# Ngô Quốc Anh

## August 9, 2015

### The third fundamental lemma in the method of moving spheres

Filed under: Uncategorized — Ngô Quốc Anh @ 4:03

In this post, we proved the following result (appeared in a paper by Y.Y. Li published in J. Eur. Math. Soc. (2004))

Lemma 1. For $n \geqslant 1$ and $\nu \in \mathbb R$, let $f$ be a function defined on $\mathbb R^n$ and valued in $[-\infty, +\infty]$ satisfying

$\displaystyle {\left( {\frac{\lambda }{{|y - x|}}} \right)^\nu }f\left( {x + {\lambda ^2}\frac{{y - x}}{{|y - x{|^2}}}} \right) \leqslant f(y), \quad \forall |y - x| > \lambda > 0.$

Then $f$ is constant or $\pm \infty$.

Later, we considered the equality case in this post and proved the following result:

Lemma 2. Let $n\geqslant 1$, $\nu \in \mathbb R$ and $f \in C^0(\mathbb R^n)$. Suppose that for every $x \in \mathbb R^n$ there exists $\lambda(x)>0$ such that

$\displaystyle {\left( {\frac{\lambda(x) }{{|y - x|}}} \right)^\nu }f\left( {x + {\lambda(x) ^2}\frac{{y - x}}{{|y - x{|^2}}}} \right) =f(y), \quad \forall |y - x| > 0.$

Then for some $a \geqslant 0$, $d>0$ and $\overline x \in \mathbb R^n$

$\displaystyle f(x) = \pm a{\left( {\frac{1}{{d + {{\left| {x - \overline x } \right|}^2}}}} \right)^{\frac{\nu }{2}}}.$

In this post, we consider the third result which can be stated as follows: