Topological Bounds for Fourier Coefficients and Applications to Torsion

Abstract: Let $\Omega \subset \mathbb{R}^2$ be a bounded convex domain in the plane and consider \begin{align*} -\Delta u &=1 \qquad \mbox{in}~\Omega \ u &= 0 \qquad \mbox{on}~\partial \Omega. \end{align*} If $u$ assumes its maximum in $x_0 \in \Omega$, then the eccentricity of level sets close to the maximum is determined by the Hessian $D^2u(x_0)$. We prove that $D^2u(x_0)$ is negative definite and give a quantitative bound on the spectral gap $$ \lambda_{\max}\left(D^2u(x_0)\right) \leq - c_1\exp\left( -c_2\frac{diam(\Omega)}{inrad(\Omega)} \right)$$ for universal $c_1, c_2$ This is sharp up to constants. The proof is based on a new lower bound for Fourier coefficients whose proof has a topological component: if $f:\mathbb{T} \rightarrow \mathbb{R}$ is continuous and has $n$ sign changes, then $$ \sum_{k=0}^{n/2}{ \left| \left\langle f, \sin{kx} \right\rangle \right| + \left| \left\langle f, \cos{kx} \right\rangle \right| } \gtrsim_n \frac{ | f|^{n+1}_{L^1(\mathbb{T})}}{ | f|^{n }_{L^{\infty}(\mathbb{T})}}.$$ This statement immediately implies estimates on higher derivatives of harmonic functions $u$ in the unit ball: if $u$ is very flat in the origin, then the boundary function $u(\cos{t}, \sin{t}):\mathbb{T} \rightarrow \mathbb{R}$ has to have either large amplitude or many roots. It also implies that the solution of the heat equation starting with $f:\mathbb{T} \rightarrow \mathbb{R}$ cannot decay faster than $\sim\exp(-(# \mbox{sign changes})^2 t/4)$.