Rapid decay of a shifted torus integral

Statement

Let $v$ be a Diophantine direction on $\mathbb T^n$ , and let smooth $F(s,y)$ be supported in a fixed bounded $s$ -interval. Assume $\int\langle F(s,\cdot)\rangle_y\,ds=0$ . For $M\ge1$ , define

J_MF(U,y)=\int_{\mathbb R}F(U+z,y+Mzv)\,dz.

For every fixed pair of integers $m,p\ge0$ , there is a finite integer $r=r(m,p)$ and a constant independent of $M$ such that

\|J_MF\|_{C^m_{U,y}}\le C_{m,p}M^{-p}\|F\|_{C^r_{s,y}}.

This uses the zero-mean directional inverse.

Repeated integration by parts

Let $L=v\cdot\nabla_y$ , and let $F^\circ=F-\langle F\rangle_y$ . The omitted mean has zero full-line integral. Integrating the total $z$ -derivative of $L^{-1}F^\circ(U+z,y+Mzv)$ gives

J_MF=(-M^{-1})^p\int_{\mathbb R} \partial_s^pL^{-p}F^\circ(U+z,y+Mzv)\,dz.

All boundary terms vanish. A fixed finite loss for each application of $L^{-1}$ , together with $p$ source derivatives, gives a finite $r$ ; for example, if one inverse costs $\ell$ derivatives, $r=m+p+p\ell$ suffices. The fixed-shift form handles additional $U$ derivatives without an $M$ factor.

Quantifiers

The constant and required derivative order can grow with $p$ . A family with uniform bounds at every fixed derivative order therefore gives decay faster than every inverse power of $M$ . A single finite regularity bound gives only the corresponding finite range of powers. No estimate uniform in $p$ is asserted.