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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.