Frostman-Type Framework

• Frostman-type framework is a collection of measure-theoretic tools that relate the scaling behavior of measures to various geometric dimensions.
• It extends the classical Frostman lemma to non-Euclidean, fractal, and discrete settings, enabling optimal Sobolev embeddings and robust capacity estimates.
• Applications span multifractal analysis, trace theory, eigenfunction estimates, and random graph analysis, providing actionable insights into geometric measure theory.

A Frostman-type framework encompasses a family of measure-theoretic tools and principles foundational to modern geometric measure theory, analysis on fractals, and the theory of function spaces over singular or non-Euclidean geometries. At its core, the Frostman lemma relates the scaling behavior of measures supported on sets to their (Hausdorff, box, or intermediate) dimension, and more broadly informs the construction of extremal and test measures for embedding, multifractal, and capacity-theoretic problems. In recent decades, the Frostman paradigm has been extended and generalized across several axes: to fractal measures (upper Ahlfors-regularity), the slice method for intermediate dimensions, optimal Sobolev embeddings in metric and measure spaces, non-Euclidean or unimodular discrete settings, correlation and Fourier dimensions, and to the construction of joint and two-scale Frostman measures. These innovations enable the identification and estimation of fine geometric properties of sets and measures, and support sharp embedding theorems in the analysis of function spaces.

1. Classical Frostman Principle and Measure Decay

The classical Frostman lemma asserts that for any compact set $E \subset \mathbb{R}^d$ and parameter %%%%1%%%%, there exists a probability measure $\mu$ on $E$ such that

$$\mu(B(x,r)) \leq C r^s \quad \forall x \in E,\, r>0$$

if and only if the $s$-dimensional Hausdorff measure $\mathcal{H}^s(E)>0$ (Falconer et al., 27 May 2025, Dobronravov, 2022). This measure witnesses that $\dim_{H} E \geq s$, and the optimal exponent of decay recovers the exact Hausdorff dimension: $$\dim_{H} E = \sup\{s:\exists\, \mu \text{ with } \mu(B(x, r)) \leq C r^s ~ \forall x, r \}.$$ Equivalent forms employ energy integrals, notably

$$I_s(\mu) = \iint |x-y|^{-s} d\mu(x)d\mu(y) < \infty \implies \dim_H(E)\geq s$$

(Dobronravov, 2022, Falconer et al., 27 May 2025), and the capacity of $E$ for the Riesz kernel of order $s$ is positive precisely when $\mathcal{H}^s(E)>0$.

For absolutely continuous measures or those with densities, the Frostman condition applies locally, and upper $d$-Ahlfors regular measures (e.g., $d$-dimensional Hausdorff measure on a smooth $d$-plane) satisfy an estimate $\mu(B(x,r)) \leq C r^d$ for all $x, r$ (Cianchi et al., 2019).

2. Extensions: Sobolev Embeddings and Rearrangement-Invariant Spaces

The Frostman-type framework plays a central role in the characterization of optimal Sobolev embeddings in non-Euclidean or measure-theoretic settings. For domains $\Omega \subset \mathbb{R}^n$ equipped with a Frostman (upper Ahlfors-regular) measure $\mu$ of order $d \in (0,n]$, sharp embeddings are obtained by reducing the problem to one-dimensional weighted Hardy inequalities. For Sobolev spaces $W^{m,p}(\Omega)$ and target rearrangement-invariant spaces $Y(\Omega,\mu)$, the embedding

$$\|u\|_{Y(\Omega,\mu)} \leq C \| D^m u\|_{X(\Omega)}, \quad u \in W^{m,p}(\Omega)$$

is controlled by the behavior of $\mu$ via a critical exponent $\alpha = 1 - (d-mp)/p$ (Cianchi et al., 2019). The fast-decay ($mp < d \leq n$) and slow-decay ($0< d < mp$) regimes correspond to distinct types of one-dimensional Hardy operators. In limiting cases (e.g., $p = n/m$), the target spaces become augmented Lorentz–Zygmund spaces $L^{\infty, q;(-1)}$ which are strictly larger than $L^\infty$ yet capture endpoint embeddings. Classical Lebesgue measures (case $d=n$) and their sharp-Sobolev embedding manifestations (e.g., Brezis–Wainger) are recovered as special instances of this framework.

3. Multi-Scale, Intermediate, and Joint Frostman Measures

Recent advances extend the Frostman paradigm to characterize intermediate (Falconer–Fraser–Kempton) and multi-scale dimensions. A two-scale (or joint) Frostman measure provides simultaneous control over two different scaling regimes, often interpolating between Hausdorff and box-counting exponents. If $E\subset\mathbb{R}^d$ is compact, $\theta \in (0,1)$, and $0<h<s<\overline{\dim}_\theta E$, a joint Frostman measure $\mu_\delta$ on $E$ satisfies

$$\mu_\delta(B(x, r)) \leq \begin{cases} C\, \delta^{s-h} r^h & 0< r < \delta, \ C\, r^s & \delta \leq r \leq \delta^\theta, \end{cases}$$

for all $x\in E$ (Nicolas et al., 14 Feb 2025, Angelini et al., 6 Nov 2025). The extra factor $\delta^{s-h}$ allows interpolation across annular scales and is crucial in deducing lower bounds for the dimensions of typical slices (Marstrand-type projection theorems at intermediate scales). The two-scale construction (with exponents $s$ and $t$) provides a unified existence theory for measures reflecting both fine (micro-scale) and coarse (meso-scale) geometric properties (e.g., via a dyadic refinement exponent $\mathcal{D}(E)$) (Angelini et al., 6 Nov 2025).

4. Generalizations: Discrete and Unimodular Settings

The Frostman framework has further been adapted to unimodular random rooted discrete spaces, central to random graphs and point process theory (Baccelli et al., 2018). Here, equivariant weight functions play the role of measures, and the unimodular Frostman lemma states that there exists a weight $w$ with

$$w(N_r(v)) \leq r^\alpha \quad \text{a.s., for all } v \in D,\, r \geq M$$

if and only if the unimodular Hausdorff content at exponent $\alpha$ is positive, providing both upper and lower bounds on Hausdorff and Minkowski dimensions via growth of weights. Analogues of the mass distribution principle and Billingsley’s lemma are then used to relate the dimension of random discrete spaces to the polynomial rate of volume growth, with applications to random walks, trees, point processes, and percolation clusters.

In these spaces, covering and capacity arguments carry over by substituting classical measures with equivariant weight functions and integrating via the Mass-Transport Principle.

5. Frostman-Type Conditions in Multifractal and Mixed Settings

Frostman-type constructions in multifractal analysis monitor the local scaling exponents of measures and support the multifractal formalism for single and multiple measures, including non-Gibbsian or mixed cases (Menceur et al., 2018). Here, an auxiliary control function $\varphi(r)$ replaces $\log r$ in the definition of generalized Hausdorff and packing measures, leading to dimension functions $\dim_\varphi^q(E)$. The upper and lower scaling sets

$$\underline{\alpha}_j(x) = \liminf_{r\to 0} \frac{\log \mu_j(B(x,r))}{\varphi(r)}$$

play a Frostman-type role in defining multifractal spectra. The formalism accommodates both exact and inhomogeneous scaling (e.g., via slowly varying $\varphi$), as well as large deviation requirements. Classical Gibbs measure situations are retrieved as special cases.

6. Applications: Trace Theory and Eigenfunction Estimates

Explicit Frostman-type measures support optimal trace inequalities for Sobolev functions on lower-dimensional sets. For instance, on arbitrary planar rectifiable curves $\Gamma \subset \mathbb{R}^2$, one constructs sequences $\{\mu_k\}$ of measures (1-regular sequences) with uniform upper and lower mass bounds at dyadic scales, enabling a dyadic decomposition of the trace space and explicit representation of the quotient norm for the restriction of $W_p^1(\mathbb{R}^2)$ to $\Gamma$ (Tyulenev, 2020).

On Riemannian manifolds, eigenfunction restriction estimates depend on Frostman-type ball-growth at scale: if $\mu$ is a probability measure on $\Gamma \subset M$ with $\mu(B_g(x,r))\leq C r^s$, one obtains $L^p$–norm estimates for Laplace–Beltrami eigenfunctions restricted to $\Gamma$ reflecting the dimension $s$ and with exponent sharpness depending on the Frostman condition (Eswarathasan et al., 2019). This unifies restriction theory for submanifolds and generic fractal sets.

7. Broader Impact, Optimality, and Technical Innovations

The Frostman-type framework underpins a wide array of developments:

Context or Generalization	Key Feature	Reference
Sobolev embeddings on fractal measures	One-dimensional Hardy reduction and augmented target spaces	(Cianchi et al., 2019)
Intermediate and two-scale measures	Simultaneous fine- and coarse-scale Frostman conditions	(Nicolas et al., 14 Feb 2025, Angelini et al., 6 Nov 2025)
Discrete/unimodular spaces	Equivariant weight functions and random covering principles	(Baccelli et al., 2018)
Multifractal formalism, mixed measures	Control functions $\varphi(r)$ and joint spectra	(Menceur et al., 2018)
Hausdorff/correlation/box dimension	Integral and Fourier-analytic Frostman-type equivalences	(Falconer et al., 27 May 2025)
Generalized sum-over-disjoint balls	Supercovering and monotonicity for signed or vector-valued measures	(Dobronravov, 2022)

Several advances resolve technical obstacles in higher-order analysis, e.g., the use of symmetrization and sharp interpolation (Calderón–Zygmund K-functional) for embedding problems, and supercovering arguments for establishing Hausdorff measures and capacities for measures of bounded variation (Dobronravov, 2022). The optimality of exponents in Hardy-type inequalities is verified by radial power-law examples and concentration on $d$-planes.

In sum, the Frostman-type framework provides a universal language for deducing, witnessing, and bounding geometric and analytic dimensions, supporting sharp functional inequalities, dimension-theoretic projections, measure-theoretic images, and multifractal phenomena with broad applicability across analysis, probability, and geometric measure theory.