Knowlpedia

Theorem

Exact plurisubharmonic Beurling–Malliavin extension

Uniform Hilbert-transform and transverse Hessian bounds make a linewise Poisson extension plurisubharmonic after adding a multiple of the imaginary norm.

Statement

Let ω:Rd→R≤0\omega:\mathbb R^d\to\mathbb R_{\le0} be C2C^2 and compactly supported. Assume every affine line ℓ\ell satisfies

∥H[(ω∣ℓ)′]∥∞≤C1,\|H[(\omega|_\ell)']\|_\infty\le C_1,

and, for every line direction y^\widehat y and unit v^⊥y^\widehat v\perp\widehat y,

1π∫R⟨D2ω(x+ty^)v^,v^⟩ dt≥−C2.\frac1\pi\int_{\mathbb R} \langle D^2\omega(x+t\widehat y)\widehat v,\widehat v\rangle\,dt \ge-C_2.

If C≥max⁡(C1,C2)C\ge\max(C_1,C_2), then

u(x+iy)=Eω(x+iy)+C∣y∣u(x+iy)=E\omega(x+iy)+C|y|

is continuous and , and

u(x)≤u(x+iy)≤u(x)+2C∣y∣.u(x)\le u(x+iy)\le u(x)+2C|y|.
Division of labor between the hypotheses

The bound controls complex disks centered on Rd\mathbb R^d. The transverse Hessian integral controls the off the real locus via the .

References
  1. Alex Cohen, “Fractal uncertainty in higher dimensions,” 2024. arXiv record. Relevant: Proposition 3.1.