Tag: geometry

Let ${u_k}_{k=1}^infty$ be a sequence of ($L^2$ normalized) mutually orthogonal eigenfunctions of the operator $-Delta$ on the sphere $mathbb{S}^n$ (here $Delta$ is the Laplace Beltrami operator). Let $u$ be a smooth (real valued) function on the sphere. It is a well-known result that we can write $u=sum_{k=1}^infty c_k u_k$ for some (real) constants $c_k$. My question is: Is the convergence of this sum uniform?

I am trying to prove that the optimal constant in the Poincare inequality is $lambda_1=n$. That is to say, I am trying to prove the inequlity $int_{mathbb{S}^n} |nabla u|^2 geq n int_{mathbb{S}^n} |u|^2$. Here is what I have done so far:

First, integrate by parts on the LHS so that it suffices to prove $int_{mathbb{S}^n} -uDelta u geq n int_{mathbb{S}^n} |u|^2$. Then use $u=sum_{k=1}^infty c_k u_k$ and assume that the convergence is uniform. Then we can switch the order of the sum with the derivative and integral (and use the fact that ${u_k}$ are orthonomal) so that
begin{align*}
int_{mathbb{S}^n} -left(sum_{k=1}^infty c_k u_kright)Delta left(sum_{j=1}^infty c_j u_jright)&=
int_{mathbb{S}^n} -left(sum_{k=1}^infty c_k u_kright)left(sum_{j=1}^infty c_j Delta u_jright)=
int_{mathbb{S}^n} -left(sum_{k=1}^infty c_k u_kright)left(sum_{j=1}^infty lambda_j c_j u_jright)
\&= sum_{j,k}c_k c_j lambda_jint_{mathbb{S}^n} u_k u_j= sum_{j,k}c_k c_j lambda_j delta_{jk}=sum_{j}c_j^2 lambda_jgeq lambda_1 sum_j c_j^2.
end{align*}
By the same logic, the last sum is equal to $int |u|^2$.

Now obviously, this proof requires some argument showing that the sum commutes with $Delta$ and the integral but I have not been able to find a reference that the sum converges uniformly. My thought is that this would follow from some basic facts in Harmonic analysis though I am no expert in that field. Would anyone be able to provide a reference for this?