# CANCELED: Hardy’s inequalities: a unification and applications to isoperimetric inequalities

Jon Wellner, UW
Monday, March 9, 2020 - 2:30pm
Denny 110
Over the period 1915 to 1925, G. H. Hardy proved and refined two
interesting inequalities as follows.  First the continuous (or integral form) inequality:
for a non-negative function ψ  on  (0,\infty) and p>1
\begin{eqnarray*}
\int_0^{\infty} \left ( \frac{1}{x} \int_0^x \psi (y) dy \right )^p dx \le \left ( \frac{p}{p-1} \right )^p \int_{\mathbb R} \psi^p (x) dx .
\end{eqnarray*}
On the other hand,
the discrete (or series form) inequality is:  for a sequence {a_n} of non-negative real numbers and p>1
\begin{eqnarray*}
\sum_{n=1}^\infty \left ( \frac{1}{n} \sum_{k=1}^n a_k \right )^p \le
\left ( \frac{p}{p-1} \right )^p \sum_{n=1}^\infty a_n^p .
\end{eqnarray*}
It seems that the long-standing lack of such a unified version
has much to do with the history of the development of probability theory within mathematics.

In this talk I will discuss a unification of these two inequalities in probability terms:
for any distribution function F on \mathbb R, non-negative function ψ on \mathbb R, and p>1
\begin{eqnarray*}
\int_{\mathbb R} \left ( \frac{1}{F(x)} \int_{(-\infty, x]} \psi (y) dF(y) \right )^p d F(x)
\le \left ( \frac{p}{p-1} \right )^p \int_{\mathbb R} \psi^p (x) d F(x) .
\end{eqnarray*}

I will indicate how Hardy's inequality can be used to derive
isoperimetric constants and optimal Poincaré inequalities.

This talk is based on joint work with C. A. J. Klaassen and separate joint work with
Adrien Saumard.

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