Uncertainty principles with error term in Gelfand-Shilov spaces

In this note, an alternative approach to establish observability for semigroups based on their smoothing properties is presented. The results discussed here are closely related to those recently obtained in [arXiv:2112.01788], but the current proof allows to get rid of several technical assumptions by following the standard complex analytic approach established by Kovrijkine combined with an idea from [arXiv:2201.02370].


Introduction and results
Uncertainty principles are frequently used in control theory to prove observability for certain abstract Cauchy problems. Often this is done via the so-called Lebeau-Robbiano method, where an uncertainty principle for elements in the spectral subspace, a so-called spectral inequality, is combined with a dissipation estimate, see [25,5,20,11]. The aforementioned spectral inequalities were studied for several differential operators, see, e.g., [10,4,9,16,8,7,19,21,6] and the references cited therein. Suitable dissipation estimates to treat also semigroups generated by some quadratic differential operators were provided in [5,3,16,7].
Then, for every measurable set ω ⊂ R d satisfying with some γ ∈ (0, 1) and for every ε ∈ (0, 1], we have Here, B(x, ρ(x)) denotes the open Euclidean ball of radius ρ(x) > 0 centered at x. Note that (1.4) is automatically satisfied if δ < 1 with, say, η = 1/2 and In [15], the same result is proved but under more technical assumptions, namely that ρ is a Lipschitz contraction with a uniform positive lower bound. On the other hand, the case s = 1, which also allows µ = 1, is treated in [15] but is not in the scope of the method we discuss here. However, [15] does not present any application in terms of observability for this case.
Our proof, as well as the one in [15], follows the approach from [13,12]. The main idea of the latter is to localize certain Bernstein-type inequalities on so-called good elements of some covering of R d . Since in the setting of Theorem 1.1 there is no Bernstein-type inequality at disposal, the definition of good elements replaces, in some sense, the missing Bernstein-type inequalities needed. The proof then reduces to a local estimate for (quasi-)analytic functions. For the latter, [15] uses an estimate for quasianalytic functions proven in [22], see also the L 2 -version in [15,Proposition 5.10], and a suitable estimate for the so-called Bang degree. By contrast, we rely on the more standard approach for (complex) analytic functions from [13,12] by estimating suitable Taylor expansions. This is combined with ideas introduced in [8,7] that incorporate the quadratic decay guaranteed by (1.2) to reduce the considerations to a bounded subset of R d . This allows us to obtain a more streamlined proof while getting rid of the mentioned technical assumptions in [15].
If (T (t)) t≥0 is a strongly continuous contraction semigroup on L 2 (R d ) satisfying the Gelfand-Shilov smoothing effects (1.1), then f = T (t)g with g ∈ L 2 (R d ) and t ∈ (0, t 0 ) satisfies (1.2) with Thus, choosing the constant ε in Theorem 1.1 appropriately, we are able to apply [18,Lemma 2.1] in literally the same way as in the proof of [15,Theorem 2.11] and thereby obtain the following observability result, which reproduces [15, Theorem 2.11]. We omit the proof for brevity.
More precisely, in this case the density γ is allowed to be variable and exhibit a certain subexponential decay, so that ω may even have finite measure. Note that semigroups satisfying (1.1) can also be studied using forced symmetrization (of µ and ν), see [2], but by this approach one in any case loses special characteristics of the particular operator at hand. In the present setting, we are able to obtain a variant of Theorem 1.1 where the density γ is allowed to exhibit a polynomial decay, but the result seems not to be sufficient to give observability as in Corollary 1.2, see Theorem 2.5 and Remark 2.6 below. If even k = m = 1, the Shubin operator corresponds to the harmonic oscillator, for which also a sharper observability constant can be obtained. Indeed, [7, Theorem 6.1] shows that the observability constant can then be chosen to vanish as T → ∞. We expect that such results hold also for the general Shubin operators. However, this would require setting up a spectral inequality for these operators, which seems out of reach at the moment Acknowledgements. The authors are grateful to Ivan Veselić for introducing them to this field of research and to Christopher Strothmann who referred to the reference [24] for the asymptotics of the series used in the proof of Theorem 1.
Then, (1.2) with n = 1 and β = 0 implies that We therefore just work with (2.3) for the remaining part.
The proof of the theorem now follows the following lines: Inspired by [9], the estimates (2.3) imply, in particular, that f is analytic, see Lemma A.1. Moreover, by (2.2) the contribution of f outside of the ball B(0, r) can be subsumed into the error term. The ball B(0, r) is then covered with balls of the form B(x, ρ(x)) by Besicovitch covering theorem. Based on (2.3) and following [13,12] and [15], these balls are classified into good and bad ones, where on good balls local Bernstein-type estimates are available and the contribution of bad balls can again be subsumed into the error term, see Lemma 2.1. Following again [13,12], on good balls the local Bernstein-type estimates allow to bound suitable Taylor expansions of f , which by analyticity of f lead to local estimates of the desired form, see Lemmas 2.2-2.4. Summing over all good balls finally concludes the proof.

Good and bad balls.
Similarly as in [15], we now define the so-called good elements of the covering. We do this in such a way that we have some localized Bernstein-type inequality on all good elements. More precisely, we say that Q k , Although we can not show that the mass of f on the good balls covers some fixed fraction of the mass of f on the whole of R d , inequality (2.3) nevertheless implies that the mass of f on the bad balls is bounded by εD 2 1 /2. Hence, the contribution of the bad elements can likewise be subsumed into the error term. This is summarized in the following result.

Lemma 2.1. We have
Proof. Since it suffices to show that To this end, we first note that Q 0 ⊂ R d \ B(0, r) and, thus, f 2 L 2 (Q0) ≤ εD 2 1 /2 by estimate (2.2). Let now Q k , k ∈ K 0 , be bad, that is, there is m ∈ N 0 such that Summing over all bad Q k with k ∈ K 0 and using (2.3) then gives where we used that |β|=m 1/β! = d m /m!. This proves (2.4).
Now, as in [9], see also [13,12,10], we use the definition of good elements to extract a pointwise estimate for the derivatives of f .
Reordering the terms and summing over all m ∈ N 0 in order to get rid of the x-dependence, we then obtain for all x ∈ Q k . We observe that Thus, integrating (2.7) over x ∈ Q k and using that Q k is good gives leading to a contradiction. Hence, there is and taking square roots proves the claim.

The local estimate.
In order to estimate f on each (good) Q k , k ∈ K 0 , we use a complex analytic local estimate that goes back to [23,13,12]. It has later been used in [9] and implicitly also in [10,26,4,16]. We rely here on a particular case of the formulation in [9]. Lemma 2.3 (see [9,Lemma 3.5]). Let k ∈ K 0 , and suppose that the function f | Q k : Q k → C has an analytic extension F : On good Q k , the hypotheses of Lemma 2.3 are indeed satisfied, and the pointwise estimates (2.5) can be used to obtain a suitable upper bound for the quantity M k .

Lemma 2.4. Let Q k be good. Then, the restriction f | Q k has an analytic extension
where K ′ ≥ 1 is a constant depending only on s and where for the last inequality we again used that |β|=m 1/β! = d m /m!. Taking into account that Q k + D 8ρ(x k ) ⊂ x k + D 10ρ(x k ) and that f is analytic by Lemma A.1, this shows that the Taylor expansion of f around x k defines an analytic extension F k : Q k + D 8ρ(x k ) → C of f with bounded modulus and that Now, suppose first that |x k | ≤ r 0 with r 0 ≥ 1 as in (1.4). Then, (2.6), and the definition of q m , it follows that where we have taken into account that w(x) ≥ 1 for all x ∈ R d .
On the other hand, if |x k | ≥ r 0 , then for all x ∈ Q k we have by (1.4) the lower bound |x| ≥ |x k | − ρ(x k ) ≥ (1 − η)|x k | > 0 and, thus, Using again (2.8) and (2.6) then gives We conclude that for both cases |x k | ≤ r 0 and |x k | ≥ r 0 we have We estimate the series using the asymptotics where K ′ is a constant depending only on s. Hence, We are now in position to prove our main result.
In particular, where K ′′ ≥ 1 is constant depending on R, r 0 , η, ν, s, and the dimension d. Summing over all good Q k gives Together with Lemma 2.1 this proves the theorem with K ≤ K ′′ log(1/γ)+log κ.
A slight adaptation of the proof of Theorem 1.1 allows us to consider a variable density γ = γ(x) with a polynomial decay. Theorem 2.5. Let D 1 , D 2 , µ, ν, and δ be as in Theorem 1.1 above, and suppose that f satisfies (1.2). Let ω ⊂ R d be any measurable set satisfying with some γ 0 ∈ (0, 1) and some a > 0. Then where K ≥ 1 is a constant depending on γ 0 , R, r 0 , η, ν, s, a, and the dimension d.
Proof. We have |Q k |/|Q k ∩ ω| ≤ (1 + |y k | a )/γ 0 , and it is easily checked that for all k ∈ K 0 . Since r is as in (2.1), this shows whereK ≥ 1 is a constant depending on r 0 , η, and a. In light of the inequality log , the theorem now follows in the same way as Theorem 1.1.
Remark 2.6. The exponent in (2.10) depends on ε essentially by (log(1/ε)) 2 , as compared to just log(1/ε) in (1.6) of Theorem 1.1. To the best of our knowledge, the proof of the observability estimate from [15,18] does not work with the kind of dependence in (2.10).
Appendix A. Analyticity The following result establishes that Gelfand-Shilov smoothing effects as in (1.1) guarantee analyticity of the functions in the range of the semigroup. This is required in order to follow the complex analytic approach discussed in the main text.
with some constants C 1 , C 2 > 0. Then, f is analytic in R d .
Proof. Choose σ ∈ (0, 1] with 2C 2 σ < 1. Let y ∈ R d , and let B = B(y, τ ) be an open ball around y with τ < σ/d. We show that the Taylor series of f around y converges in B and agrees with f there. To this end, it suffices to establish cf. [ We proceed similarly as in the proof of [9, Lemma 3.2]: Since B satisfies the cone condition, by Sobolev embedding there exists a constant c > 0, depending only on τ and the dimension, such that g L ∞ (B) ≤ c g W d,2 (B) for all g ∈ W d,2 (B), see, e.g., [1,Theorem 4.12]. Applying this to g = ∂ α f | B with |α| = m ∈ N 0 , we obtain Taking the square root and using the hypothesis gives We clearly have In view of the choice of σ, we thus further estimate and since τ is chosen such that dτ /σ < 1, this shows (A.1) and, hence, completes the proof.