Interpolation Refinement of Gap Bounds

• Interpolation refinement of gap bounds is a technique using interpolation methods to reduce the gap between lower and upper analytic estimates in PDEs and optimization problems.
• It utilizes multiscale covering arguments, Caccioppoli inequalities, and embedding methods to enhance regularity and integrability under varying phase conditions.
• This approach extends to numerical, algebraic, and geometric settings, yielding sharper, parameterized bounds in areas from singular PDE analysis to conformal bootstrap.

Interpolation refinement of gap bounds encompasses a variety of phenomena and methodologies wherein interpolation techniques are leveraged to optimize, tighten, or structurally interpolate the “gap” between lower and upper bounds in analytic inequalities, regularity theory for PDEs, polynomial representation problems, approximation theory, and convex optimization, among others. The most developed and technically intricate instances arise in the analysis of singular parabolic double-phase problems, where the interpolation refinement precisely quantifies how improved regularity or integrability on solutions leads to strictly larger admissible “gap” between competing exponents in growth conditions.

1. Paradigmatic Setting: Singular Parabolic Double-Phase Problems

The singular parabolic double-phase equation, as found in recent works by Kim–Oh, is of the form

$$u_t - \operatorname{div}(|Du|^{p-2}Du + a(x,t)|Du|^{q-2}Du) = -\operatorname{div}(|F|^{p-2}F + a(x,t)|F|^{q-2}F)$$

in spacetime domains $$\Omega_T = \Omega \times (0,T) \subset \mathbb{R}^n \times \mathbb{R}$$, with $p$ and $q$ satisfying $\frac{2n}{n+2} < p \leq 2 < q$ and a modulating coefficient $a(x,t) \in C^{\alpha,\alpha/2}(\Omega_T)$, $0<\alpha \leq 1$, controlling the double-phase feature. Weak solutions $u$ are sought in appropriate parabolic energy classes.

The “gap” refers to $q-p$, quantifying the degeneracy between the $p$-phase and $q$-phase terms. The central objective is to determine under what hypotheses on $u$ and $a$ one may admit the largest possible $q-p$ while retaining higher integrability (regularity) of $Du$—that is, under which gap bound local reverse Hölder and higher integrability results hold, and how these gap bounds can be refined by interpolating between endpoint cases using additional information on $u$.

2. Classical and Interpolative Gap Conditions

Historically, the admissible $q-p$ (i.e., the gap) for obtaining higher gradient integrability was a rigid quantity, with sharp thresholds known for specific cases:

• In the elliptic, bounded-solution setting, the gap bound reduces to $q \leq p + \alpha$ for $a(\cdot) \in C^{\alpha}$.
• In the parabolic case with weak $u$ (e.g., $u \in L^2$), previous results required $q \leq p + 2\alpha/(n+2)$ or similar bounds (Kim et al., 17 Nov 2025).
• For minimizers with Hölder continuity, the precise threshold for the double-phase model is $q < p + \frac{\alpha}{1-\alpha}$ (Filippis et al., 2021).

The “interpolation refinement” asserts that the maximum admissible gap can be parameterized as a one-parameter family—interpolating between lower and upper thresholds—by exploiting improved integrability or regularity on $u$. For instance, assuming $u \in L^\infty(\Omega_T)$, one obtains

$q \leq p + \frac{\alpha[p(n+2) - 2n]}{4},$

while for $u \in C(0,T;L^s(\Omega))$, $s \geq 2$,

$$q \leq p + \frac{\alpha \mu_s}{n+s}, \qquad \mu_s = \frac{[p(n+2)-2n]s}{4}.$$

The endpoint $s=2$ recovers the classical bound; as $s \to \infty$, one recovers the $L^\infty$ result (Kim et al., 4 Jan 2026, Kim et al., 17 Nov 2025).

3. Analytical Techniques and Interpolative Mechanism

The core of the interpolation refinement methodology is a multiscale, phase-adapted covering argument, combined with Caccioppoli-type inequalities, parabolic and Gagliardo–Nirenberg embeddings, and bespoke reverse Hölder estimates. The analytic structure is as follows:

• Employ intrinsic cylinders—different scalings for $p$-phase and $(p,q)$-phase regions—defined by suitable local energy conditions.
• Use a stopping-time argument and disjoint Vitali-type covers to decompose super-level sets of the gradient energy.
• Derive phase-specific energy inequalities and parabolic Poincaré-type inequalities on these cylinders, ensuring that only $p$ or $q$-dominated phases persist under the refined gap bound; the “intermediate” (degenerate) case is ruled out precisely when the admissible gap holds.
• Glue local reverse Hölder inequalities via a covering-plus-Fubini-in-$\Lambda$ argument to achieve global higher integrability of $Du$.

The admissible $q-p$ is shown to interpolate smoothly as a function of the spatial-time integrability exponent $s$, with the transition captured by $\mu_s$ and ultimately by the regularity of $u$ (Kim et al., 4 Jan 2026).

4. Connections to Interpolation in Nonautonomous Integrals and Function Spaces

Closely related interpolative gap refinements arise in the calculus of variations for nonautonomous integrals with $(p,q)$-growth. Under a priori Hölder continuity of the minimizer, De Filippis–Mingione introduced a method whereby mixed local/nonlocal approximations and fractional Sobolev estimates yield sharp interpolation-type bounds on $q/p$, tied to the Hölder exponent. The main result for the classical double-phase integrand $F(x,z) = |z|^p + a(x)|z|^q$, $a \in C^{0,\alpha}$, is

$\frac{q}{p} < 1 + \frac{\alpha}{1 - \alpha},$

which is sharp and results from interpolating analytic regularity and spatial growth constraints using fractional Sobolev embeddings and Gagliardo–Nirenberg interpolants (Filippis et al., 2021).

5. Interpolation Refinement in Broader Contexts: Numerical and Algebraic Gap Bounds

The interpolation refinement paradigm also features prominently:

• In numerical conformal bootstrap, optimal polynomial interpolation at equilibrated nodes systematically tightens bounds on operator dimensions (“gap bounds”), with the interpolation error directly controlling the relaxation of positivity conditions; the gap bound is thus improved exponentially in the number of nodes, via potential-theoretic density optimization (Chang et al., 17 Sep 2025).
• In the algebraic analysis of univariate polynomial representations, Birkhoff ("lacunary") interpolation arguments convert linear independence questions of power sums into combinatorial gap bounds, closing the previous $\Omega(\sqrt{d})$ vs $O(d)$ lower-bound gap to $\Omega(d)$—thereby realizing the true optimality of sum-of-powers decompositions for polynomials of degree $d$ (Garcia-Marco et al., 2015).

6. Geometric Interpolation Refinements: Approximation of Manifolds

In geometric approximation, gap bounds measured via interpolation between triangulated surfaces and smooth manifolds achieve sharp constants using local feature size (lfs) and empty-ball size (ebs). For a $d$-simplex of radius $r$, the positional error between a point $x$ on the simplex and its closest point $\tilde{x}$ on the manifold satisfies

$$|x\tilde x| \leq \frac{r^2}{2\,ebs(\tilde x)} + O\!\left(\frac{r^4}{ebs(\tilde x)^3}\right),$$

which is optimal and improves previous constants by exploiting the sharp geometric structure of the embedding—an “interpolation” between combinatorics, geometry, and analysis (Khoury et al., 2019).

7. Implications and Structural Significance

Across these domains, the interpolation refinement of gap bounds shares core structural properties:

• Admissible “gaps” (differences between exponents, parameters, or quantities) can be enlarged beyond static/endpoint values by finely measuring intermediate regularity, integrability, or analytic behavior.
• The strength of the bound varies monotonically with the interpolating parameter (e.g., integrability exponent $s$, regularity $\alpha$), recovering classical endpoints and providing a parameterized family interpolating between them.
• The interpolative mechanism typically involves intrinsic scaling, stopping-time selection, energetic or combinatorial decomposition, and exploitation of mixed local/nonlocal analytic features.
• Such refinements reveal that improved a priori estimates (beyond mere energy classes) yield strictly stronger regularity, integrability, or approximation results and enable sharper threshold phenomena not accessible by classical techniques (Kim et al., 4 Jan 2026, Kim et al., 17 Nov 2025, Filippis et al., 2021, Khoury et al., 2019, Garcia-Marco et al., 2015, Chang et al., 17 Sep 2025).

In summary, interpolation refinement of gap bounds constitutes a cross-disciplinary analytical motif yielding optimal and quantitatively sharp results in regularity theory for PDEs, analytic and algebraic interpolation problems, numerical optimization bounds, and geometric approximation, often by interpolating between endpoint phenomena through technical and conceptual innovations in analytic interpolation frameworks.