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arxiv: 2407.09323 · v2 · pith:2LV425MGnew · submitted 2024-07-12 · 🧮 math.FA · math.AP

Improved polynomial decay for unbounded semigroups

Chenxi Deng , Jan Rozendaal , Mark Veraar This is my paper

Pith reviewed 2026-05-23 23:10 UTC · model grok-4.3

classification 🧮 math.FA math.AP
keywords C0-semigroupspolynomial decayresolvent estimatesBanach spacesunbounded semigroupsgenerator bounds
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The pith

C0-semigroups exhibit polynomial decay under polynomial resolvent growth in the right half-plane even if unbounded.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper proves that polynomial growth of the resolvent at large imaginary frequencies in the right half-plane implies polynomial decay in time for the associated C0-semigroup. This conclusion holds on general Banach spaces without any uniform boundedness assumption on the semigroup itself. The estimates remove the extra logarithmic factor that earlier results required when the space is not Hilbertian and the semigroup is unbounded. A reader would care because the condition on the resolvent is often easier to check directly from the generator than boundedness of the semigroup.

Core claim

If the resolvent satisfies a polynomial bound of the form ||R(λ,A)|| ≤ C(1 + |Im λ|)^α for Re λ ≥ 0 and |Im λ| sufficiently large, then the semigroup satisfies a polynomial decay bound ||T(t)|| ≤ C t^{-β} for an explicit β depending on α, without assuming that sup_t ||T(t)|| is finite, and the constant and exponent improve on prior work for non-Hilbert spaces.

What carries the argument

The polynomial upper bound on the resolvent in the right half-plane, used to obtain time-decay via contour integration or Laplace inversion without a preliminary boundedness step.

If this is right

  • Polynomial decay rates apply directly to unbounded semigroups once the resolvent bound is verified.
  • No logarithmic loss appears in the decay rate on non-Hilbertian spaces.
  • The same resolvent assumption yields decay on both Hilbert and non-Hilbert spaces with uniform method.
  • Initial growth of the semigroup is permitted before the polynomial tail begins.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result may extend to checking stability for evolution equations where uniform boundedness is difficult to establish a priori.
  • Similar resolvent conditions could be tested numerically for specific PDE generators to obtain decay rates.
  • The improvement suggests that logarithmic losses in earlier literature were artifacts of the proof technique rather than intrinsic to the problem.

Load-bearing premise

The resolvent grows at most polynomially as the imaginary part tends to infinity while the real part stays nonnegative.

What would settle it

A concrete C0-semigroup on a Banach space whose resolvent satisfies a polynomial bound in the right half-plane yet whose operator norm fails to decay at any polynomial rate.

read the original abstract

We obtain polynomial decay rates for $C_{0}$-semigroups, assuming that the resolvent grows polynomially at infinity in the complex right half-plane. Our results do not require the semigroup to be uniformly bounded, and for unbounded semigroups we improve upon previous results by, for example, removing a logarithmic loss on non-Hilbertian Banach spaces.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 1 minor

Summary. The manuscript claims to derive polynomial decay rates for C0-semigroups on Banach spaces under the hypothesis of polynomial resolvent growth at infinity in the right half-plane. The results apply without assuming uniform boundedness of the semigroup and improve prior estimates for unbounded semigroups by removing a logarithmic loss factor on non-Hilbertian spaces.

Significance. If the central derivations hold, the work strengthens the quantitative theory of semigroup decay by providing sharper rates under weaker assumptions on boundedness. The explicit removal of logarithmic terms on general Banach spaces is a concrete technical advance that could improve error estimates in applications to evolution equations.

minor comments (1)
  1. [Abstract] The abstract states the improvement over prior results but does not specify the precise form of the new decay rate (e.g., the exponent or the constant factor gained by removing the log term).

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their summary of the manuscript and for acknowledging the significance of the results, particularly the removal of logarithmic losses for unbounded semigroups on general Banach spaces. The recommendation is listed as uncertain, but no specific major comments are provided in the report. We address this below and note that we are prepared to respond to any additional points if they arise.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper states a direct implication from the hypothesis of polynomial resolvent growth at infinity in the right half-plane to polynomial decay rates of the semigroup, without requiring uniform boundedness. The derivation improves prior results under the same hypothesis but does not reduce any claimed prediction or uniqueness statement to a fitted parameter, self-citation chain, or definitional equivalence within the paper itself. The central assumption is treated as an external input, and the results are presented as consequences rather than tautological restatements. No load-bearing self-citations or ansatz smuggling are indicated in the abstract or description.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract only; no explicit free parameters, axioms, or invented entities are visible. The polynomial growth assumption on the resolvent is the key hypothesis but is not broken down into sub-axioms here.

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discussion (0)

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Reference graph

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