Plurisubharmonic Beurling–Malliavin proposition

Statement

Suppose $\omega:\mathbb R^d\to\mathbb R_{\le0}$ satisfies the vanishing, three-derivative, and radial-line growth hypotheses of the higher-dimensional Beurling–Malliavin theorem. Then there is a continuous plurisubharmonic $u:\mathbb C^d\to\mathbb R$ such that

u(x)\le\omega(x),\qquad u(x)=0\quad(|x|\le2),

|u(x_1)-u(x_2)|\le C_{\mathrm{Lip}}|x_1-x_2|, \qquad u(x)\le u(x+iy)\le u(x)+\rho|y|,

where $C_{\mathrm{Lip}},\rho\lesssim_d \max(C_{\mathrm{reg}},C_{\mathrm{gr}})$ .

Why modification is needed

The original weight need not satisfy the transverse Hessian condition in the exact PSH-BM proposition. A dyadic modification produces a minorant with comparable regularity and controlled line integrals, to which the exact extension applies.

Role

This proposition solves the potential-theoretic half of the multiplier problem. The analytic BM proposition then replaces $u$ by the logarithmic size of an entire function.

References

Alex Cohen, “Fractal uncertainty in higher dimensions,” 2024. arXiv record. Relevant: Proposition 2.2 and §§3–4.