Hölder Stability Inequalities Overview

• Hölder Stability Inequalities are quantitative refinements of classical Hölder inequalities that measure deviations using explicit deficit terms in Lp and geometric contexts.
• They establish precise stability estimates in elliptic, parabolic, and spectral problems, controlling perturbations and guiding symmetry and regularity analyses.
• Their methods leverage deficit function analysis, symmetrization techniques, and weighted functional inequalities to derive robust bounds across analytic and probabilistic models.

Hölder Stability Inequalities are quantitative refinements of classical Hölder-type inequalities, focusing on explicit, often optimal, measures of stability or deficit from the equality case. They provide control over how perturbations in structure—whether of functions, measures, domains, or operators—translate into quantitative bounds in analytic, geometric, and probabilistic frameworks. These inequalities have been established, sharpened, and characterized in both abstract $L^p$ settings and highly structured geometric–analytic environments, and play a key role in elliptic and parabolic regularity theory, spectral problems, and geometric analysis.

1. Quantitative Hölder Inequalities and Deficit Terms

The classical Hölder inequality asserts for $1 < p < \infty$, $q = p/(p-1)$, and $f \in L^p$, $g \in L^q$: $\|fg\|_1 \leq \|f\|_p \|g\|_q,$ with equality if and only if $|f|^p$ and $|g|^q$ are proportional almost everywhere. Hölder stability inequalities provide an explicit "deficit term" quantifying deviation from equality. Aldaz (Aldaz, 2013) establishes: $$\|fg\|_1 \leq \|f\|_p \|g\|_q \left( 1 - \frac{1}{M} D(f,g)^2 \right),$$ where $M = \max\{p, q\}$, and

$$D(f,g) = \left\| \frac{|f|^{p/2}}{\|f\|_p^{p/2}} - \frac{|g|^{q/2}}{\|g\|_q^{q/2}} \right\|_2$$

measures the $L^2$-distance between the normalized power functions. Equality is attained if and only if $|f(x)|^p/\|f\|_p^p = |g(x)|^q/\|g\|_q^q$ almost everywhere, precisely the classical equality cases.

Notably, for $p = 2$ (thus $q = 2$), this reduces to the standard form of Cauchy–Schwarz stability: $$\|fg\|_1 \leq \|f\|_2 \|g\|_2\left(1 - \tfrac12\| f/\|f\|_2 - g/\|g\|_2 \|_2^2\right).$$ The deficit term directly quantifies how far the functions are from being multiples in $L^2$-geometry. The constant $M$ is optimal, as verified on discrete models (Aldaz, 2013).

2. Geometric and Functional Analytic Stability: Domains and Eigenfunctions

Hölder-type stability arises in sharpening isoperimetric, spectral, and symmetry results. In "Hölder stability for Serrin's overdetermined problem" (Ciraolo et al., 2014), quantitative stability is given in domain symmetry for semilinear elliptic problems: $$\Delta u + f(u) = 0 \text{ in } \Omega, \quad u=0 \text{ on } \partial \Omega,$$ with $f$ Lipschitz, $\Omega$ convex (or more generally $(C,\theta)$-domain), and $u_\nu$ the inward normal derivative on $\partial\Omega$. Define the Lipschitz seminorm $[u_\nu]_{\partial\Omega}$ and radii

$$r_i = \min_{x\in\partial\Omega} |x-O|, \quad r_e = \max_{x\in\partial\Omega} |x-O|$$

about a near-center $O$. The sharp stability inequality is: $r_e - r_i \leq C [u_\nu]_{\partial\Omega}^\tau$ with explicit $\tau = 1/(1+\gamma)$, where $\gamma$ is computable from cone Harnack constants, domain geometry, and nonlinearity parameters. This result gives polynomial (Hölder) control of closeness-to-sphericity based on the boundary oscillation of $u_\nu$, improving upon earlier logarithmic estimates.

The proof combines the method of moving planes, barrier constructions, boundary and cone Harnack inequalities, and quantitative reflection arguments, iterating in orthogonal directions to identify an approximate center and concentric trapping balls (Ciraolo et al., 2014).

3. Reverse and Stability Hölder Inequalities in Metric-Measure and Spectral Geometry

In "A reverse Hölder inequality for first eigenfunctions of the Dirichlet Laplacian on RCD(K,N) spaces" (Gunes et al., 2021), sharp reverse Hölder inequalities are established for first eigenfunctions $u$ of the Dirichlet Laplacian on non-smooth metric-measure spaces with synthetic Ricci lower bounds ("RCD(K,N)" spaces): $$\|u\|_{L^q(\Omega)} \leq \left( \frac{\int_0^{r(a)} z^q\,dm_{K,N}}{\int_0^{r(a)} z^p\,dm_{K,N}} \right)^{1/p} \|u\|_{L^p(\Omega)},\quad q\ge p>0.$$ Where $z$ is a one-dimensional model eigenfunction on the corresponding model interval. If equality occurs for any $q>p$, the space must be a spherical suspension model, indicating sharp rigidity.

A new quantitative stability theorem is also established: if $\|u\|_{L^q}$ is $\varepsilon$-close to the model value for an unbounded set of $q$, the diameter of the space is $O(\sqrt{\varepsilon})$-close to the model, quantified explicitly via: $$B \left( \pi\sqrt{(N-1)/K} - \mathrm{diam}(X) \right)^2 \leq C \varepsilon$$ for $B,B>0$ and $C=C(N,v,\lambda_1(\Omega))$. Proof utilizes one-dimensional rearrangement, Rayleigh quotient monotonicity, and quantitative isoperimetric control (Gunes et al., 2021).

4. Stability in Hölder Regularity via Weighted Functional Inequalities

Cho–Kim (Cho et al., 1 Mar 2025) developed a full equivalence between analytic and probabilistic Hölder stability properties for symmetric Dirichlet forms on metric measure spaces—including local, nonlocal, and mixed-type forms. Their framework uses weighted functional inequalities and a new tail condition (TJE($\Theta$)) for the jump kernel: $$J(x,B(x,r)^c) \leq C_\mathrm{JE} \frac{\Theta(x,r)}{\phi(x,r)}\quad 0<r\le R_0.$$ Weighted Poincaré, cut-off Sobolev, and Faber–Krahn inequalities (with scaling function $\phi$ and admissible weight $\Theta$) are shown to be quantitatively equivalent to stable Hölder regularity estimates (EHR($\Theta$), PHR($\Theta$)) for both harmonic and caloric functions, as well as near-diagonal lower bounds for the heat kernel. The main equivalence theorem precisely characterizes when such regularity is stable under perturbation, referencing explicit metric, measure, and tail parameters.

This mutual implication structure is summarized as:

Functional Inequality	Regularity/Heat Kernel Estimate
Cut-off Sobolev + Poincaré	Elliptic/Parabolic Hölder regularity
Tail bound for jump kernel	Near-diagonal heat kernel lower bounds

Under volume-doubling and the tail condition, all these properties are equivalent up to changes in constants (Cho et al., 1 Mar 2025).

5. Connections to Nonlocal and Probabilistic Regularity

The stability of Hölder estimates is particularly relevant for pure-jump and mixed-type Dirichlet forms describing stable Lévy processes and their trace processes on exterior or Lipschitz domains. In these settings, the jump kernel can exhibit strong boundary singularities (e.g., $|x-y|^{-d-\alpha}$ plus an additional singularity as $x\to\partial D$). The weighted framework and deficit terms control up-to-boundary and interior regularity quantitatively (Cho et al., 1 Mar 2025). This yields robust regularity results for both symmetric stable and censored operator models, encompassing solutions to both Dirichlet and Neumann-type nonlocal PDEs.

6. Technical Schemes Underlying Hölder Stability

Several methodologies recur in the derivation and proof of stability estimates:

• Deficit function analysis: Utilizing $L^2$-distances after nonlinear normalization (Mazur map, rearrangement) to explicate the "gap" from extremality (Aldaz, 2013).
• Symmetrization and rearrangement: Employing one-dimensional and decreasing rearrangements to reduce high-dimensional stability to model settings (Gunes et al., 2021).
• Barrier and covering arguments: Iterative application of weak/strong Harnack inequalities over chains of balls/cones to convert local information into global quantitative symmetry (Ciraolo et al., 2014).
• Weighted Caccioppoli and mean-value inequalities: Adapted to handle singular or weighted kernels in Dirichlet forms (Cho et al., 1 Mar 2025).
• Explicit dependence on geometric and analytic parameters: All constants and exponents are tracked with respect to domain regularity, curvature, jump measure tails, and spectral data.

These structural and analytical ingredients constitute the contemporary framework of Hölder stability inequalities as they arise in nonlinear analysis, geometric PDE, and probability theory.