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Effective quasianalytic Remez inequalities on tame sets

This paper establishes an effective Remez inequality for functions in quasianalytic Denjoy-Carleman classes on tame fat compact sets by replacing polynomial degree with the Bang degree, thereby deriving explicit quantitative consequences such as Lojasiewicz, Harnack, and Markov-type inequalities.

Original authors: Armin Rainer

Published 2026-06-09

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CC BY 4.0

Original authors: Armin Rainer

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to guess the height of a mountain range (a mathematical function) based on a few scattered measurements you took while hiking. In the world of mathematics, there is a famous rule called the Remez inequality. It basically says: "If you know how high the mountain is in a small, specific patch of land, you can put a very strict limit on how high the entire mountain could possibly be, even in the parts you haven't measured."

For a long time, this rule only worked perfectly for polynomials (simple, smooth curves like $x^2$ or $x^3$ ). But what if the mountain is made of much more complex, wiggly material? What if it's a "quasianalytic" function—a type of smooth curve that is so well-behaved that knowing its shape at one point tells you everything about the whole curve, but it's not quite a simple polynomial?

This paper, by Armin Rainer, extends that "guessing the mountain" rule to these complex, wiggly functions, but with a twist: it works on a very specific, "tame" family of shapes (like blobs with jagged edges or cusps, but not infinitely chaotic ones).

Here is a breakdown of the paper's main ideas using everyday analogies:

1. The "Bang Degree": A New Ruler for Complexity

In the old days, to use the Remez rule for polynomials, you needed to know the degree of the polynomial (e.g., is it a $x^2$ curve or a $x^{100}$ curve?). The higher the degree, the more "wiggles" the curve can have, and the harder it is to predict the whole from a part.

For these complex functions, the author introduces a new ruler called the Bang degree.

The Analogy: Imagine the polynomial degree is like counting the number of gears in a clock. The Bang degree is like counting the "wiggles" allowed by the specific type of smooth material the function is made of. It's a number calculated from the function's "recipe" (its derivatives) and its current size.
The Catch: If the function gets very small (close to zero), this "Bang degree" can get huge, making the prediction harder. The paper carefully accounts for this.

2. The "Tame" Territory

The paper doesn't work on just any shape. It focuses on "tame sets."

The Analogy: Think of a "tame" set as a shape you could draw with a pen without lifting it, even if the shape has sharp corners, cusps, or weird indentations. It's not a fractal that goes on forever in a chaotic way; it's a "well-behaved" blob.
The paper proves that if your "mountain" sits on one of these tame blobs, the Remez rule still holds. This includes shapes found in nature and geometry that are "definable" in a specific logical system (o-minimal structures), which essentially means they aren't infinitely messy.

3. The Main Result: The "Propagation of Smallness"

The core theorem is a Quantitative Propagation-of-Smallness Principle.

The Analogy: Imagine you have a rubber sheet (the function) stretched over a tame blob. If you know the sheet is at least a certain height in a small patch, the paper gives you a formula to calculate exactly how high the sheet can possibly get on the rest of the blob.
The formula depends on:
1. The Bang degree (how wiggly the material is).
2. The geometry of the blob (how big and weirdly shaped it is).
3. How much of the blob you measured (the size of the patch).

4. What Else Can We Do With This? (The "Spillover" Effects)

Once you have this powerful ruler, the paper shows you can use it to solve other problems, like measuring the "volume" of things or how fast things change.

Sublevel Sets (The "Valley" Volume): If you ask, "How much of the blob is covered by water if the water level is $t$ ?" (i.e., where the function is less than $t$ ), the paper gives a precise limit on the volume of that water.
Lojasiewicz Inequalities (The "Slope" Rule): This is about how steep the slope is near a zero (where the function hits the ground). The paper gives a formula saying: "If you are close to the ground, the slope must be at least this steep." It replaces vague "there exists a constant" statements with exact numbers.
Harnack Inequalities (The "Temperature" Balance): If a function is like a temperature map that never hits zero, this rule says the hottest spot and the coldest spot on the blob can't be too far apart. The paper gives the exact ratio.
Markov Inequalities (The "Speed" Limit): This limits how fast the function can change (its derivative) based on its overall size. It's like saying, "If the car's average speed is low, it can't have a moment where it's going 200 mph."
Oscillatory Integrals (The "Wave" Decay): This deals with waves (like sound or light) passing through these shapes. The paper predicts how fast these waves die out (decay) as they travel, which is crucial for understanding how signals behave in complex environments.

Summary

In simple terms, Armin Rainer has taken a classic tool for predicting the behavior of simple curves and upgraded it to handle complex, "wiggly" curves on weirdly shaped, but "tame," territories. He didn't just say "it works"; he wrote down the exact math formulas (with explicit constants) so that anyone can calculate the limits of these functions without guessing. This is a "quantitative" breakthrough, turning vague mathematical concepts into precise, usable engineering tools for these specific types of functions.

Technical Summary: Effective Quasianalytic Remez Inequalities on Tame Sets

1. Problem Statement

The classical Remez inequality provides a bound on the maximum norm of a polynomial on a domain $K$ in terms of its maximum norm on a subset $E \subset K$ with positive measure. Specifically, for polynomials of degree $d$ , the ratio $\|p\|_K / \|p\|_E$ is controlled by a constant depending on $d$ and the relative measure of $E$ . This paper addresses the extension of such inequalities to quasianalytic Denjoy–Carleman classes $C^M$ on a broad family of compact sets with "tame geometry" in $\mathbb{R}^n$ .

The core challenge is that quasianalytic functions are not determined by a finite degree but by a sequence of derivative bounds $M = (M_j)$ . Consequently, the "degree" of the function must be replaced by an effective complexity parameter, the Bang degree, which depends on the weight sequence $M$ and the size of the function. Furthermore, the geometric setting must extend beyond convex bodies to include sets with cusps and subanalytic structures, requiring a unified framework that handles non-convex, "fat" compact sets.

2. Methodology

The author employs a synthesis of techniques from approximation theory, o-minimal geometry, and real analysis:

Quasianalytic Weights and Bang Degree: The paper utilizes a sequence of weights $\mu = (\mu_j)$ to define the class $C^M$ . The Bang degree $D_\mu(b)$ is introduced as a non-negative integer derived from the sequence $\mu$ and a lower bound $b$ on the function's norm. This integer serves as the effective degree in the inequalities.
Admissible Sets ( $\mathcal{K}(\mathbb{R}^n)$ ): The geometric framework relies on a collection of sets defined by Pierzchala's notion of admissibility. These sets admit a uniform polynomial parametrization with controlled Jacobians. This class includes all convex bodies, fat compact sets with Hölder boundaries, and all fat compact sets definable in the o-minimal structure $\mathbb{R}_{CM}$ (generated by restricted $C^M$ functions).
Reduction to Univariate and Convex Cases: The proof strategy involves reducing the multivariate problem on admissible sets to:
1. A univariate Remez inequality for vector-valued functions (generalizing results by Nazarov, Sodin, and Volberg).
2. A multivariate inequality on convex bodies.
3. A geometric decomposition of admissible sets into images of star-shaped sets under polynomial maps.
Composition Estimates: A critical technical step involves bounding the derivatives of the composition of a $C^M$ -smooth function with a polynomial curve. This requires careful estimation using Bernstein's lemma and Cauchy estimates to determine how the weight sequence shifts under composition.

3. Key Contributions and Results

3.1. The Main Remez Inequality

The central result (Theorem 1.1 and Theorem 4.13) establishes that for a function $f$ in a quasianalytic class $C^M$ on a tame set $K$ , the maximum norm is bounded by:
$\|f\|_K \le C_N \left( \frac{|K|^{\nu_K}}{|K|^{\nu_K} - (|K| - |E|)^{\nu_K}} \right)^N \|f\|_E$
where:

$N = 2 D_{a_K \mu}(b_0/M_0)$ is the Bang degree, depending on the geometry of $K$ (via $a_K$ ) and the size of $f$ (via $b_0$ ).
$\nu_K \in (0, 1]$ is a geometric constant depending only on $K$ .
$C_N$ is an explicit constant depending on $N$ and the weight sequence.
$a_K$ is a constant related to the diameter of $K$ and the degree of the polynomial maps parametrizing $K$ .

Unlike polynomial Remez inequalities, the degree $N$ is not fixed but depends on the function's magnitude relative to the weight sequence.

3.2. Quantitative Consequences

The paper derives several explicit inequalities from the main Remez result:

Volume of Sublevel Sets: An explicit bound on the measure of sets where $|f(x)| \le t$ , showing that small values occupy a volume controlled by the Bang degree (Corollary 1.6).
Reverse Hölder Inequalities: Comparison of $L^p$ and $L^q$ norms for $0 < q < p \le \infty$ , with constants depending explicitly on $N$ and the geometry of $K$ (Corollary 1.7).
Integrability: Proof that $|f|^{-\alpha}$ is integrable for $0 < \alpha < 1/N$ (Corollary 1.8).
Mean Oscillation: Bounds on the mean oscillation of $\log |f|$ , though the paper notes this does not generally imply $f$ is in BMO due to the dependence of $N$ on the function size (Corollary 1.9).
Lojasiewicz Inequalities: Explicit bounds for the order of vanishing and gradient estimates near the zero set of $f$ . For instance, $|\nabla f(x)| \ge c |f(x)|^\alpha$ with explicit $c$ and $\alpha$ depending on the Bang degree and the "thickness" of the set $K$ (Corollaries 1.11, 1.12, 1.13).
Harnack Inequalities: Bounds comparing the maximum and minimum of non-vanishing functions on $K$ , with explicit constants derived from the Remez inequality applied to both $f$ and $1/f$ (Corollary 1.14).
Markov Inequalities: Upper bounds for the derivatives of $f$ in terms of the norm of $f$ , utilizing the bound on the number of zeros provided by the Bang degree (Corollary 1.15).
Oscillatory Integrals: Decay estimates for integrals of the form $\int e^{i\lambda f(x)} g(x) dx$ , where the phase $f$ is quasianalytic. The decay rate is $\lambda^{-1/(N+2)}$ , determined by the Bang degree of the second derivative (Corollaries 1.16, 1.17).

3.3. Vector-Valued Generalization

A significant technical contribution is the extension of these results to vector-valued maps $f: L \to \mathbb{R}^m$ . The paper emphasizes that the vector-valued case does not follow trivially from the scalar case component-wise, as the "degree" $N$ must account for the simultaneous behavior of all components to avoid overestimation (Remark 1.2).

4. Significance and Claims

The paper claims to establish a quantitative propagation-of-smallness principle for quasianalytic functions on a wide class of non-convex, tame sets. Its significance lies in:

Explicit Constants: Unlike many results in o-minimal geometry or quasianalytic theory where constants are existential or unspecified, this work provides explicit, computable constants depending on the Bang degree and geometric parameters.
Unification: It unifies the classical polynomial Remez inequality, its convex multivariate generalizations, and previous univariate quasianalytic results into a single framework applicable to "tame" sets (including subanalytic and restricted $C^M$ definable sets).
Effective Complexity: The introduction of the Bang degree as a complexity parameter analogous to polynomial degree allows for the translation of algebraic geometric intuition (e.g., number of zeros, degree of vanishing) into the analytic setting of quasianalytic functions.
Applications: The derived inequalities (Lojasiewicz, Harnack, Markov, oscillatory integrals) provide effective tools for analysis in settings where analyticity is too restrictive but smoothness is insufficient, specifically within the context of o-minimal structures.

The author maintains a modest tone regarding the limitations, noting that the Bang degree depends on the function's size (shrinking to infinity near zeros) and that the resulting bounds for mean oscillation do not yet establish the BMO property for general quasianalytic functions. The work is presented as a foundational step toward effective quantitative analysis in tame geometry.