ANALYTICITY AND OBSERVABILITY FOR FRACTIONAL ORDER PARABOLIC EQUATIONS IN THE WHOLE SPACE

. In this paper, we study the quantitative analyticity and observability inequality for solutions of fractional order parabolic equations with space-time dependent potentials in R n . We ﬁrst obtain a uniformly lower bound of analyticity radius of the spatial variable for the above solutions with respect to the time variable. Next, we prove a globally H¨older-type interpolation inequality on a thick set, which is based on a propagation estimate of smallness for analytic functions. Finally, we establish an observability inequality from a thick set in R n , by utilizing a telescoping series method


Introduction and main results
It has been proved in [12] and [33] independently that the following observability inequality ∀T > 0, ∃C = C(n, T, E) > 0 so that R n |u(T, x)| 2 dx ≤ C Here | · | denotes the n-dimensional Lebesgue measure, and Q L (x) stands for the cube in R n centered at x ∈ R n with a side L > 0. The arguments in [12,33] are indeed adapted from the Lebeau-Robbiano strategy (see, e.g., [5,9,19,21,29,32]). Let us now recall it in an abstract way: Let H be a self-adjoint operator so that −H generates a C 0 semigroup {e −tH } t≥0 in L 2 (R n ), and let {π N } N ≥1 be a family of orthogonal projection operators on L 2 (R n ). Assume that there are constants b > a > 0 and C > 0 so that the spectral inequality and the dissipative inequality hold for all N ≥ 1 and f ∈ L 2 (R n ). Then, the following observability inequality is true: ∀T > 0, ∃C = C(n, T, E) > 0 so that Recently, Lebeau and Moyano in [20] have proved a similar spectral inequality for the Schrödinger operator ∆ + V (·) in R n , with V (·) being a spatially analytic function vanishing at infinity. More precisely, if E is a thick set, then there exists C > 0 so that where π N is a spectral projection from the total space to a low-frequency subset (see [20] for the more accurate definition). Thus, the observability inequality from the thick set holds for all solutions of the heat equation with such a kind of potentials ∂ t u − ∆u = V (x)u in R + × R n , u(0, ·) ∈ L 2 (R n ).
The following question arise naturally: Does the corresponding observability inequality from the thick set hold true for the heat operator with space-time dependent potentials ∂ t u − ∆u = a(t, x)u in R + × R n , u(0, ·) = u 0 ∈ L 2 (R n ).
Up to now, to the best of our knowledge, it is still open in the existing literature.
It is worth mentioning that the observability inequality from the equi-distributed set (which is stronger than the thick set) is proved in [8] for the heat operator with bounded potentials in R n .
In this paper, we concern the above question in a more general case: We would like to replace the Laplace operator by a fractional one ∂ t u + Λ s u = a(t, x)u in R + × R n , u(0, ·) = u 0 ∈ L 2 (R n ), (1.1) where Λ s = (−∆) s/2 , s > 0, is the fractional Laplacian defined through the Fourier transform Λ s f (ξ) = |ξ| s f (ξ), ∀ξ ∈ R n .
The Fourier transform f is here given by Due to the space-time dependent potential considered in the present paper, it is not clear that how to adapt the Lebeau-Robbiano approach for the equation (1.1). To overcome this difficulty, we will utilize a different strategy developed in [3,11] for parabolic equations in bounded domains. This new strategy is mainly based on a locally quantitative estimate of propagation of smallness for analytic functions.
In this paper, we try to generalize this approach for the fractional order parabolic equations in the whole space. To this end, we first study the analyticity for solutions of the equation (1.1) under the following spatial analyticity assumption on potentials a(·, ·): (A) There exist constants C and R > 0 so that The first result of this paper can be stated as follows.
Theorem 1.1. Assume that (A) is true and s > 1. Then there exist constants c > 0, C > 0 such that for all solutions u of (1.1) and for all t > 0, where the norm · A R is defined by (2.5) below.
The above estimate shows that the L 2 -norm of u(t, ·)e cR|·| is finite for every t > 0. This, according to the well-known Paley-Wiener theorem [31], Theorem IX.13, p.18, implies that the solution u(t, x) can be extended to a holomorphic function u(t, z) in the strip S σ = {z ∈ C n : |Imz| < σ} with σ = cR. Thus, u(t, ·) has a uniformly lower bound of analyticity radius for any positive time.
Similar results have been proved in [11] for 2m-order (m is an integer) parabolic equations in bounded domains with zero Dirichlet boundary conditions. The proof in [11] relies on Schauder's estimates, while (1.2) will be proved by certain tools in Fourier analysis, due to the nonlocal nature of the equation (1.1).
Another novelty of the above estimate (1.2) is that it gives an explicit linear dependence for the spatial analyticity radius of the solution in terms of that of potentials a(·, ·). This observation seems to be new according to the existing literature.
Next, based on an interpolation inequality on a thick set for a function f satisfying e cR|·| f (·) L 2 (R n ) < ∞, we could prove a Hölder-type interpolation inequalities of unique continuation for solutions of (1.1). Theorem 1.2. Let s > 1, (A) hold and E be a thick set in R n . Then, there exist constants C and C such that for all solutions u of (1.1), holds for all t > 0 and θ ∈ (0, e −C max{1,R −1 } ), where Finally, with regards to the observability inequality for solutions to (1.1), we have the following result. Theorem 1.3. Let s > 1, (A) hold, and E ⊂ R n be a thick set. Then there exists a constant C > 0 such that for all T > 0 and all solutions of (1.1) We mention that, in the regime s > 1, the observability inequality in Theorem 1.3 has been established in [1], Remark 1.13 and [30], Theorem 2.8 for some fractional heat equations (including the particular case that the potential a = 0). Theorem 1.3 generalizes the results in [1,30] to parabolic equations with space-time potentials. Thus, it partially solves the question raised above under the assumption of spatial analyticity, while the complete answer is quite open.
As mentioned before, Theorem 1.3 cannot follow directly from the Lebeau-Robbiano strategy. Here we prove it by the interpolation inequality (1.3). Note that this approach has been used successfully in a recent work [8]. To prove (1.3), we shall establish the following unique continuation inequality for analytic functions (see Sect. 2.1 for a definition of G σ and Thm. 3.5 below) The restriction s > 1 is essential for our purpose. 1 In fact, if 0 < s ≤ 1, then the fractional heat equation is not null controllable on a thick set E (say, E is the complement of a nonempty open set, see [18,24,27]). It is well known that the observability and controllability for linear PDEs are equivalent by the duality argument. We also refer the reader to, e.g., [9,13,22,23,[27][28][29] for null-controllability results for fractional heat equations on bounded domains.
We also note that there is a growing interest on observability inequalities of parabolic equations associated with the Schrödinger operator −∆ + V (x), where the potential V (x) → +∞ as x → ∞. In particular, if V (x) = |x| 2k , k is an integer, then the solution is analytic and decreasing exponentially as x → ∞. It is shown that the observability inequality holds on some sensor sets of decaying density, which are larger set classes than thick sets, see e.g. [7,25,26].
Throughout the paper, we use A B to denote A ≤ CB for some universal constant C > 0. If both A B and B A hold, then we write A ∼ B.
This paper is organized as follows. In Section 2, we prove Theorem 1.1. In Section 3, we first establish some interpolation inequalities for analytic functions, and then use them to prove Theorem 1.2, with the aid of Theorem 1.1. Finally, we prove Theorem 1.3 in Section 4.

Auxiliary results
For every σ > 0, we define the Banach space G σ = G σ (R n ), consisting of analytic functions in S σ = {z ∈ C n : |Imz| < σ}, endowed with the norm This kind of analytic functions, according to the Paley-Wiener theorem, is related to the function whose Fourier transform decays exponentially at infinity. The proof of the following lemma is inspired by Problem 76 in [31], p. 132.
Proof. We first claim that In fact, by the Fourier inversion, Replacing x by x + iy, we find for every |y| < σ. By the Plancherel theorem, which proves (2.2). Next, we prove (2.1). We note that (2. 2) implies f G σ e σ|ξ| f (ξ) L 2 ξ (R n ) , if we use the simple fact that |y · ξ| ≤ |y||ξ| ≤ σ|ξ|. Thus, it remains to show e sup |y|<σ e y·ξ f (ξ) L 2 ξ (R n ) . By a scaling argument, it suffices to consider the case σ = 1, namely, In the case that n = 1, this holds clearly, see [16], p. 5285. But for the higher dimensional case we need more analysis. In fact, for every ξ ∈ R n , the Lebesgue measure This implies that for all ξ ∈ R n Integrating (2.4) over ξ ∈ R n , and using the Fubini theorem, we infer that This proves (2.3) and completes the proof.
For every σ > 0, we introduce an analytic function space A σ endowed with the norm Proof. By the Taylor expansion Recalling the definition of G σ norm, and using (2.6), we obtain This completes the proof.
Let {e −tΛ s } t≥0 be the semigroup generated by the fractional Laplacian −Λ s in L 2 (R n ). This semigroup can be expressed by the Fourier transform as Proof. By (2.1) and the Plancherel theorem, we have The desired bound follows from the fact that where we used s > 1 in the last inequality.
Using the semigroup {e −tΛ s } t≥0 , we can rewrite the evolutional differential equation (1.1) as an integral equation Proposition 2.5. Assume (A) holds for some R > 0. Then for every u 0 ∈ L 2 (R n ), there exists a unique solution u of (2.7) satisfying where C > 0 is a constant independent of u 0 .
Proof. Thanks to Lemmas 2.3 and 2.4, there exist constants C 0 , C 1 > 0 such that for any t > 0 We now consider the linear mapping We claim that ΓB ⊂ B. In fact, if u ∈ B, then by (2.8) and (2.9) we have Note that in the second inequality above, we have used (2.8), (2.9), and the inequality In the third inequality above, we have used the definition of C 2 and the norm of X.
Hence, Γ : B → B is a contraction mapping, and (2.7) has a unique solution u in B satisfying the bound which implies that This gives the desired bound and completes the proof, since T ∈ (0, (R/2) s ] is arbitrary.
. Then there exists a constant C > 0 so that for all u 0 ∈ G b , the solution of (2.7) satisfies Proof. The proof is the same as above, and so it is omitted here. In fact, it suffices to use instead of (2.8) along the above lines.

Proof of Theorem 1.1
We shall utilize Proposition 2.6 repeatedly to obtain a lower bound of analyticity radius, which is independent of the time variable. We begin with the following lemma.
Lemma 2.7. Assume that (A) holds for some R > 0, then there exist constants c, C > 0 so that the solution u of (2.7) satisfies Proof. Let t ∈ (0, T ] with T = ( R 2 ) s . For every m ≥ 1, we make the decomposition Applying Proposition 2.6 on each interval [ j−1 m t, j m t], j = 1, 2, · · · , m, we get Combining these inequalities, we infer that which is in fact equivalent to Hence, it follows from (2.10) that for some c, C > 0 This implies the desired bound and completes the proof.
Proof of Theorem 1.1. Let t 0 = ( R 2 ) s . It follows from Lemma 2.7 that Similarly, we have for all τ ≥ 0 that ) u(τ, ·) L 2 (R n ) . (2.11) By the classical energy estimate we deduce from (2.11) that for all t ≥ t 0 ) u 0 L 2 (R n ) . (2.12) By Lemma 2.7 again, we have for all t ∈ (0, t 0 ] Together with (2.12) and (2.13), we infer that for any t > 0 This, along with Lemma 2.1, indicates the desired bound (1.2), and completes the proof.

Proof of Theorem 1.2
We first prove a global interpolation inequality on thick sets for analytic functions, and then prove Theorem 1.2. To this end, we first recall a local interpolation inequality for analytic functions.
Here and in the sequel, we use B R to denote a ball in R n with a radius R.
Remark 3.2. As a consequence of (3.1), we could derive its L p -version (1 ≤ p < ∞), which will be useful later. Let 1 ≤ p < ∞ and 1 p + 1 p = 1. By the Hölder inequality, we have Inserting them into (3.1) we obtain Clearly, (3.2) also holds in the case that p = ∞.
Based on Lemma 3.1, we establish an interpolation inequality of unique continuation for functions in G σ , with an explicit interpolation index.
In the sequel, let Q L be a closed cube with a side L > 0 centered at some point in R n . We do not specify the center point, since the estimates below are uniformly for cubes with different centers. Lemma 3.3. Let 2 ≤ p ≤ ∞, L, σ > 0 and ω ⊂ Q L be a subset of positive measure. Then there exist two constants C = C(p, n, |ω|, L, σ) > 0 and C = C (n, |ω|, L) > 0 so that Proof. Arbitrarily take f ∈ G σ with some σ > 0. Clearly, f is analytic on R n . Also, by the definition of M , we have for some θ 0 ∈ (0, 1). Thus the lemma holds in this case. Now we consider the reminder case that σ < L. We first claim that there exists a point x 0 ∈ Q L so that for some c 0 = c 0 (n, L, |ω|) > 0. In fact, let k ≥ 1 be an integer, we split Q L as disjoint (except for the boundary) small cubes Q L k (x i ), i = 1, 2, · · · , k n . Note that if the center of Q L is given, then x i are determined uniquely. Then Since the number of small cubes is k n , we infer that This proves the claim (3.4).
We present the following Hölder-type inequality of unique continuation on thick sets for analytic functions in G σ .
In particular, by letting p = 2, we obtain the following result.
Before proving Theorem 3.5, we give two remarks below.
Remark 3.7. The inequality (3.21) fails in the case that 1 ≤ p < 2. Indeed, giving now 1 ≤ p < 2, since To this end, for every s > 0, we define a function Then f s ∈ L p (R n ) if and only if s > n p . Moreover, by [14], Proposition 6.1.5, p. 6, if s < n, then the Fourier transform f s satisfies that Then f s ∈ G 1 4 if s ∈ ( n 2 , n). Since p < 2, we can always choose s 0 so that s 0 ≤ n p and s 0 ∈ ( n 2 , n). Then f s0 ∈ G 1 4 but f s0 / ∈ L p (R n ), this shows that (3.23) fails to hold for σ = 1 4 . We conclude the same result for the general σ > 0 after a scaling argument.
Remark 3.8. We recall here some previous works on the interpolation inequality (3.22).
In [20], E is thick, but no explicit dependence of θ on σ, proved by Carleman estimates. In [34], E is the complement set of a ball, θ ∼ e −1/σ , proved by the three-ball inequality of analytic functions. In [6,15], E is a Borel set satisfying the thickness condition, θ ∼ e −1/σ , proved by harmonic measure estimate.
Note that the complement set of every ball is a thick set, and that every Borel set is a Lebesgue measurable set (but the converse is not true). Thus, Corollary 3.6 recovers all the results in [6,15,20,34] in a unified way.

Proof of Theorem 1.3
In this section, we first present an abstract approach to ensure an observability inequality (which is actually motivated from [4], Thm. 11). Then we apply it to solutions of (1.1) (i.e., giving the proof of Theorem 1.3).
We now replace the space-time norm in (4.1) by a space norm at the final time.
With the aid of Corollary 4.2, we can prove the observability inequality for (1.1) by interpolation inequalities as in Theorem 1.2.
Proof of Theorem 1.3. Multiplying (1.1) with u and integrating over x ∈ R n , we obtain Applying Grönwall's inequality, recalling that the potential a is bounded, we see that for some C > 0 R n |u(t 2 , x)| 2 dx ≤ Ce C(t2−t1) R n |u(t 1 , x)| 2 dx, ∀t 2 ≥ t 1 . (4.14) We finish the proof by two cases as follows.