Higher-dimensional Beurling–Malliavin multiplier theorem

Statement

Let $\omega:\mathbb R^d\to\mathbb R_{\le0}$ vanish on $B_2$ , and suppose that for $0\le a\le3$

|D^a\omega(x)|\le C_{\mathrm{reg}}\langle x\rangle^{1-a},

while its radial-line growth functional satisfies

\int_0^\infty\frac{G^*(r)}{1+r^2}\,dr\le C_{\mathrm{gr}}.

For every $\sigma>0$ , there is $f\in L^2(\mathbb R^d)$ with

\operatorname{supp}\widehat f\subseteq B_\sigma, \qquad |f(x)|\le C e^{c\sigma\omega(x)}, \qquad |f(x)|\ge\tfrac12\quad(x\in B_{r_{\min}}).

One may take $c\asymp_d\max(C_{\mathrm{reg}},C_{\mathrm{gr}})^{-1}$ , $r_{\min}\asymp_d\min(\sigma,\sigma^{-1})$ , and an explicit $C$ depending on $d$ and $\sigma$ .

Proof chain

The PSH-BM proposition constructs a plurisubharmonic minorant of the weight. The analytic BM proposition converts it to an entire $L^2$ function. Finally the Paley–Wiener theorem gives the required Fourier support.

Why this is weaker than the one-dimensional theorem

The classical theorem needs only Lipschitz regularity and a logarithmic integral. The higher-dimensional result imposes symbol-type bounds through third derivatives; these are nevertheless satisfied by the weights built from line-porous sets.

References

Alex Cohen, “Fractal uncertainty in higher dimensions,” 2024. arXiv record. Relevant: Theorem 1.4.