Topological Bounds for Fourier Coefficients and Applications to Torsion

Let $\mathrm{\Omega }\subset {\mathbb{R}}^2$ 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 $x_0\in \mathrm{\Omega }$, then the eccentricity of level sets close to the maximum is determined by the Hessian $D^{2}u(x_0)$. We prove that $D^{2}u(x_0)$ is negative definite and give a quantitative bound on the spectral gap $${\lambda }_{\max }(D^{2}u(x_0))\le -c_1\exp (-c_2\frac{diam(\mathrm{\Omega })}{inrad(\mathrm{\Omega })})$$ 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 $f:\mathbb{T}\to \mathbb{R}$ is continuous and has $n$ sign changes, then $$\sum _{k=0}^{n\mathrm{/}2}\mid ⟨f,\sin kx⟩\mid +\mid ⟨f,\cos kx⟩\mid {\gtrsim }_n\frac{\mathrm{\mid }f{\mathrm{\parallel }}_{L^1(\mathbb{T})}^{n+1}}{\mathrm{\parallel }f{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(\mathbb{T})}^n}\mathrm{.}$$ 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}$$ has to have either large amplitude or many roots. It also implies that the solution of the heat equation starting with $f:$\mathbb{T}$ \rightarrow $\mathbb{R}$$cannotdecayfasterthan$\sim\exp(-(# \mbox{sign changes})^2 t/4)$.