Weighted Partial Integrability Functional

• Weighted Partial Integrability Functional is a quantitative measure that assesses integrability in weighted analytic systems by relating L^p thresholds to the size of admissible structures.
• The framework establishes sharp critical thresholds and cellular decompositions, enabling precise canonical expansions for polyharmonic functions and vector fields.
• Numerical algorithms compute weighted fractions of invariant tori, facilitating the analysis of transitions from integrable to chaotic behavior in dynamical systems.

The weighted partial integrability functional quantifies structural and quantitative regimes of integrability in weighted analytic systems, including polyharmonic functions, degenerate PDEs, and analytic vector fields with distinguished invariant measures or densities. For polyharmonic functions, it parametrizes the threshold for nontrivial solutions under weighted $L^p$ constraints; for vector fields with Jacobi multipliers, it determines the weighted phase-space fraction foliated by ergodic invariant tori. In both settings, the theory achieves explicit thresholds, canonical decompositions, and numerical algorithms to analyze persistence, breakdown, and uniqueness phenomena.

1. Definition and Principal Frameworks

The weighted partial integrability functional appears in multiple settings:

1. Polyharmonic functions on the unit disk $\D$ (Borichev-Hedenmalm (Borichev et al., 2012)): The functional

$I_{p,\alpha}(u) = \int_{\D} |u(z)|^p (1 - |z|^2)^\alpha\, dA(z)$

measures the weighted $L^p$ integrability of $N$-harmonic functions $u$ ($\Delta^N u = 0$ in $\D$). The associated space is $$PH^N_\alpha(D) = \{u : I_{p,\alpha}(u) < \infty,\, \Delta^N u = 0\}$$.

2. Weighted invariant tori for analytic vector fields with Jacobi multipliers (Sardón–Zhao (Sardón et al., 11 Nov 2025)): Let $V$ admit a nonvanishing analytic Jacobi multiplier $\rho$, i.e., $\operatorname{div}(\rho V) = 0$. The weighted partial integrability functional

$$m_{\rho}(V) = \frac{\int_{\mathcal{T}_{\rho}(V)}\rho\,\Omega}{\int_{M}\rho\,\Omega}$$

tracks the weighted fraction of phase space (total measure $d\mu_\rho = \rho\,\Omega$) foliated by ergodic invariant $(n-1)$–tori.

Both functionals formalize the tradeoff between integrability, geometry, and the "size"—in a weighted sense—of the integrable set.

2. Critical Thresholds and Existence–Uniqueness Dichotomies

Polyharmonic Functions

A central result is the explicit, piecewise-affine “critical curve” $\beta(N, p)$: $$\beta(N,p) = \min_{0 \le j \le N}\, b_{j,N}(p),\quad b_{0,N}(p) = -1-(N-1)p$$

$$b_{j,N}(p) = \max\{-1-(j+N-1)p,\,-2+(j-N+1)p\},\quad 1 \le j \le N$$

This determines:

• For $\alpha \le \beta(N, p)$,

$PH^N_\alpha(D) = \{0\}$

i.e., only the zero function is admissible.

• For $\alpha > \beta(N, p)$, nontrivial $N$-harmonic functions with $I_{p,\alpha}(u) < \infty$ exist.

This threshold is sharp, justified via Hardy–Littlewood-type ellipticity estimates and induction on Almansi expansions.

Jacobi Multiplier Vector Fields

For analytic vector fields $V$ with Jacobi multiplier $\rho$:

• If $V$ is (weighted) Arnold–integrable and nondegenerate, then under small analytic perturbations with $\operatorname{div}(\rho V) = 0$, the measure $m_{\rho}(V) \to 1$, i.e., almost all phase space is covered by invariant tori.
• As perturbations increase, $m_{\rho}(V)$ decreases, quantifying the breakdown of integrability and the destruction of toroidal foliations.

A plausible implication is that weighted integrability thresholds govern the persistence of quasi-periodic structures in weighted dynamical systems.

3. Canonical and Cellular Decomposition Schemes

Refined Almansi (Cellular) Decomposition

Every $u \in PH^N_\alpha(D)$ can be written uniquely as

$$u = w_0 + M[w_1] + \cdots + M^{N-1}[w_{N-1}],\quad M[f](z) = (1 - |z|^2) f(z)$$

Each $w_j \in PH^{N-j}_{\alpha + jp}(D)$ satisfies $L_{N - j - 1}[w_j] = 0$, where

$$L_\varepsilon = (1 - |z|^2)\Delta + 2\varepsilon(z\partial_z + \bar z \bar\partial_z) - \varepsilon(\varepsilon + 1)$$

This decomposition reflects a "cellular" structure: the region $\{\alpha > \beta(N, p)\}$ is tiled into open cells, each characterized by the active indices in the expansion.

Entangled Region

The entangled region $$E_N = \{(p, \alpha) : 0 < p < 1/N,\, \alpha \le -1 - Np\}$$ is characterized by the vanishing of all but the last cellular terms; outside $E_N$, additional lower powers of $M$ are admissible.

In the context of weighted vector fields, every symmetry field must be tangent to invariant tori under suitable nondegeneracy and resonance, meaning the symmetry algebra is tightly controlled by the integrability structure.

4. Quantitative Algorithms and Computation

For analytic vector fields with Jacobi multipliers, $m_{\rho}(V)$ can be computed numerically:

1. Integrate orbit equations for a grid of initial points.
2. Compute finite-time maximal Lyapunov exponents.
3. Classify orbits as quasi-periodic ($|\lambda_{\max}| < \varepsilon_{\mathrm{tol}}$) or chaotic.
4. Form the weighted sum:

$$\widehat m_{\rho}(V) = \frac{\sum_{i=1}^N \rho(u_0^{(i)}) R_i}{\sum_{i=1}^N \rho(u_0^{(i)})}$$

This method robustly estimates the weighted fraction of regular (integrable) phase-space structures up to a small numerical error. Representative examples exhibit transitions from fully integrable ($\alpha = 0$, $\widehat m_{\rho} = 1$) through partial integrability ($\alpha = 0.1$, $\widehat m_{\rho} \approx 0.69$), to regimes with no surviving tori ($\alpha = 0.5$, $\widehat m_{\rho} \approx 0$).

5. Geometric Tiling and Uniqueness in PDEs

The $(p,\alpha)$–plane is partitioned by lines $\alpha = a_{j,N}(p)$ into cells. Within each cell, the algebraic dimension of $PH^N_\alpha(D)$ and the structure of admissible expansions are constant. The entangled region is an exception, in which only the highest-power cellular term can survive.

In Dirichlet problems for $\Delta^N$, uniqueness of solutions under weighted integrability is equivalent to $\alpha \le \beta(N, p)$. For larger $\alpha$, nontrivial solutions exist even with vanishing boundary data. Local uniqueness near an arc is also characterized by sharp threshold conditions, generalizing Holmgren’s theorem.

6. Applications, Illustrative Examples, and Further Directions

Sharpness of all thresholds and decompositions is demonstrated by explicit model kernels, notably $U_{j,N}(z) = (1 - |z|^2)^{N + j - 1}/|1 - z|^{2j}$, with classification governed by $U_{j,N} \in L^p_\alpha(D) \iff \alpha > b_{j,N}(p)$.

In vector field theory, the continuous nature of $m_{\rho}(V)$ under analytic perturbations and its explicit dependence on $\rho$ generalizes classical Kolmogorov–Arnold–Moser measure estimates to weighted volume-preserving flows.

A plausible implication is the extension of cellular decomposition and integrability fraction concepts to more general analytic settings with degeneracies, weights, or partial invariance, with practical uses in distinguishing regimes of regularity and chaos in weighted PDEs and dynamical systems.