Radial-line growth functional of a weight

Definition

For a measurable weight $\omega:\mathbb R^d\to\mathbb R$ , define

G(x)=\int_{1/2}^{2}|\omega(sx)|\,ds, \qquad G^*(r)=\sup_{|x|=r}G(x).

The radial-line growth condition is

\int_0^\infty\frac{G^*(r)}{1+r^2}\,dr<\infty.

Geometric meaning

The first average smooths the weight along a short radial segment; the spherical supremum then records the worst direction at radius $r$ . The final integral measures whether these worst directional masses can be summed over scales.

One-dimensional comparison

For $d=1$ , the growth integral is comparable, up to absolute constants, to $\int_{\mathbb R}|\omega(t)|/(1+t^2)\,dt$ , the condition in the classical Beurling–Malliavin multiplier theorem.

Role of line porosity

When $\omega$ is assembled from cubes meeting a line-porous set, the line-section measure estimate controls $G^*$ on every dyadic annulus.

References

Alex Cohen, “Fractal uncertainty in higher dimensions,” 2024. arXiv record. Relevant: equations (1.9)–(1.13).