Damping functions from line porosity

Statement

Let $Y\subseteq[-3h^{-1},3h^{-1}]^d$ be $\nu$ -porous on lines from scales $\mu>1$ to $h^{-1}$ . There is $\alpha=\alpha(\nu)<1$ such that, for every $0<\sigma<1$ , $Y$ admits a damping function with spectral radius $c_1=\sigma$ . The remaining parameters have the form

c_2=c(d,\mu)\sigma^{C_d},\qquad c_3=c(\nu,d)\sigma,

and one may choose $\alpha\le1-c\nu/|\log\nu|$ .

Construction

Cover each dyadic annulus by a finitely overlapping family of cubes. Place scaled bump functions on cubes meeting $Y$ and sum them to form a negative weight. The line-section measure bound gives the growth hypothesis for the higher-dimensional Beurling–Malliavin theorem.

Final polynomial decay

The multiplier obtained from Beurling–Malliavin has rapid decay on $Y$ but only a global bound. Multiplication by a fixed Schwartz function preserves controlled Fourier support up to convolution and supplies the required $\langle x\rangle^{-d}$ decay.

References

Alex Cohen, “Fractal uncertainty in higher dimensions,” 2024. arXiv record. Relevant: Proposition 1.7 and §6.1.