Damping function for a frequency set

Definition

Let $Y\subseteq\mathbb R^d$ and $c_1,c_2,c_3,\alpha\in(0,1)$ . The set $Y$ admits a damping function with these parameters if there is $\psi\in L^2(\mathbb R^d)$ such that

\operatorname{supp}\widehat\psi\subseteq B_{c_1},\qquad \|\psi\|_{L^2(B_1)}\ge c_2,

|\psi(x)|\le\langle x\rangle^{-d}\quad(x\in\mathbb R^d), \qquad |\psi(x)|\le \exp\!\left(-c_3\frac{|x|}{(\log(2+|x|))^\alpha}\right) \quad(x\in Y).

Here $\langle x\rangle=(1+|x|^2)^{1/2}$ .

Meaning of the four conditions

Bounded Fourier support limits the spectral cost of convolving with $\psi$ . The local $L^2$ lower bound prevents the zero function. Polynomial decay makes the function globally usable, while the final estimate gives substantially faster damping on $Y$ .

Why

\alpha<1

For $\alpha>1$ , the enhanced bound can hold everywhere for a nonzero band-limited function and would not record special geometry of $Y$ . The sublinear logarithmic exponent is the quasi-analytic threshold needed in the associated quantitative unique-continuation argument.

References

Rui Han and Wilhelm Schlag, “A higher-dimensional Bourgain–Dyatlov fractal uncertainty principle,” Analysis & PDE 13 (2020), 813–863. DOI record.
Alex Cohen, “Fractal uncertainty in higher dimensions,” 2024. arXiv record. Relevant: Definition 1.5.