Abstract
In this note, an alternative approach to establish observability for semigroups based on their smoothing properties is presented. The results discussed here reproduce some of 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].
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1 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 [5, 11, 21, 26]. The aforementioned spectral inequalities were studied for several differential operators, see, e.g., [4, 6,7,8,9,10, 17, 20, 22] and the references cited therein. Suitable dissipation estimates to treat also semigroups generated by some quadratic differential operators in the sense of Hörmander [12] (that is, by operators associated to homogeneous quadratic polynomials via Weyl quantization) were provided in [3, 5, 7, 17].
A different approach was introduced in [16], based on [19]. It allows one to derive observability estimates from uncertainty principles with error term established for functions in the range of the semigroup associated to the abstract Cauchy problem. In the situation of [16], these uncertainty principles are established using Gelfand–Shilov smoothing effects. By the latter we mean that for the strongly continuous contraction semigroup \((\mathcal {T}(t))_{t\ge 0}\) on \(L^2(\mathbb {R}^d),\) there exist constants \(C \ge 1\), \(t_0 \in (0,1)\), \(\nu >0,\) and \(0<\mu \le 1\) with \(\nu +\mu \ge 1\), and \(r_1 \ge 0\), \(r_2 > 0\), such that for all \(g\in L^2(\mathbb {R}^d),\) we have
for all \(t\in (0,t_0)\) and all \(n\in \mathbb {N}\), \(\beta \in \mathbb {N}_0^d\).
In this context, we prove the following variant of [16, Theorem 2.3].
Theorem 1.1
Suppose that \(f \in C^\infty (\mathbb {R}^d)\) satisfies
with some \(D_1 > 0,\) \(D_2 \ge 1,\) \(\nu \ge 0,\) and \(0 \le \mu < 1\). Moreover, let \(\delta \in [0,1]\) with \(s = \delta \nu +\mu < 1,\) and let \(\rho :\mathbb {R}^d \rightarrow (0,\infty )\) be a measurable function satisfying
with some \(R \ge 1\) and
with some \(\eta \in (0,1)\) and some \(r_0 \ge 1\).
Then, for every measurable set \(\omega \subset \mathbb {R}^d\) satisfying
with some \(\gamma \in (0,1)\) and for every \(\varepsilon \in (0,1],\) we have
where \(K \ge 1\) is a constant depending on \(\gamma , R,r_0,\eta ,\nu ,s,\) and the dimension d.
Here, \(B(x,\rho (x))\) denotes the open Euclidean ball of radius \(\rho (x) > 0\) centered at x. Note that (1.4) is automatically satisfied if \(\delta < 1\) with, say, \(\eta = 1/2\) and \(r_0 \ge (4R)^{1/(1-\delta )}\).
The estimate (1.6) differs from the usual form of an uncertainty principle by the appearance of the term \(\varepsilon D_1^2\). We call this the error term since it can be chosen arbitrarily small.
In [16], the same result is proved but under more technical assumptions, namely that \(\rho \) is a Lipschitz contraction with a uniform positive lower bound. On the other hand, the case \(s = 1\), which also allows \(\mu = 1\), is treated in [16] but is not in the scope of the method we discuss here. However, [16] does not present any application in terms of observability for this case.
Our proof, as well as the one in [16], follows the approach from [13, 14]. The main idea of the latter is to localize certain Bernstein-type inequalities on so-called good elements of some covering of \(\mathbb {R}^d\). 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, [16] uses an estimate for quasianalytic functions proven in [23], see also the \(L^2\)-version in [16, 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, 14] by estimating Taylor expansions around suitable points. This is combined with ideas introduced in [7, 8] that incorporate the quadratic decay guaranteed by (1.2) in order to reduce the considerations to a bounded subset of \(\mathbb {R}^d\). This allows us to obtain a more streamlined proof while getting rid of the mentioned technical assumptions in [16].
If \((\mathcal {T}(t))_{t\ge 0}\) is a strongly continuous contraction semigroup on \(L^2(\mathbb {R}^d)\) satisfying the Gelfand–Shilov smoothing effects (1.1), then \(f = \mathcal {T}(t)g\) with \(g \in L^2(\mathbb {R}^d)\) and \(t \in (0,t_0)\) satisfies (1.2) with
Thus, choosing the constant \(\varepsilon \) in Theorem 1.1 appropriately, we are able to apply [19, Lemma 2.1] in literally the same way as in the proof of [16, Theorem 2.11] and thereby obtain the following observability result, which reproduces [16, Theorem 2.11]. We omit the proof for brevity.
Corollary 1.2
Suppose that \((\mathcal {T}(t))_{t\ge 0}\) is a strongly continuous contraction semigroup satisfying (1.1) with some constants \(C \ge 1,\) \(\nu \ge 0,\) \(0\le \mu < 1,\) \(r_1\ge 0,\) \(r_2 > 0,\) and let \(\delta \in [0,1]\) and \(\rho :\mathbb {R}^d \rightarrow (0,\infty )\) be as in Theorem 1.1. Then
for every measurable set \(\omega \subset \mathbb {R}^d\) satisfying (1.5) with some \(\gamma \in (0,1)\). Here, \(N \ge 1\) is a constant depending on \(\gamma , R,r_0,\eta ,\nu ,s,C,r_2,\) and the dimension d.
As shown in [2, Corollary 2.2] and [16, Lemma 5.2], the semigroup generated by the (negative) fractional (an-)isotropic Shubin operator \(-((-\Delta )^m + |x|^{2k})^\theta \) with \(k,m\in \mathbb {N}\) and \(\theta > 1/(2m)\) satisfies (1.1) with
Hence, Corollary 1.2 can be applied, which reproduces [16, Corollary 2.12]. It should however be mentioned that in the general case \(\nu = \mu = 1/2\) of Corollary 1.2 (a particular instance being a Shubin operator with \(k=m\) and \(\theta =1\)) stronger results than Corollary 1.2 in terms of the conditions on \(\omega \) are available, see [7, Theorem 3.5]. More precisely, in this case the density \(\gamma \) is allowed to be variable and exhibit a certain subexponential decay, so that \(\omega \) may even have finite measure. In the present setting, we are able to obtain a variant of Theorem 1.1 where the density \(\gamma \) 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 \(T\rightarrow \infty \). We expect that such results also hold for the general Shubin operators. However, this would require setting up a suitable spectral inequality for these operators, which seems out of reach at the moment.
2 Proof of Theorem 1.1
Let \(\varepsilon \in (0,1]\), and choose
so that
Then (1.2) with \(n = 1\) and \(\beta = 0\) implies that
We abbreviate \(w(x) = w_\delta (x) = (1+|x|^2)^{\delta /2}\). If \(\delta < 1\), we infer from [16, Lemma 5.3] that (1.2) implies
with \(\tilde{D}_2 = 8^\nu \mathrm {e}^\nu D_2 \ge 1\). If \(\delta = 1\), then (2.3) agrees with (1.2) with \(\tilde{D}_2 = D_2 \ge 1\). 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 \(L^2\)-mass of f outside of the ball B(0, r) can be subsumed into the error term, that is the second summand on the right hand side of (1.6). Using Besicovitch’s covering theorem, the region B(0, r) is then covered with (at most countably many) balls \(B(x,\rho (x))\) satisfying \(B(x,\rho (x))\cap B(0,r) \ne \emptyset \). Based on (2.3) and following [13, 14] and [16], 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, 14], on good balls the local Bernstein-type estimates allow to bound the Taylor expansions of f around suitably chosen points, 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.
We consider balls \(B(x,\rho (x))\) with \(B(x,\rho (x))\cap B(0,r) \ne \emptyset \). The latter requires that \(|x|-\rho (x) < r\) and, thus, \(|x| < r_0\) or \((1-\eta )|x| \le |x| - \rho (x) < r\), that is, \(|x| < r/(1-\eta )\). By Besicovitch’s covering theorem, see, e.g., [18, Theorem 2.7], there is \(\mathcal {K}_0 \subset \mathbb {N}\) and a collection of points \((y_k)_{k \in \mathcal {K}_0}\) with \(| y_k | < \max \{ r_0 , r/(1-\eta ) \}\) such that the family of balls \(Q_k = B(y_k,\rho (y_k))\), \(k \in \mathcal {K}_0\), gives an essential covering of \(B(0,\max \{ r_0 , r/(1-\eta ) \})\) with overlap at most \(\kappa \ge 1\). Here, the proof in [18, Theorem 2.7] and a simple calculation shows that \(\kappa \) can be chosen as \(\kappa = K_{\mathrm {Bes}}^d\) with a universal constant \(K_{\mathrm {Bes}} \ge 1\). With \(Q_0 := \mathbb {R}^d \setminus \bigcup _{k \in \mathcal {K}_0} Q_k\) and \(\mathcal {K}:= \mathcal {K}_0 \cup \{0\}\), the family \((Q_k)_{k \in \mathcal {K}}\) thus gives an essential covering of \(\mathbb {R}^d\) with overlap at most \(\kappa = K_{\mathrm {Bes}}^d \ge 1\).
2.1 Good and bad balls
Similarly as in [16], 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 \(Q_k\), \(k \in \mathcal {K}_0\), is good with respect to f if for all \(m \in \mathbb {N}_0,\) we have
where
We call \(Q_k\), \(k \in \mathcal {K}_0\), bad if it is not good.
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 \(\mathbb {R}^d\), inequality (2.3) nevertheless implies that the mass of f on the bad balls is bounded by \(\varepsilon D_1^2/2\). 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 \(Q_0 \subset \mathbb {R}^d \setminus B(0,r)\) and, thus, \(\Vert f\Vert _{L^2(Q_0)}^2 \le \varepsilon D_1^2/2\) by estimate (2.2). Let now \(Q_k\), \(k \in \mathcal {K}_0\), be bad, that is, there is \(n \in \mathbb {N}_0\) such that
Summing over all bad \(Q_k\) with \(k\in \mathcal {K}_0\) and using (2.3) then gives
where we used that \(\sum _{|\beta |=m} 1/\beta ! = d^m/m!\). This proves (2.4).
\(\square \)
Now, as in [9], see also [10, 13, 14], we use the definition of good elements to extract a pointwise estimate for the derivatives of f.
Lemma 2.2
Let \(Q_k\) be good. Then there is \(x_k \in Q_k\) such that for all \(\beta \in \mathbb {N}_0^d\) with \(|\beta | = m \in \mathbb {N}_0,\) we have
with
Proof
Assume that for all \(x \in Q_k,\) there is \(m = m(x) \in \mathbb {N}_0\) such that
Reordering the terms and summing over all \(m \in \mathbb {N}_0\) in order to get rid of the x-dependence, we then obtain
for all \(x \in Q_k\). We observe that
Thus, integrating (2.7) over \(x \in Q_k\) and using that \(Q_k\) is good gives
leading to a contradiction. Hence, there is \(x_k \in Q_k\) such that for all \(\beta \in \mathbb {N}_0^d\) with \(|\beta | = m,\) we have
and taking square roots proves the claim.
\(\square \)
2.2 The local estimate
In order to estimate f on each (good) \(Q_k\), \(k \in \mathcal {K}_0\), we use a complex analytic local estimate that goes back to [13, 14, 24]. It has later been used in [9] and implicitly also in [4, 10, 17, 27]. We rely here on a particular case of the formulation in [9].
Lemma 2.3
(see [9, Lemma 3.5]). Let \(k\in \mathcal {K}_0,\) and suppose that the function \(f|_{Q_k} :Q_k \rightarrow \mathbb {C}\) has a bounded analytic extension \(F :Q_k + D_{8\rho (x_k)} \rightarrow \mathbb {C},\) where \(D_{8\rho (x_k)} = D(0,8\rho (x_k))\times \dots \times D(0,8\rho (x_k))\subset \mathbb {C}^d\) is the complex polydisk of radius \(8\rho (x_k)\).
Then, for every measurable set \(\omega \subset \mathbb {R}^d\) with \(|Q_k\cap \omega | > 0,\) we have
with
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 a bounded analytic extension \(F_k :Q_k + D_{8\rho (x_k)} \rightarrow \mathbb {C},\) and \(M_k\) as in Lemma 2.3 satisfies
where \(K' \ge 1\) is a constant depending only on s and where
Proof
Let \(x_k \in Q_k\) be a point as in Lemma 2.2. For every \(z \in x_k + D_{10\rho (x_k)},\) we then have
where for the last inequality we again used that \(\sum _{|\beta | = m} 1/\beta ! = d^m/m!\). Taking into account that \(Q_k + D_{8\rho (x_k)} \subset x_k + D_{10\rho (x_k)}\) and that f is analytic by Lemma A.1, this shows that the Taylor expansion of f around \(x_k\) defines a bounded analytic extension \(F_k :Q_k + D_{8\rho (x_k)} \rightarrow \mathbb {C}\) of f and that
Now, suppose first that \(|x_k| \le r_0\) with \(r_0 \ge 1\) as in (1.4). Then
Using (2.8), (2.6), and the definition of \(q_m\), it follows that
where we have taken into account that \(w(x) \ge 1\) for all \(x\in \mathbb {R}^d\).
On the other hand, if \(|x_k| \ge r_0\), then for all \(x \in Q_k,\) we have by (1.4) the lower bound \(|x| \ge |x_k| - \rho (x_k) \ge (1-\eta )|x_k| > 0\) and, thus,
Using again (2.8) and (2.6) then gives
We conclude that for both cases \(|x_k| \le r_0\) and \(|x_k| \ge r_0,\) we have
We estimate the series using the asymptotics
derived in [25, Chapter 8, Eq. (8.07)]. Thereby,
where \(K'\) is a constant depending only on s. Hence,
and therefore
\(\square \)
We are now in position to prove our main result.
Proof of Theorem 1.1
By hypothesis, we have \(|Q_k| / |Q_k\cap \omega | \le 1/\gamma \). Combining this with Lemma 2.3 and the estimate for \(\log M_k\) derived in Lemma 2.4, we obtain for all good \(Q_k\) that
In particular,
where \(K'' \ge 1\) is a constant depending on \(R,r_0,\eta ,\nu ,s\), and the dimension d. Summing over all good \(Q_k\) gives
Together with Lemma 2.1 this proves the theorem with \(K \le K''\log (1/\gamma ) + \log \kappa \).
\(\square \)
A slight adaptation of the proof of Theorem 1.1 allows us to consider a variable density \(\gamma = \gamma (x)\) with a polynomial decay.
Theorem 2.5
Let \(D_1,D_2,\mu ,\nu ,\) and \(\delta \) be as in Theorem 1.1 above, and suppose that f satisfies (1.2). Let \(\omega \subset \mathbb {R}^d\) be any measurable set satisfying
with some \(\gamma _0\in (0,1)\) and some \(a > 0\).
Then
where \(K \ge 1\) is a constant depending on \(\gamma _0,R,r_0,\eta ,\nu ,s,a,\) and the dimension d.
Proof
We have \(|Q_k| / |Q_k\cap \omega | \le (1+|y_k|^a) / \gamma _0\), and it is easily checked that
for all \(k\in \mathcal {K}_0\). Since r is as in (2.1), this shows
where \(\tilde{K} \ge 1\) is a constant depending on \(r_0, \eta \), and a. In light of the inequality \(\log D_2 \le D_2^{2/(1-s)}\), the theorem now follows in the same way as Theorem 1.1.
\(\square \)
Remark 2.6
The exponent in (2.10) depends on \(\varepsilon \) essentially by \((\log (1/\varepsilon ))^2\), as compared to just \(\log (1/\varepsilon )\) in (1.6) of Theorem 1.1. To the best of our knowledge, the proof of the observability estimate from [16, 19] does not work with the kind of dependence in (2.10).
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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 [25] for the asymptotics of the series used in the proof of Theorem 1.1. Both authors have been partially supported by the DFG grant VE 253/10-1 entitled Quantitative unique continuation properties of elliptic PDEs with variable 2nd order coefficients and applications in control theory, Anderson localization, and photonics.
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Appendix A: Analyticity
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.
Lemma A.1
Let \(f \in C^\infty (\mathbb {R}^d)\) be such that
with some constants \(C_1,C_2 > 0\). Then f is analytic in \(\mathbb {R}^d\).
Proof
Choose \(\sigma \in (0,1]\) with \(2C_2\sigma < 1\). Let \(y \in \mathbb {R}^d\), and let \(B = B(y,\tau )\) be an open ball around y with \(\tau < \sigma /d\). 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. [15, Theorem 2.2.5 and Proposition 2.2.10].
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 \(\tau \) and the dimension, such that \(\Vert g\Vert _{L^\infty (B)}\le c \Vert g\Vert _{W^{d,2}(B)}\) for all \(g \in W^{d,2}(B)\), see, e.g., [1, Theorem 4.12]. Applying this to \(g = \partial ^\alpha f|_B\) with \(|\alpha | = m \in \mathbb {N}_0\), we obtain
Taking the square root and using the hypothesis gives
We clearly have
In view of the choice of \(\sigma \), we thus further estimate
Now,
and since \(\tau \) is chosen such that \(d\tau / \sigma < 1\), this shows (A.1) and, hence, completes the proof.
\(\square \)
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Dicke, A., Seelmann, A. Uncertainty principles with error term in Gelfand–Shilov spaces. Arch. Math. 119, 413–425 (2022). https://doi.org/10.1007/s00013-022-01763-9
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DOI: https://doi.org/10.1007/s00013-022-01763-9