Analytic Beurling–Malliavin proposition

Statement

Let $u:\mathbb C^d\to\mathbb R$ be plurisubharmonic, with $u|_{\mathbb R^d} \le0$ , $u=0$ on the real ball $B_2$ , Lipschitz constant $C_{\mathrm{Lip}}$ on $\mathbb R^d$ , and

u(x)\le u(x+iy)\le u(x)+\rho|y|.

Then there is an entire function $f:\mathbb C^d\to\mathbb C$ such that

|f(x+iy)|\le A e^{2\rho|y|},\qquad |f(x)|\le C e^{u(x)},\qquad f|_{\mathbb R^d}\in L^2,

and $|f(x)|\ge1/2$ on a ball of radius comparable to $\min(\rho,\rho^{-1})$ . The constant $C$ is explicit in $d,C_{\mathrm{Lip}},\rho$ .

Construction

Choose a cutoff $h$ equal to one near the origin. The Hörmander theorem solves $\bar\partial g=\bar\partial h$ with a carefully augmented plurisubharmonic weight. Then $f=h-g$ is entire. The added logarithmic term forces $g$ to be small near the origin, while an auxiliary decaying weight makes $f|_{\mathbb R^d}$ square-integrable.

From L2 to pointwise bounds

The holomorphic $L^2$ mean-value estimate converts the weighted integral estimate for $g$ and $f$ into the displayed pointwise bounds.

References

Alex Cohen, “Fractal uncertainty in higher dimensions,” 2024. arXiv record. Relevant: Proposition 2.3 and §5.