A strong unique continuation property for the heat operator with Hardy type potential

In this note we prove the strong unique continuation property at the origin for the solutions of the parabolic differential inequality \[ |\Delta u - u_t| \leq \frac{M}{|x|^2} |u|, \] with the critical inverse square potential. Our main result sharpens a previous one of Vessella concerned with the subcritical case.


Introduction
The unique continuation property (ucp) for second order elliptic and parabolic equations represents one of the most fundamental aspects of pde's with a long history and several important ramifications. In this paper we prove the strong unique continuation property (sucp) for solutions to the parabolic differential inequality where M > 0 is arbitrary. In [9] (see also [2]) Vessella proved a general sucp result for subcritical parabolic equations of the type under Lipschitz regularity assumptions on the principal part A(x, t). This provided a parabolic counterpart to the previous work of Hörmander in [5]. Comparing (1.1) with (1.2), it is obvious that our result sharpens Vessella's theorem, when the latter is specialised to the heat equation.
As it is well-known, the inverse-square potential V (x) = M |x| 2 represents a critical scaling threshold in quantum mechanics [1], and it is equally known that its singularity is the limiting case for the sucp for the differential inequality |∆u| ≤ M |x| m |u|, see the counterexample in [3]. Such potential fails to be in L n/2 loc , and in general does not have small L n/2,∞ seminorm, thus in the context of the Laplacian the sucp cannot be treated by the celebrated result of Jerison and Kenig in [4] or the subsequent improvement by Stein in the appendix to the same paper.
We recall that, in the time-independent case of the Laplacian, the sucp for the unrestricted inverse square potential was proved by Pan in [7]. One should also see Regbaoui [8] for further generalisations to variable coefficient equations and Lin, Nakamura & Wang [6] for quantitative results.
The main new ingredient in this note is the following improved Carleman estimate for the heat operator ∆ − ∂ t in a space-time cylinder that is tailor-made for the differential inequality (1.1). Such result replaces the corresponding subcritical estimate in [9,Theorem 13] (see also [2,Theorem 2]). Similarly to the time-independent case in [7] and [8], our proof of 1.1 exploits the spectral gap on S n−1 , but we emphasise that it is also not independent of the work [9], see (2.12) below.
There exists a universal constant C > 0 such that for all α > 1 large enough, such that α − n−3 2 ∈ N, one has With (1.3) in hands, we establish the following strong unique continuation result. In the sequel, parabolic vanishing to infinite order means that as r → 0 one has for all k > 0, loc be a solution in B R × (0, T ) to the differential inequality (1.1). If u parabolically vanishes to infinite order, The plan of the paper is as follows. In Section 2 we prove Theorem 1.1. Section 3 is devoted to proving Theorem 1.2. One word of caution for the reader. It is generally accepted among experts that, once a proper Carleman estimate is available, the ucp, or the sucp follow from a standard application of the former. While this is generally true, in the time-dependent setting of the present note deducing Theorem 1.2 from Theorem 1.1 requires a delicate adaptation of the analogous treatment of the subcritical case in [9]. For this reason, we have not followed the tradition of skipping details, but we have carefully presented them in the proof of Theorem 1.2.

Proof of Theorem 1.1
We begin by introducing the relevant notation. Given r > 0 we denote by B r (x 0 ) the Euclidean ball centred at x 0 ∈ R n with radius r. When x 0 = 0, we will use the simpler notation B r . A generic point in space time R n × (0, ∞) will be denoted by (x, t). For notational convenience, ∇f and div f will respectively refer to the quantities ∇ x f and div x f of a given function f . The partial derivative in t will be denoted by ∂ t f and also by f t . We indicate with C ∞ 0 (Ω) the set of compactly supported smooth functions in the region Ω in space-time. By H 2,1 loc (Ω) we refer to the parabolic Sobolev class of functions f ∈ L 2 loc (Ω) for which the weak derivatives ∇f, ∇ 2 f and ∂ t f belong to L 2 loc (Ω). For a point x ∈ R n \ {0}, we will routinely adopt the notation r = r(x) = |x| and ω = x r ∈ S n−1 , so that x = rω. The radial derivative of a function v is v r =< ∇v, x |x| >. The following simple observations will be repeatedly used in what follows. Let γ ∈ R, then in R n \ {0} we have In particular, (2.1) gives Proof. It suffices to observe that div(f gr −1 x) = f g r + gf r + (n − 1)r −1 f g.
Integrating this identity and applying (2.1) we reach the desired conclusion.
Proof of Theorem 1.1. In all subsequent integrals, for given R, T > 0 the domain of integration will be the parabolic cylinder B R × (0, T ) (or, for that matter, the whole of R n × R, in view of the support property of the integrands), but this will not be explicitly indicated.
Nor, we will explicitly write the measure dxdt in any of the integrals involved. Let u ∈ where β is to be carefully chosen subsequently. Clearly, u = r β v. Since ∆(r β ) = β(β + n − 2)r β−2 , in a standard way we find where ω ∈ S n−1 and ∆ S n−1 denotes the Laplacian on S n−1 , we have We now apply the numerical inequality (a + b) 2 We now handle each integral in the right-hand side of (2.2) separately. Our first objective is choosing β in such a way that the integral r −2α+2β−3 vv r , which multiplies the cubic term in β, vanishes. To accomplish this we observe that Lemma 2.1 gives We now substitute the value of β given by (2.4) in the remaining integrals in (2.2) obtaining the following conclusions. First, we have Next, using polar coordinates and Stokes' theorem on S n−1 , we find Using Lemma 2.1 again, we find We also claim that To see (2.7) we apply Lemma 2.1 with g = r −n+2 v and f = v r , obtaining Since the last term vanishes in view of (2.3), we conclude that (2.7) holds. Next, Lemma 2.1 again gives Note that in the third equality above we have used that r −n+1 ∆ S n−1 vv r = 0, a fact which was earlier established in (2.5). Finally, the integral which accounts for the critical term in the Carleman estimate (1.3), is handled as follows.
Recall that in the Fourier decomposition of Using this representation and Parseval's theorem, we obtain At this point, we assume that dist(β, N) = 1/2. Since for every k ∈ N ∪ {0} we have we thus infer If we now combine (2.2)-(2.9), then for any β > 1 such that dist(β, N) = 1 2 we obtain We thus find for some universalC > 0, depending only on n, that the following inequality holds WithC fixed as in (2.10), if we now choose β >C, then 2β 2 −Cβ ≥ β 2 . Furthermore, applying first the Cauchy-Schwarz inequality, and then the inequality |ab| ≤ ε 2 a 2 + 1 2ε b 2 , with ε = β/C, we obtain the bound Combining these observations, and recalling that v = r −β u, we obtain from (2.10) We now use the following crucial inequality which follows from [2, Equation (1.8)], and keeping (2.4) in mind, which in terms of α is expressed by the condition α − n−3 2 ∈ N, we finally obtain from (2.11) From this inequality, it is at this point obvious that for all α > 1 sufficiently large, such that α − n−3 2 ∈ N, the desired conclusion (1.3) follows.

Proof of Theorem 1.2
In this section we show how to obtain the sucp result in Theorem 1.2 from Theorem 1.1. With the new estimate (1.3) in hands, we can adapt to the critical differential inequality (1.1) some of the ideas that in [9, Theor. 15, p. 658-664] were developed in the subcritical context of (1.2). As we have mentioned in the introduction, this entails a delicate modification of Vessella's proof. For this reason, and for the sake of the reader's comprehension, we will present a detailed account. We begin with the following simple Caccioppoli type inequality. Proof. From (1.1), we may assume that u solves ∆u − u t = V u, where |V (x, t)| ≤ M |x| 2 . Let now φ ≡ 1 in {r/2 < |x| < r} × (−T /2, T /2), and vanishing outside {r(1 − a)/2 < |x| < br} × (−T, T ). Using φ 2 u as a test function in the weak form of the equation we obtain Since an integration by parts gives we obtain from (3.1) By the Cauchy-Schwarz inequality we obtain in a standard fashion 2 |∇u||∇φ||φ||u| ≤ 1 2 |∇u| 2 φ 2 + 2 u 2 |∇φ| 2 . Substitution in the latter inequality gives Using the bounds |∇φ| ≤ C 2 /|x|, |φ t | ≤ C 3 /T , and the fact that φ, ∇φ, φ t are supported in {r(1 − a)/2 < |x| < br} × (−T, T ), we obtain from (3.2) that the following holds, for some C 1 depending on n, a, b, T and M . The desired conclusion follows by bounding from below the integral in the left-hand side with one over the region where φ ≡ 1.

Proof of Theorem 1.2.
In what follows it will be easier for the computations to work with the symmetric time-interval (−T, T ) instead of (0, T ). Let 0 < r 1 < r 2 /2 < 4r 2 < r 3 < R/2 be fixed, and let φ(x) be a smooth function such that φ(x) ≡ 0 when |x| < r 1 /2, φ(x) ≡ 1 when r 1 < |x| < r 2 , φ(x) ≡ 0 when |x| > r 3 . We now let T 2 = T /2 and T 1 = 3T /4, so that 0 < T 2 < T 1 < T . As in [9], we let η(t) be a smooth even function such that η(t) ≡ 1 when |t| < T 2 , η(t) ≡ 0, when |t| > T 1 . Furthermore, it will be important in the sequel (see (3.14) below) that η decay exponentially near t = ±T 1 . As in (118) of [9] we take We suppose that u parabolically vanishes to infinite order in the sense of (1.4), and we want to conclude that u ≡ 0 in B R × (0, T ). We assume that this not the case and show that we reach a contradiction. Without loss of generality we can (and will) assume that (3.4) Otherwise, the result in [9] implies u ≡ 0 in B R × (−T 2 , T 2 ) and by the arguments that follow we could conclude that u ≡ 0 also for |t| > T 2 . The assumption (3.4) will be used in the very end in the equation (3.23). Henceforth, we will indicate with the letter C an all purpose constant which might change from line to line, and which could depend in some occurrences on the number T . Now, with u as in Theorem 1.2 we let v = φηu. A standard limiting argument allows to use such v in the Carleman estimate (1.3), obtaining If we combine this inequality with an application to v of the estimate (2.1) in [2], and we keep in mind that Since the differential inequality (1.1) gives then the first integral in the right-hand side of (3.5) can be absorbed in the left-hand side. Consequently, from the way φ and η have been chosen, and keeping in mind that ∇φ is supported in {r 1 /2 < r < r 1 } ∪ {r 2 < r < r 3 } and that we have in such a set |∇φ| = O(1/r), |∆φ| = O(1/r 2 ), we obtain from (3.5) We now split the second to the last term in the right-hand side of (3.7) in three parts Since |η t | ≤ C/t, the first and the third terms in the right-hand side of (3.8) are respectively estimated as follows (3.9) C and (3.10) C In order to estimate the second term in the right-hand side of (3.8), we combine it with the last integral in the right-hand side of (3.7) and observe that, since φ ≡ 1 in the region {r 1 < |x| < r 2 }, and the function η t is supported in the set At this point our objective is to establish the following estimate The proof of (3.11) will be accomplished in several steps. First, we note that it suffices to concern ourselves with the portion of the integral in the left-hand side of (3.11) over the region Using this estimate in the above inequality, we obtain Comparing the right-hand side of (3.13) with that of (3.11), it should be clear to the reader that, in order to establish (3.11), it suffices at this point to be able to bound from above in D the quantity r −2α−3 η ηT 6 (T 1 +t) 8 . We accomplish this by first observing that, thanks to the exponential vanishing of η at t = −T 1 , see (3.3), we obtain for t ∈ (−T 1 , −T 2 ), (3.14) ηT 6 (T 1 + t) 8 ≤ C, for some universal C > 0 (depending on T ). Secondly, we show that, thanks to the inequality (3.12), the following holds in the region D provided that we choose the parameter α large enough (3.15) r −2α−3 η ≤ 1.
Using the expression (3.3) for η(t), we see that (3.15) does hold in D if and only if for α sufficiently large we have in such set To prove (3.16) observe that (3.12) can be equivalently written in D as for some universal C > 0. Since for α sufficiently large we trivially have we conclude that in D we must have if α > 1 has been chosen large enough. Since T 4 = T 1 − T 2 = T 1 + t + |T 2 + t|, from (3.17) we conclude that we must have in D If we now use this bound from below along with (3.12), we find in D provided that r < r 3 ≤ 1, and that α is sufficiently large (we stress here the critical role of the power α 9/8 , versus the linear term 2α + 3, in reaching the above conclusion. This is precisely why we have introduced in (3.5) the estimate (2.1) from [2]). We have thus proved (3.16), and consequently (3.15). Combining (3.13), (3.14) and (3.15), we conclude that (3.11) holds. Using now the estimates (3.8), (3.9), (3.10) and (3.11) in (3.7), we find By Lemma 3.1 it follows that and also where the constant in the latter estimate depends also on the parameters r 2 < r 3 ≤ 1 which are finally fixed at this point. Substituting these bounds in (3.19) we obtain The integral in the left-hand side of (3.20) can be bounded from below in the following way Adding α Br 1 ×(−T 2 ,T 2 ) u 2 to both sides of the latter inequality, and using (3.21), we obtain where, recalling our initial choice r 1 < 4r 2 , we note that in the second inequality in (3.22) we have used Keeping in mind the hypothesis (3.4), we now choose α (depending on u) such that (3.23) α where C is as in (3.22). Thus, by subtracting off C B R ×(−T,T ) u 2 from both sides of (3.22), we obtain At this point, we fix α sufficiently large in such a way that (3.6), (3.17), (3.18) and (3.23) simultaneously hold. Letting 3r 1 /2 = s, we obtain from (3.24) that for some new constants C, A depending on r 2 , r 3 , R, the ratio B R ×(−T,T ) u 2 Br 2 ×(−T 2 ,T 2 ) u 2 , and α (which at this point is fixed), the following holds for all 0 < s ≤ r 2 /8 Bs×(−T,T ) Since this estimate is in contradiction with the hypothesis that u parabolically vanish to infinite order in the sense of (1.4), we have finally proved the theorem.