Linewise Poisson extension operator

Core idea

For suitable $\omega:\mathbb R^d\to\mathbb R$ , define its linewise Poisson extension $E\omega:\mathbb C^d\to\mathbb R$ by

E\omega(x+iy)=\frac1\pi\int_{\mathbb R} \frac{\omega(x+ty)}{1+t^2}\,dt.

When $y=0$ , the formula gives $E\omega(x)=\omega(x)$ .

Linewise harmonicity

For fixed $x$ and nonzero $y$ , restriction to the complex line $z\mapsto x+zy$ is the Poisson extension of the restriction of $\omega$ to the real line $x+\mathbb R y$ . Thus it is harmonic separately along every such complex line away from the real locus.

Convergence

If $\omega$ is Lipschitz and satisfies the radial-line growth condition, the defining integral is absolutely convergent and its symmetric truncations converge uniformly on compact subsets.

Levi-form structure

Transverse components of the Levi form of $E\omega$ are line integrals of the real Hessian of $\omega$ , that is, X-ray transforms. Adding $C|y|$ can therefore make the extension plurisubharmonic under explicit linewise bounds.

References

Alex Cohen, “Fractal uncertainty in higher dimensions,” 2024. arXiv record. Relevant: §3.