: In 1921, Landau wrote a letter to Hardy including a proof of the inequality
\n\\begin{align*}
\n\\sum_{n=0}^{\\infty}|\\phi(n)-\\phi(n+1)|^{p}\\geq
\n\\left(\\frac{p-1}{p}\\right)^{p}\\sum_{n=1}^{\\infty}\\frac{|\\phi(n)|^{p}}{n^{p}}
\n\\end{align*}
\nwhich holds for all finitely supported $\\phi:\\N_0\\to\\R$ such that
\n$\\phi(0)=0$ (here $1<p<\\infty$ is a fixed number). This inequality was stated before by Hardy, and therefore, it is called a \\emph{Hardy inequality}.
\nAn integral version of Hardy’s inequality states
\n$${\\displaystyle \\int _{0}^{\\infty }|\\phi'(x)|^p\\,\\mathrm{d}x \\geq \\left(\\frac{p-1}{p}\\right)^{p}\\int _{0}^{\\infty }\\frac{\\phi(x)^{p}}{x^p}\\,\\mathrm{d}x\\qquad \\forall \\phi \\in C_0^\\infty(\\R_+). }$$
\nSince then Hardy-type inequalities have received an enormous amount of attention. In this talk I will discuss recent developments related to Hardy-type inequalities.<\/p>\n","protected":false},"excerpt":{"rendered":"
: 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…
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