Removability Critic in Analysis

• Removability critic is a framework that defines when singularities or exceptional sets can be ignored without obstructing global analytic properties such as regularity and conformality.
• The criteria rely on geometric measure theory, including capacity, Hausdorff measures, and dimension thresholds, to rigorously determine the removability in conformal, Sobolev, and PDE contexts.
• Methodologies involve the use of cutoff functions, energy estimates, and analytic capacity techniques that extend local solutions globally, thereby unifying approaches across geometric function theory and integrability.

A Removability Critic analyzes mathematical criteria that determine whether singularities, exceptional sets, or parameter values in analytic, geometric, or PDE contexts are “removable”—i.e., whether they obstruct the global extension of certain structural properties (regularity, conformality, differentiability, etc.). The topic encompasses foundational results in geometric function theory, PDE theory, calculus of variations, and the analysis of mappings, and is central to understanding when apparent local singularities can be “ignored” for global analytic purposes.

1. Notions and Definitions of Removability

Removability is always defined relative to a function class and an analytic structure. Major classical and modern variants include:

• Conformal (S-removability, CH-removability):
  • S-removable sets in the complex plane: compact $K\subset\mathbb{C}$ such that every conformal embedding $$f\colon\mathbb{C}\setminus K\rightarrow\widehat{\mathbb{C}}$$ extends to a Möbius map.
  • CH-removable: $K$ such that every homeomorphism of $\widehat{\mathbb{C}}$ that is conformal off $K$ is a Möbius map (Ntalampekos, 30 Aug 2024).
• Sobolev removability:
  • $K\subset \mathbb{R}^n$ is $W^{1,p}$-removable if every continuous function $u$ with $u\in W^{1,p}(\mathbb{R}^n\setminus K)$ also lies in $W^{1,p}(\mathbb{R}^n)$ (Ntalampekos, 2017, Doherty et al., 2023).
• Removability for differentiability and Lipschitz conditions:
  • $E\subset\mathbb{R}$ is removable for differentiability (or pointwise Lipschitz condition with constant $C$) if every continuous $f$ with $f'(x)=0$ (or $L(f,x)\le C$) off $E$ must have the same property everywhere (Craig et al., 2014).
• Removability in PDEs (Elliptic/Parabolic):
  • For solutions of PDEs (e.g., Laplace or heat equations), removability refers to the possibility to extend solutions smoothly across isolated singularities or sets, under specific “smallness” or capacity conditions (Takahashi et al., 2013, Abdulla, 1 Jan 2025, Abdulla, 2023, Jost et al., 2017).
• Parameter Removability in Integrable Systems:
  • For zero-curvature representations (ZCRs) of PDEs, a spectral parameter is removable if it can be gauged away by a smooth family of gauge transformations (Kiselev et al., 2013).
• Geometric/Mapping Removability:
  • For mappings (quasiregular, $w$-curves), a closed set $E$ is removable if every mapping with suitable regularity away from $E$ extends (uniquely) with the same structure across $E$ (Egorov, 2010, Ikonen, 2 Jul 2024).

2. Key Criteria and Sharp Removability Theorems

Removability typically reduces to a fine analysis of the geometry or measure of the exceptional set, and sufficient regularity of the structures involved.

2.1. Measure and Dimension Thresholds

• Painlevé–Iwaniec–Martin (Quasiregular mappings/curves):

For bounded $K$-quasiregular $w$-curves in $\mathbb{R}^n$, if $E$ satisfies

$\dim_{\mathcal H}(E) < \frac{n}{1+K A(n)\|w\|_1},$

then $E$ is removable (Ikonen, 2 Jul 2024).

• Sobolev removability for detour sets:

If $K$ is a “Hölder detour set” and $p>n$, then $K$ is $W^{1,p}$-removable (Ntalampekos, 2017).

• Brownian graph:

The graph of Brownian motion is a.s. not $W^{1,p}$-removable for $p<\infty$ but is $W^{1,\infty}$-removable (Doherty et al., 2023).

• Hölder graphs in the plane ($W^{1,2}$):

Graphs of Hölder-$\alpha$ functions with $\alpha > 2/3$ are $W^{1,2}$-removable; for $\alpha<2/3$ there are non-removable examples (Tecu, 2010).

• Product sets:

For $E = C \times F\subset\mathbb{R}^2$, removability for $W^{1,p}$ is dictated by $\dim_H(C) < (p-2)/(p-1)$ (Rajala et al., 2021).

2.2. Capacity and Hausdorff Measure

• Intrinsic (double-phase) Hausdorff measure in non-uniformly elliptic problems:

For a quasilinear PDE driven by $H(x,z) = |z|^p + a(x)|z|^q$, removability for Hölder-continuous solutions is characterized by the vanishing of the intrinsic Hausdorff measure tailored to $H$ (Chlebicka et al., 2019).

• Metric surface characterization (S-/CH-removability):

A compact $K\subset\mathbb{C}$ is S-removable iff there exists a metric surface $(X,d)$ and a quasiconformal homeomorphism $f$ with $\mathcal{H}^1_d(f(K))=0$ (Ntalampekos, 30 Aug 2024).

2.3. PDE Singularities

• Heat equation (Kolmogorov–Petrovsky and Wiener-type criteria):

Removability at a space-time singularity is characterized in terms of the divergence of capacity series or explicit integral criteria involving geometric thinness and parabolic capacities (Abdulla, 1 Jan 2025, Abdulla, 2023).

• Elliptic and nonlinear PDEs:

Vanishing Pohozaev constant is necessary and sufficient for the removability of isolated singularities in super-Liouville and conformally invariant systems (Jost et al., 2017).

• Time-dependent singularities:

For the heat equation, a moving point singularity is removable if the local growth of the solution near the singularity is $o(|x-\xi(t)|^{2-N})$ ($N\ge3$) or $o(\log|x-\xi(t)|)$ ($N=2$) (Takahashi et al., 2013).

2.4. Parameter Removability in ZCRs

• Gauge-theoretic criterion:

In a $\mathbb{Z}_2$-graded ZCR, a spectral parameter $\lambda$ is removable if

$$\frac{\partial a_\lambda}{\partial \lambda} = d_h Q_\lambda - [a_\lambda, Q_\lambda]$$

admits a (parity-even) solution $Q_\lambda$, allowing $\lambda$ to be trivialized via smooth gauge transformations (Kiselev et al., 2013).

3. Methods and Proof Strategies

• Capacity and Cutoff Functions:

Many arguments exploit the vanishing of Sobolev or intrinsic capacities: cut-off sequences localized near $E$ allow for global extension of functions or mappings, as in Iwaniec–Martin and Egorov’s framework (Egorov, 2010).

• Modulus and Sobolev Extension:

For planar/quasiconformal removability, modulus estimates and extension theorems for Sobolev functions across negligible sets play a critical role (Ntalampekos, 30 Aug 2024).

• Caccioppoli and Energy Estimates:

For mappings subject to analytic inequalities ($|Df|^n \leq K f^*w$), absorption of lower-order terms via Caccioppoli estimates and fundamental inequalities enable extension across small sets (Ikonen, 2 Jul 2024).

• Fine Potential Theory:

For parabolic and elliptic PDEs, fine-topological thinness and capacity-criteria (Wiener test, etc.) are central to removability analysis (Abdulla, 1 Jan 2025, Abdulla, 2023).

• Functional Equations and Analytic Capacity:

Bishop’s construction of non-removable flexible curves leverages log-capacity singular homeomorphisms, showing that neither measure nor dimension alone suffice for removability (Younsi, 2017).

4. Representative Examples and Counterexamples

Set/Context	Sharp Removability Threshold	Reference
Sierpiński gasket $K$	$W^{1,p}$-removable for $p>2$	(Ntalampekos, 2017)
Hölder graph (plane)	$W^{1,2}$-removable if $\alpha>2/3$	(Tecu, 2010)
Product set $C\times F$	$p > (2 - s)/(1 - s)$, $s = \dim_H(C)$	(Rajala et al., 2021)
Brownian motion graph	$W^{1,\infty}$-removable, not $W^{1,p}$ for $p<\infty$	(Doherty et al., 2023)
Flexible Jordan curve	Zero area but non-removable for conformal homeomorphisms	(Younsi, 2017)
Heat equation singularity	Divergence of Petrovsky-type integral or Wiener-type series	(Abdulla, 1 Jan 2025)

5. Limitations, Extensions, and Open Problems

• Sharpness and endpoints:

Many criteria are sharp but typically fail at lower endpoints (e.g., $p=n$ for detour sets in $\mathbb{R}^n$ (Ntalampekos, 2017), $\alpha=2/3$ for Hölder graphs (Tecu, 2010)).

• Lack of geometric characterization:

Conformal and quasiconformal removability lack a geometric or dimensional characterization—there exist zero-area, non-removable Jordan curves (Younsi, 2017); length, area, and analytic capacity are not enough.

• Mapping and PDE analogues:

The relation between removability for various classes (quasiconformal, Sobolev, PDE) remains open, especially at critical thresholds (e.g., $p=2$ for $W^{1,2}$-removability vs. conformal removability).

• Non-uniform elliptic and weighted/variable exponent settings:

Removability has been generalized to non-uniformly elliptic PDEs using intrinsic measures, but further development in Orlicz, fractional, and degenerate frameworks is ongoing (Chlebicka et al., 2019).

• Parameter removability obstructions:

There exist essential spectral parameters (not gauge-removable) in $\mathbb{Z}_2$-graded Lax pairs, which are critical to true integrability (Kiselev et al., 2013).

• Fine-topological analogues:

For parabolic PDEs, fine-thinness provides an exact geometric language for removability, yet extensions to nonlinear or critical/parabolic equations are largely unsettled (Abdulla, 1 Jan 2025, Jost et al., 2017).

6. Broader Impact and Applications

Removability theory informs:

• Uniformization, rigidity, and conformal dynamics (e.g., He–Schramm rigidity, circle domains, Julia sets) (Younsi, 2015, Lyubich et al., 2020).
• Geometric analysis and mapping theory, especially in regularity and extension theorems for mappings with analytic constraints (Ikonen, 2 Jul 2024, Egorov, 2010).
• PDE singularities and blow-up analysis, providing necessary and sufficient conditions for the essentiality of singularities in local and global problems (Takahashi et al., 2013, Jost et al., 2017).
• Spectral theory and integrability via the analysis of (non)removability of spectral parameters in ZCRs/flat connections (Kiselev et al., 2013).

Removability criteria act as precise “critics,” filtering out inessential singularities from the global analytic structure being considered. Their optimal forms are often subtle, requiring interplay between analysis, geometry, topology, and function theory—making the paper of removability fundamental across modern analysis.