ON FRACTIONAL LEIBNIZ RULE FOR DIRICHLET LAPLACIAN IN EXTERIOR DOMAIN

. The goal of the work is to verify the fractional Leibniz rule for Dirichlet Laplacian in the exterior domain of a compact set. The key point is the proof of gradient estimates for the Dirichlet problem of the heat equation in the exterior domain. Our results describe the time decay rates of the derivatives of solutions to the Dirichlet problem.

1. Introduction. The fractional Leibniz rule plays an important role in the study of nonlinear partial differential equations. Typical bilinear estimate used frequently is the following one whereḢ s p is the homogeneous Sobolev space with norm D s f L p and D s = (−∆) s/2 is the Riesz potential. A typical domain for parameters s, p, p j , j = 1, · · · , 4, where

VLADIMIR GEORGIEV AND KOICHI TANIGUCHI
(1) is valid is s > 0, 1 < p, p 1 , p 2 , p 3 , p 4 < ∞, Classical proof can be found in [12]. The estimate can be considered as natural homogeneous version of the non-homogeneous inequality of type (1) involving (1 − ∆) s/2 in the place of D s , obtained by Kato and Ponce in [23] (for this the estimates of type (1) are called Kato-Ponce estimates, too). More general domain for parameters can be found in [3], [6], [11]. Another generalization of the fractional Leibnitz rule in Besov spaces has the form f g Ḃs p,q (H0) ≤ C f Ḃs p 1 ,q (H0) g L p 2 (R n ) + f L p 3 (R n ) g Ḃs p 4 ,q (H0) , whereḂ s p,q =Ḃ s p,q (H 0 ) is the standard homogeneous Besov space associated with the free Hamiltonian H 0 = −∆ in R n .
A natural question is to extend this estimate to the case of perturbed Hamiltonian H = − ∆| D associated with Dirichlet Laplacian in exterior domain Ω in R n , such that its complement is a compact set with C 1,1 boundary. We recall the definition of the homogeneous Besov spacesḂ s p,q (H) with norm where s ∈ R, 1 ≤ p, q ≤ ∞, and we have used the functional calculus for the selfadjoint operator H in combination with the Littlewood -Paley partition in the right side of this identity. For the details of definition ofḂ s p,q (H), we refer to [20] (and also [21]).
The fractional Leibniz rule has been studied in [21] and the bilinear estimate f g Ḃs p,q (H) ≤ C f Ḃs p 1 ,q (H) g L p 2 (Ω) + f L p 3 (Ω) g Ḃs is obtained under the following key assumption (see (2.4) in [21]) valid for any t > 0 and 1 ≤ p ≤ p 0 , where p 0 is determined by Ω. Our first result gives an answer to the question if fractional Leibniz rule holds without additional assumptions on gradient type estimates (2).
Let Ω be the exterior domain in R n of a compact set with C 1,1 boundary. Let 0 < s < 2 and p, p 1 , p 2 , p 3 , p 4 and q be such that Then there exists a constant C > 0 such that for any f ∈Ḃ s p1,q (H) ∩ L p3 (Ω) and g ∈Ḃ s p4,q (H) ∩ L p2 (Ω). Our key point in the proof is to show that (2) holds for 1 ≤ p ≤ n, therefore our main goal shall be the proof of the gradient estimate over the evolution flow of the perturbed Hamiltonian H. The gradient estimates of the evolution flow have different applications in the theory of incompressible Navier -Stokes flow (see [2], [13], [14], [15], [16], [25], [27]) as well in harmonic analysis (comparison between Sobolev or Besov type norms associated with free and perturbed evolution flow, see [1], [4], [9], [19], [20], [21], [24]) . A typical gradient estimate for the classical heat equation valid for any t > 0 and 1 ≤ p ≤ ∞. The proof follows immediately from the explicit representation formula of u. The estimate (3) becomes in the case of initial boundary value problem with Dirichlet boundary condition in domain Ω. This estimate is true for any t > 0 and 1 ≤ p ≤ ∞, when Ω is a half space or bounded domain. However, the question whether an optimal gradient estimate similar to (4) is true for the linear heat flow in arbitrary exterior domain with Dirichlet boundary condition for any t > 0 and 1 ≤ p ≤ ∞ seems to remain without complete answer due to our knowledge.
Surprisingly, more information and in particular answers to this question can be found in the case of Stokes equations in exterior domain. In fact, the estimate (4) has appropriate modification in the case of initial boundary value problem for the Stokes equation in exterior domain Ω in R n . The case of Stokes equations with Dirichlet boundary condition x ∈ Ω is studied in [27], [2], where the gradient estimate if n < p ≤ ∞ is verified. The optimality of the estimate of (5) is discussed in [27], where the authors show that estimate of type is not true. Estimate (5) for the Dirichlet problem of heat equation in Ω: is studied for several situations. The case 1 ≤ p ≤ 2 is studied in [22], [29], where the estimate (4) is proved for any t > 0 in an arbitrary open set. On the other hand, the situation in the case p > 2 is more complicated. The case of the Ornstein -Uhlenbeck semigroup, including heat semigroup as a special case, is considered for t > 0 and 1 < p < ∞ in [8]. The case of parabolic equation is considered in [26], [5] and the results obtained in these works imply that the classical bounded solution satisfies the gradient estimate (4) with p = ∞ for 0 < t ≤ T . Our first goal shall be to prove the gradient estimate (5) for the heat flow in general exterior domain with Dirichlet boundary condition. Our second main point is suggested by the following observation. One can expect that the estimate (4) shall be true at least in the special case, when Ω is an exterior of a ball, since one can have an explicit representation of the solution u. Note that the estimate (4) is stronger than the estimate (5) for p > n as t → ∞. For this, our next step shall be to show that (4) is not fulfilled when Ω is an exterior of a ball. In this case, denoting by u(t; f ) the heat flow solution to (7) with initial data f , we can show for any 1 ≤ p ≤ ∞. The right inequality follows from the gradient estimate (5). The left inequality, i.e. the positivity of the supremum gives variational characterization of the best constant in (5) and implies the optimality of the gradient estimate at least when Ω is the exterior of a ball.

2.
Assumptions and main results on gradient estimates. As stated in the introduction, our first result concerns gradient estimate for the heat flow outside Ω with Dirichlet boundary condition.
Theorem 2.1. Let n ≥ 2 and Ω be the exterior domain in R n of a compact set with C 1,1 boundary. Then, for any 1 ≤ p < ∞, there exists a constant C > 0 such that the solution u of (7) satisfies for t > 1 (8) for any f ∈ L p (Ω), where the exponent µ is given by Remark 1. Since we consider boundary with weak regularity, it is not clear whether the gradient estimate (8) with p = ∞ is true for any f ∈ L ∞ (Ω) due to our knowledge. However the gradient estimate is true for classical solutions. In fact, the classical bounded solutions to Dirichlet problem of parabolic equations in bounded or unbounded domains with sufficiently smooth boundary satisfy the local (in time) gradient estimate [26], [5], and references therein). One can establish the global estimate by using the above local gradient estimate combined with L ∞ -estimate of Lemma 3.1 below.

Remark 2.
Assuming we have C 1,1 boundary, the authors in [8] have established that there exists ω ≥ 0 so that the following estimate holds. From this estimate we can see that is fulfilled, i.e. we have (8) for small values of t and 1 < p < ∞. The estimate (8) for small values of t and for the endpoint case p = 1 can be deduced from the results in [22], [29].
Remark 3. In the case of Neumann boundary condition, the estimate (8) can be replaced by its stronger version (4), i.e. we have the same estimate as in the case of the whole space R n . Indeed, in this case, the gradient estimate (4) holds for any t > 0 and 1 ≤ p ≤ ∞ (see, e.g., [17] and [30]).
Next, we discuss the optimality of time decay rates in the gradient estimates (8) as t → ∞.
Definition 2.2. We say that the gradient estimate (8) where u m is a solution to (7) with initial data f m and the exponent µ is given by (9).
Remark 4. If we can verify the optimality of (8) in the sense of Definition 2.2, then we can assert is a well-defined positive number that gives a variational characterization of the best constant C = C(Ω, p) in (8).
Our result on the optimality is the following. To simplify the proof, we shall fix the space dimension n = 3. Theorem 2.3. Let n = 3 and Ω be the exterior domain of a ball. Then, for any 1 ≤ p ≤ ∞, the gradient estimate (8) is optimal in the sense of Definition 2.2.
Remark 5. Note that the optimality of estimate of type (8) is verified in the context discussed in [27] for arbitrary exterior domains Ω. However, optimality treated in this work means that for any δ > 0. In other words, (6) is not true for any δ > 0. This optimality is weaker than that of Definition 2.2.
Remark 6. In the case p = ∞, when n ≥ 3 and Ω is the exterior domain of a compact connected set with C 1,1 boundary, we can obtain the estimate for any t > 1, where e −tH is the semigroup generated by the Dirichlet Laplacian H = −∆ on Ω (see Appendix A). This is stronger than the optimality of Definition 2.2, since (11) implies the optimality for p = ∞ in the sense of Definition 2.2.
Remark 7. If one compare the optimality result in Theorem 2.3 and the estimate (10), then one can deduce that ω > 0 in (10), therefore the assertion (b) of Theorem 3.1 in [8] holds for some ω > 0 and it is not true for ω = 0.
Proof of Theorem 1.1. Follows directly from the gradient estimates of Theorem 2.1, combined with Theorem 2.1 and remark (ii) in section 2 in [21].
The plan of the work is the following. In section 3 we state key estimates for solutions of heat equation (7) to prove Theorem 2.1, and then give the proof of the theorem. In section 4 we show the optimality of time decay rates in gradient estimates (8) in Theorem 2.1.
3. Proof of Theorem 2.1. In this section we prove Theorem 2.1. For the purpose, we prepare key estimates for solutions of heat equations (7). The first one is the result on L p -L q -estimates.
for any t > 0 and f ∈ L p (Ω).
For the proof, we refer to Proposition 3.1 in [22] (see also Section 6.3 in [29]). The second one is the result on the gradient estimates for 1 ≤ p ≤ 2.

Lemma 3.2.
Let Ω be an open set in R n and 1 ≤ p ≤ 2. Then for any t > 0 and f ∈ L p (Ω).
For the proof, we refer to Theorem 1.2 in [22] (see also Theorem 6.19 in [29]).
Proof of Theorem 2.1. The case p = 1 is proved in Lemma 3.2. Hence, in order to obtain (8) for any 1 ≤ p < ∞, it suffices to prove the case n ≤ p < ∞ by density and interpolation argument: For any n ≤ p < ∞, there exists a constant C > 0 such that for any f ∈ C ∞ 0 (Ω). Let us choose L > 0 such that Putting As to the first term, we can obtain Since for any t > 0 by Lemma 3.1, the right hand side in (16) is estimated as for any t > 0. Therefore we obtain the required estimates (15). Thus all we have to do is to estimate the second term in (14) as follows: We divide the proof of (17) into two cases: 0 < t ≤ 1 and t > 1.
Here v 1 (t) is the solution to the Cauchy problem of heat equation in R n : and v 2 (t) is the solution to the Cauchy problem of heat equation in R n : It is easily proved that for any 0 < t ≤ 1. Hence it is sufficient to show that for any 0 < t ≤ 1. Letting e t∆ be the semigroup generated by −∆ on R n , we write v 2 (t) as for 0 < t ≤ 1 and x ∈ R n . Denoting by G(t, x − y) the kernel of e t∆ , we estimate Here we note that for any t > 0 and x ∈ R n . In particular, if |x| ≥ L + 2 and |y| ≤ L + 1, then for any 0 < s < t. Therefore we deduce that for any 0 < s < t and L < |y| ≤ L + 1. Combining (23) and (24), we obtain Recalling the definition (20) of F (s, x), and using (15) and Lemma 3.1, we estimate for any 0 < s ≤ 1. Combining the above two estimates, we deduce that for any 0 < t ≤ 1, which proves (22). Therefore the estimate (17) for any 0 < t ≤ 1 is proved by (21) and (22).
The case t > 1. In a similar way to (19) in the previous case, we decompose for t ≥ 1. Here w 1 (t) is the solution to the Cauchy problem of heat equation in R n : x ∈ R n , and w 2 (t) is the solution to the Cauchy problem of heat equation in R n : where we recall (18) and (20). It is easily proved that for any t > 1. Hence it is sufficient to show that for any t > 1. Writing w 2 (t) as for t > 1 and x ∈ R n , we estimate, in a similar way to (25), Recalling the definition (20) of F (s, x), and using (15) and Lemma 3.1, we estimate for any s > 1. Combining the above two estimates, we deduce that for any t > 1. For 1 < t < 2 we use the inequality For t > 2 and p ≥ n, we have

4.
Proof of Theorem 2.3. In this section we discuss the optimality of time decay rates of estimates (8) in Theorem 2.1. Let Ω be the exterior domain in R 3 determined by Ω = {x ∈ R 3 : |x| > 1}.
We prove the optimality in the sense of Definition 2.2 for heat equation (7) with a radial initial data f on Ω.
Proof. Let f be a radial function on Ω. Since u(t) is also radial, we write for t > 0 and r = |x|. We rewrite the problem (7) to the following problem by the polar coordinates and making change v(t, r) = (r + 1)U (t, r + 1): where g(r) = (r+1)F (r+1) and r = |x|. Then solutions v to (28) and the derivative ∂ r v can be represented as for t > 0 and r > 0. Furthermore, noting that u(t, x) = U (t, r) = r −1 v(t, r − 1), we write In order to prove the optimality, we choose appropriate initial data f m and estimate from below the quantity from Definition 2.2: for m ∈ N, where the exponent µ is defined in (9). We divide the proof into two cases: 1 ≤ p ≤ 3 and 3 < p ≤ ∞.
The case 1 ≤ p ≤ 3. We take for m ∈ N, and define the initial data as follows Here we choose the constant C m such that C m > 0 and f m L p (Ω) = 1.
Then we have g m (r) = C m , r ∈ (m, 2m], 0, otherwise, and as m → ∞. Let us denote by u m and v m the solutions to (7) and (28) with initial data f m and g m , respectively. By the equality (31), we write .
Taking t = t m = m 2 , we write where c 0 = 2 + √ 2. From the representation (29) and definition (34) of g m , the right hand side is estimated as Hence, by combining (36)-(38), we estimate from below Hence, noting from (35) that Thus the optimality for 1 ≤ p ≤ 3 is proved.
Again we take t = t m = m 2 and denote by u m and v m the solutions to (7) and (28) with initial data f m in (32) and g m in (34), respectively.
To begin with, we prove the following: For sufficiently large m ∈ N, there exists a constant C > 0, independent of m, such that for any 10 ≤ r ≤ m 1/4 and m ≤ s ≤ 2m. Writing we calculate where we used in the last step for 10 ≤ r ≤ m 1/4 and m ≤ s ≤ 2m. Since we can neglect the remainder terms in (43) if m is sufficiently large, we obtain (42).
Let us turn to estimate form below of L p -norm of ∇u m (t m ). By combining (41)  Appendix A. . In this appendix, we show the estimate (11): ∇e −tH L ∞ (Ω)→L ∞ (Ω) ≥ C for any t > 1, when n ≥ 3 and Ω is the exterior domain of a compact connected set with C 1,1 boundary. The estimate (11) follows from the known result: There exists a constant C > 0 such that for any t > 1 (see section 1 in [18] and also [31]). In fact, we suppose that for any t > 1, where C(t) → 0 as t → ∞. Then we deduce from (46) and Lemma 3.1 that for any t > 1 and f ∈ L 1 (Ω). This contradicts the fact (45). Thus (11) is true.
Appendix B. . In this appendix we prepare two fundamental inequalities. The first one is the special case of the Gagliardo -Nirenberg inequality (see [7], [28]).
Let Ω be a bounded domain in R n having the cone property. Then, for any 1 < p < ∞, there exist constants C 1 , C 2 > 0 such that for any f ∈ W 2,p (Ω).
The second one is the global W 2,p -estimate (see Theorem 9.13 in [10]).