego Córdoba (and used in a joint paper of Diego with Antonio Córdoba), is a pointwise inequality

$\displaystyle |\nabla|^\alpha ( f^2 )(x) \leq 2 f(x) |\nabla|^\alpha f(x)$

for the fractional differentiation operators ${|\nabla|^\alpha}$ applied to a sufficiently nice real-valued function ${f: {\mathbb R}^d \rightarrow {\mathbb R}}$ (e.g. Schwartz class will do), in any dimension ${d}$ and for any ${0 \leq \alpha \leq 1}$; this should be compared with the product rule ${\nabla (f^2 ) = 2 f \nabla f}$.

The proof is as follows. By a limiting argument we may assume that ${0 < \alpha < 1}$. In this case, there is a formula

$\displaystyle |\nabla|^\alpha f(x) = c(\alpha) \int_{{\mathbb R}^d} \frac{f(x)-f(y)}{|x-y|^{d+\alpha}}\ dy$

for some explicit constant ${c(\alpha) > 0}$ (this can be seen by computations similar to those in my recent lecture notes on distributions, or by analytically continuing such computations; see also Stein’s “Singular integrals and differentiability properties of functions”). Using this formula, one soon sees that

$\displaystyle 2 f(x) |\nabla|^\alpha f(x) - |\nabla|^\alpha ( f^2 )(x) = c(\alpha) \int_{{\mathbb R}^d} \frac{|f(x)-f(y)|^2}{|x-y|^{d+\alpha}}\ dy$

and the claim follows.

The first Distinguished Lecture Series at UCLA of this academic year is being given this week by my good friend and fellow Medalist Charlie Fefferman, who also happens to be my “older brother” (we were both students of Elias Stein). The theme of Charlie’s lectures is “Interpolation of functions on ${\Bbb R}^n$“, in the spirit of the classical Whitney extension theorem, except that now one is considering much more quantitative and computational extension problems (in particular, viewing the problem from a theoretical computer science perspective). Today Charlie introduced the basic problems in this subject, and stated some of the results of his joint work with Bo’az Klartag; he will continue the lectures on Thursday and Friday.

The general topic of extracting quantitative bounds from classical qualitative theorems is a subject that I am personally very fond of, and Charlie gave a wonderfully accessible presentation of the main results, though the actual details of the proofs were left to the next two lectures.

As usual, all errors and omissions here are my responsibility, and are not due to Charlie.