Critical Besov Space Analysis

• Critical Besov Space is a function space with a norm invariant under scaling, essential for studying nonlinear PDEs and energy decoupling in complex settings.
• It enables the decomposition of spaces into irreducible components where indicator functions form a basis, reflecting finite-dimensional rigidity.
• Critical exponents in these spaces delineate thresholds for nontriviality and density, linking geometric structure with analytic regularity.

A critical Besov space is a function space whose norm is precisely invariant under the natural scaling of the underlying differential equation or geometric structure. In mathematical analysis and the theory of partial differential equations, such spaces arise as natural endpoints for existence, uniqueness, and regularity results—serving as the minimal context in which the nonlinear, scaling-invariant phenomena of interest can be effectively studied. This concept applies both to classical settings (Euclidean space, domains with boundary, spaces with group structure) and to more general metric measure spaces, including fractals, glued cube complexes, and spaces supporting non-local forms of regularity. A prominent aspect of the theory is the connection between the structure of the underlying space and finite-dimensionality, decomposition into irreducible components, and the definition of sharp exponents governing nontriviality and density.

1. Finite Dimensionality and Structure of Critical Besov Spaces

For a metric measure space $(X,d,\mu)$, consider the Besov space $B_{p,p}^{\theta}(X)$ defined by

$$B_{p,p}^{\theta}(X) = \bigg\{ u \in L^p(X) : \|u\|_{B_{p,p}^{\theta}(X)}^p = \iint_{X\times X}\frac{|u(x)-u(y)|^p}{d(x,y)^{\theta p}\,\mu(B(x,d(x,y)))}\,d\mu(y)d\mu(x) < \infty \bigg\}$$

where $p\in (1,\infty)$ and $\theta > 0$ measures the (nonlocal) smoothness. The finite-dimensionality phenomenon refers to the situation where $\dim B_{p,p}^{\theta}(X) = k < \infty$ for some $k\in \mathbb{N}$. This rigidity implies:

• Every $u\in B_{p,p}^{\theta}(X)$ is a bounded, simple function, taking at most $k+1$ distinct values (outside a null set).
• Characteristic functions of certain measurable sets $\chi_{E_1},\dots,\chi_{E_k}$ form a basis for $B_{p,p}^{\theta}(X)$.
• On finite-measure spaces with $k=1$, only constants belong to $B_{p,p}^{\theta}(X)$.

This behavior is especially prominent in spaces formed by "gluing" simpler pieces, such as joining two $n$-dimensional cubes at a vertex ("bow-tie" examples) or gluing fractals (e.g., Sierpiński gaskets) at points, where the nonlocal energy penalizes variation between components so strongly that functions in $B_{p,p}^{\theta}(X)$ are forced to be constant on each component.

2. Potential-Theoretic Decomposition and Irreducibility

A main theorem in this context ((Kumagai et al., 2 Sep 2024), Theorem 1.1) states: if $B_{p,p}^{\theta}(X)$ is $k$-dimensional for $k>1$, there exists a decomposition of $X$ into measurable "irreducible components" $E_1,\ldots,E_k$ such that

• Each $E_i$ has $0 < \mu(E_i) < \infty$,
• (If $\mu(X)<\infty$) $\bigsqcup_{i=1}^{k}E_i$ covers $X$ up to a null set,
• $\chi_{E_i}\in B^{\theta}_{p,p}(X)$ and the collection $\{\chi_{E_i}\}_{i=1}^k$ forms a basis,
• $$B_{p,p}^{\theta}(X) = \bigoplus_{i=1}^k B_{p,p}^{\theta}(E_i)$$, with each $B_{p,p}^{\theta}(E_i)$ being one-dimensional (consisting of constants),
• The indicator functions $\chi_{E_i}$ have zero Korevaar–Schoen energy: $\|\chi_{E_i}\|_{KS^{\theta}_p(X)}=0$, i.e., no nontrivial potential-theoretic energy is required to "communicate" across different irreducible components.

This reveals a "potential-theoretic disconnection": when $B_{p,p}^{\theta}(X)$ is finite-dimensional, there is energetic decoupling between components, and any "energy-carrying" function decomposes as a direct sum of constants on these sets.

3. Critical Exponents: Nontriviality and Density

The paper introduces two critical exponents sharply delineating the structure of Besov spaces on $X$:

• Nontriviality exponent:

$$\theta_p(X) := \sup \{\theta>0: B_{p,p}^{\theta}(X)\text{ contains a nonconstant function}\}$$, which quantifies the largest $\theta$ for which "nontrivial" (i.e., nonconstant) functions exist.

• Density exponent:

$$\theta_p^{*}(X) := \sup \{\theta>0: B_{p,p}^{\theta}(X) \text{ is dense in }L^p(X)\}$$, which marks the threshold for density of $B_{p,p}^{\theta}(X)$ in $L^p(X)$.

Generally, $\theta_p(X)\geq \theta_p^*(X)$. For doubling metric measure spaces, $\theta_p(X)\geq 1$ always. When $X$ supports a $p$-Poincaré inequality (such as for Newton–Sobolev spaces $N^{1,p}(X)$), $\theta = 1$ is the critical exponent, but for many self-similar fractals, one encounters $\theta_p(X)>1$.

Examples highlight the dependence of $\theta_p(X)$ on both geometry and measure-theoretic properties of $X$:

• For the "bow-tie" (two glued hypercubes), if $1$\theta_p(X) = n/p > 1$, but on each individual cube, $\theta_p=1$.
• For spaces formed by gluing Sierpiński gaskets at a point, $$\theta_p(X)=\theta_p^*(X)=\frac{\operatorname{dim}_H(K)}{p}$$, where $\operatorname{dim}_H(K)$ is the Hausdorff dimension.
• For spaces formed by gluing Sierpiński carpets, a gap may appear: $\theta_p(X)>\theta_p^*(X)$ for certain $p$ values ($1<p<d_{ARC}$, with $d_{ARC}$ the Ahlfors regular conformal dimension).

4. Key Mathematical Formulations

Central methodological points and formulas include:

• Besov seminorm (integral form):

$$\|u\|_{B_{p,p}^{\theta}(X)}^p = \iint_{X \times X} \frac{|u(x) - u(y)|^p}{d(x,y)^{\theta p}\,\mu(B(x,d(x,y)))} \, d\mu(y)d\mu(x)$$

• Korevaar–Schoen energy (localized variant):

$$\|u\|_{KS_{p}^{\theta}(X)}^p= \limsup_{r\rightarrow 0^{+}} \int_X \int_{B(x,r)} \frac{|u(x)-u(y)|^p}{r^{\theta p}\mu(B(x,r))} d\mu(y)d\mu(x)$$

• Normal contraction property: If $v$ is a "normal contraction" of $u$ (i.e., $|v(x)-v(y)|\leq |u(x)-u(y)|$, $|v(x)|\leq |u(x)|$), then $v\in B_{p,p}^{\theta}(X)$ and $$\|v\|_{B_{p,p}^{\theta}(X)}\leq \|u\|_{B_{p,p}^{\theta}(X)}$$.
• Decomposition for $u\in KS_p^\theta(X)\cap L^\infty(X)$:

$$\|u \chi_{E_j}\|_{KS_{p}^{\theta}(X)}^p = \limsup_{r\to 0^{+}} \int_{E_j}\int_{B(x,r)\cap E_j}\frac{|u(x)-u(y)|^p}{r^{\theta p}\mu(B(x,r))}d\mu(y)d\mu(x)$$

This expresses the complete decoupling of potential-theoretic energy along irreducible components.

5. Examples: Cubic, Fractal, and Glued Spaces

The theory is illustrated on various canonical spaces:

• Glued cubes ("bow-tie" spaces): $X$ is the union of two $n$-cubes glued at a point. For $1$\theta_p(X)=n/p$, associated to the (Euclidean) Hausdorff dimension, but functions in $B_{p,p}^{\theta_p(X)}(X)$ are forced to be constant on each cube for large $\theta$.
• Sierpiński gasket gluing: Two gaskets glued at a point satisfy $\theta_p(X)=\frac{\dim_H(K)}{p}$, and for this exponent, $B_{p,p}^{\theta}(X)$ is two-dimensional (characteristics of the two gaskets).
• Sierpiński carpet gluing: Two carpets glued at a point yield $\theta_p(X)>\theta_p^*(X)$ for $1<p<d_{ARC}$, indicating a gap between the scale at which nonconstant functions exist and the scale at which the Besov space is dense in $L^p(X)$. This reflects complex nonlocal relationships between the pieces at the threshold exponent.

For these fractal examples, computation of $p$-walk dimension and related scaling factors are crucial, intertwining the analysis of function spaces with the fine geometric and measure-theoretic structure of $X$.

6. Broader Applications and Implications

These results have implications for analysis on metric measure and fractal spaces, Sobolev-type embedding theorems, and geometric measure theory:

• Detection of decomposition: Finite-dimensionality of $B_{p,p}^{\theta}(X)$ signals the possibility of decomposing $X$ into energetically decoupled components.
• Measurement of nonlocality: Critical exponents $\theta_p(X),\theta_p^*(X)$ quantify the threshold of nontrivial and dense function approximation, capturing the "strength" of the nonlocal energy cost in the geometry.
• Connections to quasiconformal and fractal geometry: For spaces like the Sierpiński carpet, the $p$-walk dimension and Ahlfors regular conformal dimension bridge the theory of critical Besov spaces to conformal gauge and weight studies.
• Classical versus fractal settings: On spaces supporting a $p$-Poincaré inequality, critical Besov spaces behave as classical Sobolev spaces with full density and richness; fractal or glued spaces display rigidity and collapse in critical regimes.

7. Summary

The theory of critical Besov spaces on metric measure spaces (including fractals and glued geometric complexes) reveals that when these spaces become finite-dimensional, the underlying space necessarily decomposes into irreducible components with energetically decoupled behavior. Critical exponents $\theta_p(X),\theta_p^*(X)$ strictly quantify the transition between "flexibility" (existence/density of nonconstant functions) and "rigidity" (collapse to constants and simple functions). These structural phenomena have deep connections to potential theory, analysis on fractals, and the geometry of spaces with highly nonlocal interaction, providing a precise analytic framework that matches the geometric and measure-theoretic properties of the space itself (Kumagai et al., 2 Sep 2024).