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A polynomial gap below linear growth of Kreiss bounded $C_0$-semigroups on Hilbert spaces
Loris Arnold
TL;DR
The paper addresses whether resolvent estimates force sublinear polynomial growth for Kreiss bounded C0-semigroups on Hilbert spaces. It develops a Fourier–Plancherel and Hilbert-space argument yielding a polynomial gap below linear growth, with an explicit exponent depending on the Kreiss constant. The result also improves an estimate for Renardy’s perturbed wave equation, while prior examples prevent any universal exponent for the entire class.
Problem
Existing results gave linear or logarithmically improved growth bounds, while examples with growth arbitrarily close to linear rule out a universal polynomial exponent.
Method
The proof combines a Fourier–Plancherel representation, a bounded triangular convolution operator, adjacent-interval compression, and dyadic iteration with quadratic orbit estimates.
Results
Every Kreiss bounded C0-semigroup has a polynomial growth bound with a strictly positive exponent depending only on its Kreiss constant.
Takeaways & Limitations
Kreiss boundedness entails a genuine polynomial gap below linear growth, and the result improves the estimate for Renardy’s perturbed wave equation.
Takeaways & Limitations
The paper does not claim that the quantitative dependence of the exponent on the Kreiss constant is optimal.
Abstract
from arXiv · showhide
We prove that every Kreiss bounded $C_0$-semigroup $(T_t)_{t\geq0}$ on a Hilbert space satisfies \[ \|T_t\|\leq C(1+t)^{1-\varepsilon_K}, \qquad t\geq0, \] where $\varepsilon_K>0$ depends explicitly only on the Kreiss constant. This improves the previously known estimate $O(t/\sqrt{\log(t+1)})$ and shows that every Kreiss bounded $C_0$-semigroup has a genuine polynomial gap below linear growth. In view of the examples of Eisner and Zwart with growth arbitrarily close to linear, no universal positive exponent can hold for the whole class of Kreiss bounded $C_0$ semigroups on Hilbert spaces.
1. Introduction
The paper studies how resolvent estimates control growth of Kreiss bounded C0-semigroups on Hilbert spaces. It proves a polynomial improvement below linear growth whose exponent depends on the particular semigroup’s Kreiss constant, while prior examples rule out a universal exponent.
- Background: Kreiss bounded semigroups on Hilbert spaces were previously known to satisfy linear growth, with logarithmic refinements also available.The introduction places the result against the established estimate O(t) and subsequent logarithmic improvements.
- Background: No fixed positive exponent can improve the linear bound for all Kreiss bounded semigroups, because examples have growth arbitrarily close to linear.This obstruction is attributed to constructions of Eisner and Zwart, with a corresponding discrete obstruction also noted.
- Main result: Theorem 1.1 proves that every Kreiss bounded C0-semigroup admits a polynomial growth bound with a strictly positive exponent depending only on its Kreiss constant.The quantitative dependence is explicit, although the paper does not claim that it is optimal.
- Proof strategy: The proof uses Fourier–Plancherel theory to convert the Kreiss resolvent estimate into a bounded triangular convolution operator on L2(R; H).The operator is compressed to adjacent time intervals, and Cauchy–Schwarz yields a self-improving estimate.
- Proof strategy: Iteration over dyadic scales gives polynomial growth for an auxiliary quantity, while quadratic orbit estimates imply Theorem 1.1.The argument is presented as a Hilbertian mechanism that may extend beyond this problem.
- Application: The paper also applies the improved estimate to Renardy’s perturbed wave equation.This application is identified as an improvement over an earlier estimate.
Notation and conventions.
The paper fixes a Fourier-transform convention on Hilbert-valued functions and outlines the organization of its preparatory estimates, main theorem, application, and appendix.
- The Fourier transform is specified for functions in L1(R; H) ∩ L2(R; H).
- Under this convention, translating f by u has a corresponding Fourier-side shift operation.
- The paper develops preparatory estimates, proves Theorem 1.1, applies it to Renardy’s perturbed wave equation, and records a Fourier representation in the appendix.
2. Preparatory lemmas
The preparatory section introduces an operator on L2(R; H), derives Fourier–Plancherel estimates from the Kreiss condition, and establishes lemmas controlling semigroup trajectories.
- For 0 < r < 1, the paper defines an operator Cr from L2(R; H) to L2(R; H).
- Fourier–Plancherel and the Kreiss condition provide an estimate used in the preparatory argument.
- The proof uses extension by zero from L2(Jm; H) and restriction to L2(Im; H).
- For w(s) = ∥T_{t−s}x∥^2, Lemma 2.2 supplies an estimate for each 1 ≤ m ≤ t when T_tx ≠ 0.
- Lemma 2.3 establishes the integrated bound ∫_0^t ∥T_sx∥^2 ds ≤ At^2∥x∥^2 for t ≥ 1.
3. Proof of Theorem 1.1
The proof iterates a dyadic lower bound for an integrated trajectory quantity and combines it with an upper estimate to obtain polynomially sublinear semigroup growth.
- The proof reduces to t ≥ 2 and nonzero T_tx, then introduces the trajectory quantity w(s).
- Applying Cauchy–Schwarz and the preparatory estimate yields a recursive inequality for the integrated quantity W.
- The recursion uses the factor 1 + 1/(4e^2C_K^2) in the dyadic iteration.
- W(2^ℓ) ≥ (1 + θ_K)^ℓW(1) = 2^{ℓδ_K}W(1) for admissible integers ℓ.
- The final estimate is ∥T_tx∥^2 ≤ 2δ_KAD^2t^{2−δ_K}∥x∥^2, followed by square roots and a supremum over unit vectors.
4. Application to Renardy’s perturbed wave equation
The paper applies its growth theorem to Renardy’s perturbed wave equation on the two-dimensional torus, improving an earlier proposition with an explicit exponent depending on the resolvent constant.
- The application is formulated on the two-dimensional torus T^2 using the second-order Sobolev space W^{2,s}(T^2).
- The operator A is defined using the Laplacian on T^2 and a multiplication operator M on L2(T^2).
- Prior work establishes that −A generates a C0-group on the relevant Hilbert space.
- Theorem 1.1 yields an improvement of the earlier proposition for this perturbed wave equation.
- The improved estimate has some ε0 > 0 and C > 0, with ε0 chosen explicitly from the constant C_RV in the prior resolvent estimate.
Appendix A. Properties of the Operator Cr
Appendix A establishes that the operator C_r is bounded on L2(R; H) for 0 < r < 1 and identifies it with a resolvent multiplier via Fourier analysis.
- The operator C_r is represented through a Bochner integral involving the semigroup, shifts, and the factor e^−ru.
- Fact 1: C_r belongs to B(L2(R; H)) for every 0 < r < 1.The proof uses shifts, pointwise extensions of the semigroup, strong continuity, and a Bochner-integral representation.
- For f ∈ L1(R; H) ∩ L2(R; H), Fourier analysis gives (C_r f)(ξ) = −R(−r + iξ, A) f̂(ξ).The Laplace representation of the resolvent applies because ω0(T) ≤ 0, so Re(λ) > 0 is sufficient.
- The operators U and V are bounded on L2(R; H), with U bounded by Fourier–Plancherel and V bounded by Kreiss boundedness.
- The identity Uf = Vf, initially established on L1(R; H) ∩ L2(R; H), extends uniquely to all of L2(R; H) by density.