Knowlpedia

Theorem

Directional inverse with a Diophantine bound

A zero-mean smooth periodic function has a unique zero-mean directional primitive with finite derivative loss under a polynomial divisor bound.

Statement

Let vv be a with constants c,τc,\tau. Every smooth zero-mean ff on Tn\mathbb T^n has a unique smooth zero-mean solution of v⋅∇u=fv\cdot\nabla u=f, given by

u^(0)=0,u^(m)=f^(m)2πi v⋅m(m≠0).\widehat u(0)=0,\qquad \widehat u(m)=\frac{\widehat f(m)}{2\pi i\,v\cdot m}\quad(m\ne0).
Finite loss estimate

Fix an even integer s>τ+ns>\tau+n. For every integer k≥0k\ge0,

∥u∥Ck≤Ck,s,n,c,τ,v∥f∥Ck+s.\|u\|_{C^k}\le C_{k,s,n,c,\tau,v}\|f\|_{C^{k+s}}.

Indeed, smooth Fourier coefficient decay bounds the coefficient of ∂βf\partial^\beta f by C∥f∥Ck+s(1+∣m∣)−sC\|f\|_{C^{k+s}}(1+|m|)^{-s} for ∣β∣≤k|\beta|\le k. Dividing by the directional eigenvalue costs at most c−1(1+∣m∣)τc^{-1}(1+|m|)^\tau; the resulting series is absolutely summable because s−τ>ns-\tau>n. This proves the estimate and smooth reconstruction at every fixed order. Any two solutions differ only in the zero mode, so the zero-mean normalization gives uniqueness.

This inverse is a Fourier multiplier. It preserves a prescribed pullback frequency lattice and therefore the corresponding torus descent condition. It need not preserve compact support within the auxiliary torus.