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Improved polynomial decay for unbounded semigroups

This paper establishes improved polynomial decay rates for $C_0$ -semigroups under polynomial resolvent growth conditions in the right half-plane, specifically advancing previous results by eliminating logarithmic losses for unbounded semigroups on non-Hilbertian Banach spaces without requiring uniform boundedness.

Original authors: Chenxi Deng, Jan Rozendaal, Mark Veraar

Published 2026-05-20

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Chenxi Deng, Jan Rozendaal, Mark Veraar

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

The Big Picture: Taming a Wild System

Imagine you are trying to predict how a complex machine behaves over time. In mathematics, this machine is called a semigroup. It's a system that starts with an initial state (like a ball being dropped) and evolves according to specific rules (gravity).

Usually, mathematicians love systems that are "well-behaved." They want to know: Does this system eventually calm down and stop moving? If so, how fast?

For a long time, researchers had two main ways to answer this:

The "Perfect" Machine: If the machine is perfectly stable (bounded), we know exactly how fast it slows down based on how its internal gears (the "resolvent") behave.
The "Wild" Machine: If the machine is unstable or grows wildly at first (unbounded), the rules get messy. Previous methods to predict how fast it slows down were either too slow (pessimistic) or required the machine to be very smooth and perfect, which isn't always true in the real world.

This paper introduces a new, sharper tool. It allows mathematicians to predict exactly how fast these "wild" machines will calm down, even if they start out chaotic, without needing to assume they are perfectly smooth.

The Core Problem: The "Friction" vs. The "Kick"

To understand the paper, let's use the analogy of a swing.

The Swing (The System): Imagine a swing on a playground.
The Push (The Initial Data): You give the swing a push. How hard you push and how smooth your push is matters.
The Wind (The Resolvent): Imagine there is a weird wind blowing.
- In "perfect" systems, the wind is gentle and predictable.
- In the systems this paper studies, the wind is gusty and unpredictable. It might blow harder as the swing moves faster (growing polynomially). This makes the swing's path chaotic and hard to predict.

The Goal: We want to know: If I give the swing a push, how long until it stops swinging?

The Old Way vs. The New Way

The Old Way (Previous Research)

In the past, if the wind was gusty (the system was unbounded), mathematicians could only make a safe guess if the person pushing the swing was extremely skilled (very smooth initial data).

Even then, their prediction had a "safety margin" that was too big. It was like saying, "The swing will stop in about 10 minutes, but maybe it takes a little bit longer, so let's add a logarithmic penalty."

The "Logarithmic Loss": Think of this as a tiny, annoying tax on your prediction. It meant the math wasn't precise. It said, "It will stop, but we aren't 100% sure of the exact speed, so we'll add a tiny bit of extra time to be safe." This was especially true for complex, non-standard playgrounds (non-Hilbertian spaces).

The New Way (This Paper's Breakthrough)

The authors (Deng, Rozendaal, and Veraar) found a way to remove that annoying tax.

They developed a new method to analyze the "gusty wind" (the resolvent growth). Instead of just looking at the wind's speed, they looked at the shape of the wind using a special mathematical lens called Fourier Multipliers and Besov Spaces.

The Analogy: Imagine you are trying to predict the path of a leaf in a storm.
- Old method: You guess the leaf will land somewhere, but you have to add a "maybe" factor because the wind is weird.
- New method: The authors realized that if you look at the leaf's starting position with the right level of detail (using a specific type of mathematical "smoothness" called Real Interpolation Spaces), you can predict the landing spot exactly, without the "maybe" factor.

What Did They Actually Prove?

Removing the "Tax": For systems that grow wildly (unbounded semigroups), they proved that you can predict the decay rate (how fast it stops) without the extra logarithmic penalty that previous math required.
Handling Rough Starts: They showed that even if the starting push isn't perfectly smooth, as long as it has a certain level of "roughness" (measured by a specific mathematical scale called $p$ ), the system will still calm down at a predictable polynomial rate.
The "Sharp" Result: They found the exact "endpoint" of the prediction. Before, there was a gap in the math where we didn't know if the prediction was possible. They filled that gap.

The "Secret Sauce": How They Did It

The authors didn't just tweak the old formulas; they changed the playground entirely.

The Old Tool: They used standard rulers to measure the system.
The New Tool: They used a specialized microscope (Besov spaces and weighted Fourier multipliers).
- This microscope allows them to see the connection between the "gusty wind" (resolvent) and the "starting push" (initial data) much more clearly.
- By using this microscope, they could cancel out the chaos of the wind, proving that the system settles down faster and more predictably than anyone thought possible for these specific types of "wild" machines.

Summary in One Sentence

This paper provides a more precise mathematical map for predicting how fast chaotic, unstable systems calm down, successfully removing the "safety tax" (logarithmic loss) that previously made these predictions less accurate.

What This Means (Strictly Based on the Paper)

No Clinical Uses: The paper does not mention hospitals, medicine, or specific engineering applications like bridges or planes. It is purely about the abstract math of how systems evolve over time.
The Scope: It applies to "Banach spaces" (a broad category of mathematical spaces). It works for systems where the "wind" (resolvent) grows at a polynomial rate.
The Result: If you have a system that is allowed to grow wildly but has a specific type of "wind" behavior, you can now calculate its decay rate with higher precision than before, specifically by removing the extra logarithmic factor that was previously unavoidable.

Technical Summary: Improved Polynomial Decay for Unbounded Semigroups

Problem Statement
This paper investigates the asymptotic behavior of solutions to the abstract Cauchy problem $\dot{u}(t) = Au(t)$ on a Banach space $X$ , where $A$ generates a $C_0$ -semigroup $(T(t))_{t \geq 0}$ . The study focuses on the scenario where the semigroup is not uniformly bounded (i.e., $\|T(t)\|$ may grow exponentially) and the resolvent operator $R(\lambda, A) = (\lambda - A)^{-1}$ is not uniformly bounded on the right half-plane $\mathbb{C}_+$ . Specifically, the authors assume the resolvent exhibits polynomial growth at infinity:
$\|R(\lambda, A)\|_{\mathcal{L}(X)} \leq C(1 + |\lambda|)^\beta \quad (\lambda \in \mathbb{C}_+),$
for some $\beta > 0$ . The goal is to derive polynomial decay rates for the semigroup orbits $\|T(t)x\|_X$ for sufficiently smooth initial data $x$ , relating the decay rate to the resolvent growth rate $\beta$ and the geometric properties of the Banach space $X$ .

Methodology
The authors employ a Fourier multiplier approach applied to the resolvent on the imaginary axis, extending techniques previously developed for uniformly bounded semigroups. The methodology diverges from prior work (e.g., [31, 32]) in two primary ways:

Besov Space Multipliers: For the first main result, the authors utilize Fourier multiplier theory mapping between Besov spaces $B^s_{p,p}(\mathbb{R}; Y)$ and Lebesgue spaces $L^{p'}(\mathbb{R}; X)$ . This allows them to obtain endpoint estimates that were previously inaccessible. They leverage the connection between Besov spaces and real interpolation spaces to cancel out the resolvent growth.
Weighted Space Multipliers: For the second main result (specifically for $\rho=0$ ), the authors consider Fourier multipliers between weighted Lebesgue spaces $L^p(\mathbb{R}, w; Y)$ and $L^q(\mathbb{R}, v; X)$ . This approach facilitates results involving fractional domains $D((-\!A)^\tau)$ but requires a priori assumptions regarding the mapping properties of the Fourier transform between these weighted spaces (specifically Hardy–Littlewood type/cotype conditions).

Key technical tools include:

Real Interpolation: The use of spaces $D_A(\tau, q) := (X, D(A^m))_{\tau/m, q}$ to characterize initial data.
Fourier Type: Utilizing the property that $X$ has Fourier type $p \in [1, 2]$ (boundedness of the Fourier transform from $L^p$ to $L^{p'}$ ).
Hardy–Littlewood Type/Cotype: Employing weighted Fourier transform bounds to handle specific geometric cases.

Key Contributions and Results

1. Main Theorem (Theorem 1.1 / Theorem 3.3):
For a Banach space $X$ with Fourier type $p \in [1, 2]$ , and a generator $A$ satisfying the polynomial resolvent bound with exponent $\beta$ , the authors establish the following decay rate for initial data in the real interpolation space $D_A(\tau, p)$ :
$\|T(t)x\|_X \leq C_\rho t^{-\rho} \|x\|_{D_A(\tau, p)}$
where the required smoothness exponent is:
$\tau = (\rho + 1)\beta + \frac{1}{p} - \frac{1}{p'}.$

Improvement over [32]: For $\rho > 0$ and $p \in [1, 2)$ , this result removes the logarithmic loss ( $\log(t)^\sigma$ ) present in previous estimates (e.g., [32, Eq. 1.5]), achieving the sharp endpoint exponent.
Generalization: The result holds for general Banach spaces (since all have Fourier type $p=1$ ) and improves upon previous results for $p \in (1, 2]$ .
Extension to $\rho=0$ : The theorem provides an endpoint result for $\rho=0$ in the interpolation space setting.

2. Fractional Domain Result (Theorem 1.1 / Theorem 4.2):
Under additional geometric assumptions (specifically, if $X$ has Hardy–Littlewood type $p$ or Hardy–Littlewood cotype $p'$ with $p > 1$ ), the decay estimate holds for initial data in the fractional domain $D((-\!A)^\tau)$ rather than the interpolation space:
$\|T(t)x\|_X \leq C_\rho t^{-\rho} \|x\|_{D((-\!A)^\tau)}.$

For $\rho = 0$ , this fully extends the classical result (1.4) to the case of unbounded semigroups with the exponent $\tau = \beta + \frac{1}{p} - \frac{1}{p'}$ .
For $\rho > 0$ , this result is consistent with the interpolation space result since $D((-\!A)^\tau) \subseteq D_A(\tau, \infty)$ .

Significance and Claims
The paper claims to improve upon existing literature by:

Removing Logarithmic Loss: Eliminating the logarithmic factor in decay estimates for unbounded semigroups on non-Hilbertian Banach spaces with Fourier type $p \in [1, 2)$ , a gap left open by previous work [31, 32].
Handling Unbounded Semigroups: Providing a unified framework that does not require the semigroup to be uniformly bounded, addressing cases arising in damped wave equations with sign-changing damping, Schrödinger operators with unbounded potentials, and delay differential equations.
Sharpness: The authors note that the exponent $\tau$ is sharp for $p=2$ and $\rho=0$ , and for general $p$ as $\beta \to 0$ , based on modifications of known counterexamples (e.g., Arendt's example).

The authors explicitly state that for general Banach spaces with Fourier type $p \in [1, 2)$ and $\rho=0$ , it remains an open question whether the decay holds with $D_A(\tau, p)$ replaced by the fractional domain $D((-\!A)^\tau)$ . The results are presented as improvements on the state-of-the-art for polynomial stability in the absence of uniform boundedness.