Quantitative Propagation of Smallness

• Quantitative propagation of smallness is the phenomenon where controlling a PDE solution or its gradient on a small set yields explicit bounds on its behavior over a larger domain.
• Key methods such as three-sphere inequalities, Carleman estimates, and Hausdorff content techniques enable precise estimates across elliptic, parabolic, and transmission problems.
• Applications include sharp spectral estimates, effective control theory, and robust unique continuation results, highlighting its importance in modern PDE analysis.

Quantitative Propagation of Smallness Estimates

Quantitative propagation of smallness describes the phenomenon wherein control over the "smallness" of a solution to a PDE, or its gradient, on a subset with positive measure or suitable Hausdorff content, leads to explicit bounds on the solution in a larger domain. This concept is intricately connected to unique continuation, three-sphere inequalities, absolute monotonicity, log-convexity, frequency function methods, and Carleman estimates. It has become central in the analysis of elliptic, parabolic, and transmission problems on various geometries, enabling precise spectral and control-theoretic applications.

1. Core Principles and Three-Sphere Inequalities

The foundational device for quantitative propagation of smallness is the three-sphere (three-ball) inequality. For a harmonic function $u$ in the unit ball (or, more generally, a solution of a uniformly elliptic PDE with Lipschitz or analytic coefficients), the classical estimate is

$$\sup_{|x|\leq r} |u(x)| \leq C \left( \sup_{|x|\leq r'} |u(x)| \right)^{\theta} \left( \sup_{|x|\leq R} |u(x)| \right)^{1-\theta}$$

for $0$\theta = \frac{\log(R/r')}{\log(R/r)} \in (0,1)$, and $C$ dependent on dimension and aspect ratios (Lippner et al., 2013). In discrete settings (harmonic functions on lattices), Lippner–Mangoubi establish analogous results using $L^2$ growth functions defined along random walks, with exact log-convexity and the optimal exponent $\theta$ surviving discretization up to an exponentially decaying error term in the number of steps.

These inequalities generalize to solutions of elliptic equations with lower-order terms, higher-order systems (e.g. buckling plates (Tian et al., 2023)), parabolic problems with transmission (coefficient jumps) (Francini et al., 2017), and to domains with piecewise smooth coefficients (Cârstea et al., 2019). The constants in the inequalities explicitly reflect ellipticity, geometric distortion, and coefficient regularity.

2. Hausdorff Content, Critical Sets, and Gradient Estimates

Propagation of smallness from a set of small measure or fractal dimension is governed by the Hausdorff content of the smallness set. For solutions $u$ of $\operatorname{div}(A \nabla u)=0$ with Lipschitz $A$, Logunov–Malinnikova established the principle: $$|u(x)|\leq C\epsilon^{\gamma} \quad\text{in } B_{1/2}$$ when $|u|<\epsilon$ on $E\subset B_{1/2}$ with $|E|>0$, and $C$, $\gamma$ depend on $A$ and $|E|$ (Logunov et al., 2017). This extends, via combinatorial doubling-index induction and effective critical-set estimates [Cheeger–Naber–Valtorta], to smallness on subsets of positive $(n-1+\delta)$-dimensional Hausdorff content. Recent work has sharpened this threshold:

• Foster–Gallegos (Foster et al., 28 Aug 2025) demonstrate that for harmonic $u$ in $B_1\subset \mathbb R^n$, control of $|\nabla u|$ on a set $E$ of positive $(n-2+\delta)$-Hausdorff content (for arbitrarily small $\delta>0$) propagates to quantitative smallness on all $B_{1/2}$, resolving conjectures on codimension thresholds for unique continuation. The proof exploits refined frequency-function inductive arguments and hyperplane combinatorics.
• In 2D, Zhu (Zhu, 2023) and (Wang, 12 Mar 2024) obtain analogous propagation inequalities with explicit dependence on ellipticity and the content of the observation set, applicable to both the function and its gradient.

A plausible implication is that propagation thresholds for gradients match the maximal codimension of critical sets for the PDE under consideration.

3. Absolute Monotonicity, Log-Convexity, and Discrete Settings

Absolute monotonicity in discrete contexts underpins log-convexity and propagation of smallness. For discrete harmonic functions $u$ on $\mathbb Z^d$, the $L^2$ growth functions $Q_u(n) = \mathbb E_0[u(X_n)^2]$ (where $X_n$ is the discrete-time random walk) are absolutely monotonic sequences: $\Delta^k Q_u(0)\geq 0 \;\;\forall k,$ implying exact log-convexity on the logarithmic scale (Lippner et al., 2013). Application of Newton series and power-series convexity arguments yield discrete analogues of three-sphere inequalities, with fully explicit constants and exponents matching continuous settings in certain regimes.

This concept extends to analytic and periodic settings (see (Kenig et al., 2019) for propagation in elliptic homogenization), and is tightly related to the finite-dimensionality of polynomial harmonic functions and the control of vanishing orders.

4. Carleman Estimates, Transmission Problems, and Piecewise Coefficients

Carleman estimates are central for propagation of smallness in PDEs with rough or discontinuous coefficients. For transmission problems across interfaces with Lipschitz jumps (elliptic or parabolic), local Carleman estimates yield three-region or three-ball inequalities that allow propagation of smallness from one side of an interface to the other (Francini et al., 2020, Francini et al., 2017, Cârstea et al., 2019). The transmission conditions (continuity of solution and flux) ensure control over boundary and interior norms via weighted inequalities.

In parabolic settings, the propagation of smallness manifests as Hölder-type interpolation: smallness propagates from the inner and outer spatial regions across time-slices, facilitated by parabolic Lipschitz bounds. The precise interpolation exponents reflect underlying Carleman parameters and geometric data.

Piecewise Lipschitz operators (with jumps) admit "global" propagation results: quantitative bounds on solutions or their gradients are achieved throughout the domain, with control depending on interface regularity (e.g. $C^2$ geometry), ellipticity, and the ratio of domain volume to minimal separation from the interface (Cârstea et al., 2019).

5. Applications: Spectral Estimates, Control Theory, and Quantitative Invertibility

Propagation of smallness facilitates sharp spectral projectors and control-theoretic inequalities:

• Spectral inequalities for $-\Delta$ and Schrödinger operators on compact manifolds: $L^2\to L^\infty$ bounds for finite-band spectral projectors are derived using chains of three-ball inequalities, covering the manifold by small balls where propagation applies (Burq et al., 2021, Balc'h et al., 22 Mar 2024, Burq et al., 2019). This yields the classical bound

$$\|\Pi_{[\lambda,\lambda+1]}\|_{L^2\to L^\infty} \leq C\lambda^{(n-1)/2}.$$

for Laplace–Beltrami eigenfunctions.

• Observability and null-controllability for the heat equation: global smallness, or even control on sets of positive Lebesgue or Hausdorff measure, leads to cost estimates for control localized to sparse (possibly fractal) sets (Burq et al., 2019, Balc'h et al., 22 Mar 2024).
• Spectral estimates for Schrödinger equations with unbounded potentials and control on thick or generalized thick sets: explicit dependence of constants on the density and growth of the sensor set, as well as the spectral interval, are established via sharp Cauchy uniqueness and holomorphic three-ball inequalities (Malinnikova et al., 6 May 2025, Wang, 12 Mar 2024).
• Quantitative invertibility and approximation for nonlocal operators: propagation-of-smallness estimates underlie robust inversion and Runge-type approximation for truncated Hilbert and Riesz transforms via harmonic extension methods (Rüland, 2017).

6. Optimality, Constants, and Limitations

Key features determining the propagation rates and constants:

• The optimality of the exponent $\theta$ in three-sphere inequalities is established for both discrete and continuous harmonic functions (Lippner et al., 2013).
• The constants $C$ and the exponents $\gamma,\alpha$ depend polynomially or exponentially on geometric ratios, ellipticity, coefficient bounds, and the measure or Hausdorff content of the smallness set.
• In analytic domains or for equations with analytic coefficients, sharper (e.g. Cartan–Pólya or Remez-type) inequalities are available, replacing measure with capacity.
• Thresholds for propagation from small sets are precise: e.g., for gradients, codimension two is sharp in the harmonic case (Foster et al., 28 Aug 2025), while for more general elliptic equations a small dimensional loss $c_n>0$ remains (Logunov et al., 2017).
• For operators with singular lower-order terms, rates are weakened and depend on integrability and geometric separation (Malinnikova et al., 2010).

A plausible implication is that further generalizations—e.g., to systems or higher codimension—require deeper geometric or analytic tools, and failure of propagation may occur for "too small" sets (below the critical Hausdorff dimension threshold).

7. Connections to Nodal Geometry, Unique Continuation, and Runge Approximation

Propagation of smallness also provides quantitative bounds on the measure (Hausdorff content) of nodal and critical sets of eigenfunctions, harmonic functions, and solutions to PDEs with analytic or piecewise smooth coefficients (Logunov et al., 2017, Tian et al., 2023). Coupled with combinatorial covering arguments, these bounds inform sharp Runge-type approximation costs and quantitative invertibility, with exact exponential dependence on domain, regularity, and vanishing order (Rüland, 2017, Cârstea et al., 2019).

In summary, quantitative propagation of smallness estimates form a rigorous backbone for modern analyses of unique continuation, control problems, spectral theory, and nodal geometry in elliptic and parabolic PDEs, with deep ramifications for inverse problems, approximation, and geometric analysis.