Critical Integrability Index

• Critical Integrability Index is a precise threshold in analysis and physics that demarcates distinct regimes of functional, geometric, or dynamical behavior.
• It is defined through explicit analytic and algebraic formulas across domains such as Sobolev embeddings, pluripotential theory, and spectral optimization.
• Applications span improved regularity in PDEs, conserved quantum currents in sigma models, and optimal spectral bounds, guiding integrability transitions.

The critical integrability index is a central concept across harmonic analysis, complex geometry, PDEs, spectral theory, and mathematical physics, serving in each context as a precise threshold that separates distinct regimes of functional, geometric, or dynamical behavior. This index quantifies, via explicit analytic or algebraic formulas, the borderline at which various phenomena—ranging from Sobolev embeddings and maximum principles to quantum integrability and spectral optimization—undergo a qualitative change. Its manifestation depends acutely on the underlying objects (operators, spaces, singularities, spectral families) and the nature of admissible functions or conserved quantities.

1. Critical Integrability Index in Harmonic and Functional Analysis

The classical paradigm for the critical integrability index appears in the study of Sobolev inequalities, Riesz potentials, and sharp Adams/Moser–Trudinger-type embeddings. For the Riesz potential $I_a f(x)$ and the fractional Laplacian $(-\Delta)^{a/2}$ on $\mathbb{R}^n$, the gain of integrability mapped by these operators is governed by the critical index $p = n/a$. At this threshold, standard $L^q$ embeddings fail and are replaced by optimal exponential-integrability inequalities: $$I_a : L^{n/a}(\mathbb{R}^n) \to \text{exp-}\text{Orlicz}\left(\frac{n}{n-a}\right),$$ where the critical integrability exponent $\frac{n}{n-a}$ demarcates the Orlicz class dictating integrability of $\exp(c|u|^{n/(n-a)})$ for $u$ in the critical Sobolev space $W^{a, n/a}(\mathbb{R}^n)$ (Fontana et al., 2017). This index is sharp and universal: incrementally lowering $(-\Delta)^{a/2}$0 results in higher Lebesgue integrability, while attempting to raise it above $(-\Delta)^{a/2}$1 renders the embedding false.

2. Critical Integrability Index in Pluripotential Theory

In pluripotential theory, especially concerning plurisubharmonic (psh) singularities and multiplier ideals, the integrability index $(-\Delta)^{a/2}$2 of a singular psh function $(-\Delta)^{a/2}$3 is defined as

$(-\Delta)^{a/2}$4

serving as a complex-analytic measure of the strength of a singularity and generalizing the log-canonical threshold in algebraic geometry. For multi-circled (radial) singularities, $(-\Delta)^{a/2}$5 admits a convex-analytic expression: $(-\Delta)^{a/2}$6 where $(-\Delta)^{a/2}$7 is the directional Lelong number, and its value encodes the optimal integrability exponent for $(-\Delta)^{a/2}$8 near the origin (Rashkovskii, 2011). There exist sharp inequalities of Skoda and Demailly type relating the integrability index to higher Lelong numbers: $(-\Delta)^{a/2}$9 with equality cases precisely characterized for monomial and multi-circled singularities, demonstrating the fine structure of the critical integrability index in complex analysis.

3. Critical Integrability in PDEs and Regularity Theory

In nonlinear, particularly degenerate and singular PDEs, the critical integrability index governs the threshold beyond which regularity theory is fundamentally altered. For the singular porous medium equation (or fast diffusion system), the critical exponent

$\mathbb{R}^n$0

in dimension $\mathbb{R}^n$1 marks the transition: if $\mathbb{R}^n$2, local boundedness and higher integrability of gradients require enhanced $\mathbb{R}^n$3 integrability of solutions, with the self-improving estimate

$\mathbb{R}^n$4

holding for $\mathbb{R}^n$5 determined explicitly in terms of the problem data, but the gain $\mathbb{R}^n$6 as $\mathbb{R}^n$7 (Bögelein et al., 16 Jan 2025). This critical index thus delineates the fast-diffusion range with delicate regularity properties and specifies the extra summability needed for higher integrability.

For maximum principles in elliptic and nonlocal PDEs, critical integrability exponents arise in the sharp dependence of the strong maximum principle on the integrability of coefficients. For instance, for the Schrödinger operator $\mathbb{R}^n$8 on $\mathbb{R}^n$9,

$p = n/a$0

is precisely the critical case: one-sided (weak) maximum principles hold, but the strong maximum principle fails, and full positivity only returns for $p = n/a$1 (Li et al., 2019). Analogous thresholds appear for first-order perturbations and fractional Laplacians, with matching critical exponents $p = n/a$2.

4. Critical Integrability Index in Quantum Integrability and Sigma Models

In two-dimensional quantum field theory, specifically sigma models, the (quantum) integrability index, as defined by Komatsu–Mahajan–Shao (KMS), provides a systematic, conformal-data-driven method to diagnose the existence of quantum-conserved higher-spin currents (Tian et al., 2020, Komatsu et al., 2019). For each spin $p = n/a$3,

$p = n/a$4

where $p = n/a$5 is the multiplicity of primary operators of given dimension and spin. $p = n/a$6 serves as a lower bound: if $p = n/a$7, there are at least $p = n/a$8 quantum-conserved spin-$p = n/a$9 currents. The critical integrability index—sometimes informally termed the "minimal spin $L^q$0 with $L^q$1"—distinguishes integrable models (with $L^q$2) from non-integrable ones (with $L^q$3). This method efficiently predicts known integrability properties, e.g., for $L^q$4, $L^q$5, and flag sigma models.

In $L^q$6 SYM theory, a related critical integrability index $L^q$7 appears as a parameter interpolating between the free and fermion-loop-deformed Baxter Q-operator at the Wilson-Fisher fixed point, encoding the effect of fermionic corrections on anomalous dimensions within the integrable spin-chain description (Velizhanin, 2022). Here, $L^q$8 links critical indices, deformation theory, and spectral integrability.

5. Critical Integrability in Spectral Optimization and Hamiltonian Dynamics

In the context of optimal spectral bounds (e.g., sums of Sturm-Liouville eigenvalues), the critical integrability index arises as the Lebesgue exponent $L^q$9 for which the spectral optimization problem

$$I_a : L^{n/a}(\mathbb{R}^n) \to \text{exp-}\text{Orlicz}\left(\frac{n}{n-a}\right),$$0

leads to critical systems of ODEs whose associated Hamiltonians become polynomial, i.e., meromorphically integrable, precisely for $$I_a : L^{n/a}(\mathbb{R}^n) \to \text{exp-}\text{Orlicz}\left(\frac{n}{n-a}\right),$$1 (Tian et al., 2023).

Polynomial Hamiltonian systems of this type are integrable in the sense of Liouville if and only if $$I_a : L^{n/a}(\mathbb{R}^n) \to \text{exp-}\text{Orlicz}\left(\frac{n}{n-a}\right),$$2 (i.e., $$I_a : L^{n/a}(\mathbb{R}^n) \to \text{exp-}\text{Orlicz}\left(\frac{n}{n-a}\right),$$3, the $$I_a : L^{n/a}(\mathbb{R}^n) \to \text{exp-}\text{Orlicz}\left(\frac{n}{n-a}\right),$$4 case) or $$I_a : L^{n/a}(\mathbb{R}^n) \to \text{exp-}\text{Orlicz}\left(\frac{n}{n-a}\right),$$5 with uniform spectral parameters; otherwise, dynamical chaos and non-integrability prevail. Thus, in this domain, the critical integrability index $$I_a : L^{n/a}(\mathbb{R}^n) \to \text{exp-}\text{Orlicz}\left(\frac{n}{n-a}\right),$$6 governs the existence of commuting Hamiltonian flows and solvability by quadrature.

6. Summary Table: Critical Integrability Indices Across Domains

Context	Index Expression	Transition Characterized
Riesz/Fractional Sobolev Embedding	$$I_a : L^{n/a}(\mathbb{R}^n) \to \text{exp-}\text{Orlicz}\left(\frac{n}{n-a}\right),$$7	From $$I_a : L^{n/a}(\mathbb{R}^n) \to \text{exp-}\text{Orlicz}\left(\frac{n}{n-a}\right),$$8 to exponential Orlicz spaces
Plurisubharmonic Singularities	$$I_a : L^{n/a}(\mathbb{R}^n) \to \text{exp-}\text{Orlicz}\left(\frac{n}{n-a}\right),$$9 via Lelong numbers	$\frac{n}{n-a}$0-integrability of $\frac{n}{n-a}$1
Porous Medium Equation	$\frac{n}{n-a}$2	Self-improving gradient integrability
Schrödinger Operator Maximum Principle	$\frac{n}{n-a}$3	Weak vs strong maximum principle
Sigma Model Quantum Integrability	$\frac{n}{n-a}$4	Onset of quantum-conserved currents
Spectral Optimization Hamiltonians	$\frac{n}{n-a}$5	Polynomial (integrable) vs nonintegrable
SYM/QFT Spin Chains	$\frac{n}{n-a}$6	Q-function deformation and spectrum shift

Each instance reflects a precise, quantifiable transition governed by the critical integrability index, with sharp inequalities, explicit constants, and boundary behaviors entirely dictated by these indices.

7. Connections and Concluding Remarks

The critical integrability index serves as a universal organizing principle for sharp regularity, embedding, integrability, and integrability-breaking thresholds in analysis and mathematical physics. Its role is fundamentally structural: it encodes the scaling, symmetry, and algebraic properties of underlying operators or dynamical systems. The precise determination of these indices leads not only to sharp functional inequalities but also to definitive characterizations of phenomena such as anomalous conservation laws, spectral transitions, or the onset of chaotic dynamics. Across domains, the critical integrability index is both a technical and conceptual tool of the first importance (Fontana et al., 2017, Rashkovskii, 2011, Bögelein et al., 16 Jan 2025, Tian et al., 2020, Komatsu et al., 2019, Tian et al., 2023, Velizhanin, 2022, Li et al., 2019).