Colloquium: Tuesday, November 22, 2 pm. Speaker: Yehuda Pinchover (Technion). Title: “Some aspects of Hardy-type inequalities”.
: In 1921, Landau wrote a letter to Hardy including a proof of the inequality
\begin{align*}
\sum_{n=0}^{\infty}|\phi(n)-\phi(n+1)|^{p}\geq
\left(\frac{p-1}{p}\right)^{p}\sum_{n=1}^{\infty}\frac{|\phi(n)|^{p}}{n^{p}}
\end{align*}
which holds for all finitely supported $\phi:\N_0\to\R$ such that
$\phi(0)=0$ (here $1<p<\infty$ is a fixed number). This inequality was stated before by Hardy, and therefore, it is called a \emph{Hardy inequality}.
An integral version of Hardy’s inequality states
$${\displaystyle \int _{0}^{\infty }|\phi'(x)|^p\,\mathrm{d}x \geq \left(\frac{p-1}{p}\right)^{p}\int _{0}^{\infty }\frac{\phi(x)^{p}}{x^p}\,\mathrm{d}x\qquad \forall \phi \in C_0^\infty(\R_+). }$$
Since then Hardy-type inequalities have received an enormous amount of attention. In this talk I will discuss recent developments related to Hardy-type inequalities.
