Quantitative unique continuation

Definition

A quantitative unique-continuation estimate for a class $\mathcal F\subseteq L^2(\mathbb R^d)$ and an observation set $U\subseteq\mathbb R^d$ is an inequality

\|f\mathbf 1_U\|_2\ge c\|f\|_2, \qquad f\in\mathcal F,

with an explicit constant $c>0$ determined by stated geometric and analytic parameters. It strengthens qualitative unique continuation, which would only say that vanishing on $U$ forces $f=0$ .

Spectral version used in fractal uncertainty

The class $\mathcal F$ may consist of functions whose Fourier transforms are supported in a prescribed set and whose frequency-side decay is controlled by a damping function. Observation sets formed by placing one small cube in each unit cube then give a uniform lower bound. Iterating the resulting single-scale estimate across the holes of a porous set produces a fractal uncertainty principle.

Scope

The name also covers Carleman-estimate and spectral-inequality results in PDE. The defining feature is quantitative control, not one particular proof method.

References

Benjamin Jaye and Mishko Mitkovski, “Quantitative uniqueness properties for (L^2) functions with fast decaying, or sparsely supported, Fourier transform,” IMRN (2022). DOI record.
Rui Han and Wilhelm Schlag, “A higher-dimensional Bourgain–Dyatlov fractal uncertainty principle,” Analysis & PDE 13 (2020). DOI record.