Topological bounds for Fourier coefficients and applications to torsion

Let Ω⊂R2 be a bounded convex domain in the plane and consider−Δu=1inΩu=0on∂Ω. If u assumes its maximum in x0∈Ω, then the eccentricity of level sets close to the maximum is determined by the Hessian D2u(x0). We prove that D2u(x0) is negative definite and give a quantitative bound on the spectral gapλmax(D2u(x0))≤−c1exp⁡(−c2diam(Ω)inrad(Ω))for universalc1,c2>0. 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:T→R is continuous and has n sign changes, then∑k=0n/2|〈f,sin⁡kx〉|+|〈f,cos⁡kx〉|≳n|f‖L1(T)n+1‖f‖L∞(T)n. 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):T→R has to have either large amplitude or many roots. It also implies that the solution of the heat equation starting with f:T→R cannot decay faster than ∼exp⁡(−(#sign changes)2t/4).