On Landau-Kato inequalities via semigroup orbits

Let $\omega>0$. Given a strongly continuous semigroup $\{e^{tA}\}$ on a Banach space and an element $f\in\mathbf{D}(A^2)$ satisfying the exponential orbital estimates $$\|e^{tA}f\|\leq e^{-\omega t}\|f\| \quad\text{and}\quad \|e^{tA}A^2f\|\leq e^{-\omega t}\|A^2f\|,\quad t\geq0,$$ a dynamical inequality for $\|Af\|$ in terms of $\|f\|$ and $\|A^2f\|$ was derived by Herzog and Kunstmann (Studia Math., 2014). Here we provide an improvement of their result by relaxing the exponential decay to quadratic, together with a simple and direct way recovering the usual Landau inequality. Herzog and Kunstmann also demanded an analogue, again via semigroup orbits, for the Kato type inequality on Hilbert spaces. We provide such a result by using Hayashi-Ozawa machinery [Proc. Amer. Math. Soc., (2017)] which in turn relies on Hilbertian geometry.


Introduction
The classical Landau inequality for functions defined on R + ∶= (0, ∞) states . The constant 4 can be replaced by 2 if one considers the Hilbert space L 2 (R + ) instead of L ∞ (R + ); this is a classical result of Hardy, Littlewood, and Pólya.Let A be the generator of a strongly continuous contractive 1 semigroup on a Banach space X, with norm ⋅ = ⋅ X for notational convenience.If f ∈ D(A 2 ) {0}, we also have the abstract Landau inequality If X is in addition a Hilbert space, the abstract Hardy-Littlewood-Pólya inequality was obtained by Kato in [Kat71] (see also Hayashi and Ozawa [HO17] for an elegant proof based on Hilbertian geometry).For related results and a recent survey of such norm inequalities, see for example Kallman-Rota Typical candidates for φ(t) are e −t and 1 1 + t + t 2 2 .Assuming proper evolutionary bounds (measured by above function φ) along two "semigroup orbits", the following theorem extends and improves the corresponding result of Herzog and Kunstmann in [HK14] for rigid exponential decay.
Theorem 1.1.Let {e tA } t≥0 be a strongly continuous semigroup on a Banach space X.Suppose that for some f ∈ D(A 2 ) {0} and ω > 0, and for all t ≥ 0, Then, for the quantities Remark 1.2.When φ(t) = e −t , the dynamical inequality (1.5) was obtained by Herzog and Kunstmann in [HK14].Moreover, the following relation was justified: In view of the bounds (1.3), it is natural to require that φ is decreasing and convex.
Remark 1.3.The precision of polynomial decay for operator semigroups is important in applications to evolution equations, see e.g.Borichev-Tomilov [BT10].
Note that b and c depend on the locality parameter ω, although it is factored out in b 2 ≤ 4ac.Thus (1.5)-(1.6)can be understood as the ω-localised version of (1.1).

Proof of (1.5)
We follow the beautiful proof of (1.5) given by Herzog and Kunstmann in [HK14] when φ(t) = e −t .For f ∈ D(A 2 ) {0} we have the following integral representation In the last step we used the integration by parts.Then by the triangle inequality and the orbital decay assumptions for e tA on f and A 2 f, we have (2.1) In the last step we used again the integration by parts, together with φ(0) = −φ ′ (0) = 1.Rewriting (2.1) using the notations (1.4) then gives (1.5).
Remark 3.1.In above reasoning, the order relation (1.7) is not needed.
Remark 3.2.In [HK14] with where 0, x = 0, was obtained via minimising the left hand side of (1.5).However, to evaluate the quality of (3.1), a rather involved limiting argument was used to recover (1.6).

Kato inequality via semigroups orbits
As Open Problem, Herzog and Kunstmann demanded at the end of [HK14] an analogue of Theorem 1.1 for (1.2).The aim of this section is to provide such a result by using the Hilbertian machinery of Hayashi and Ozawa in [HO17].
Thus we are working in a Hilbert space H with norm ⋅ and inner product (⋅ ⋅).The following lemma based on Hilbertian geometry is elementary but very useful.Lemma 4.1.For all f 1 , f 2 and f 3 that are elements in H, we have Proof.By straightforward expansion of the involved norms and inner products.
For any λ ≥ 0 and f ∈ D(A2 ) {0}, taking in Lemma 4.1 and dropping out the nonnegative term once we assume the following "restricted dissipativity" Note that the inequality (4.1) is equivalent to the Kato inequality (1.2).Now, suppose for some f ∈ D(A 2 ) {0} one has a family of orbital estimates and its infinitesimal version (4.2).The Kato inequality (1.2) then follows.Hence we proved the following "semigroup orbits" result in the spirit of [HK14].
Theorem 4.2.Let {e tA } t≥0 be a strongly continuous semigroup on a Hilbert space H with norm ⋅ .Then, the family of semigroup orbital estimates (4.3) for some element f ∈ D(A 2 ) {0} implies the Kato inequality (1.2) for the same f.
Remark 4.3.The orbital estimates (1.3) require an exponential decay of the semigroup bound on A 2 f and f (and by homogeneity, on λf for all λ > 0), whereas (4.3) requires only the contractivity of the semigroup on (A + λ)f for all λ > 0.

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