The sharp second order Caffareli-Kohn-Nirenberg inequality and stability estimates for the sharp second order uncertainty principle
aa r X i v : . [ m a t h . F A ] F e b The sharp second order Caffareli-Kohn-Nirenberginequality and stability estimates for the sharp secondorder uncertainty principle
Anh Tuan Duong ∗ and Van Hoang Nguyen † February 3, 2021
Abstract
In this paper we prove a class of second order Caffarelli-Kohn-Nirenberg inequal-ities which contains the sharp second order uncertainty principle recently establishedby Cazacu, Flynn and Lam [13] as a special case. We also show the sharpness of ourinequalities for several classes of parameters. Finally, we prove two stability versionsof the sharp second order uncertainty principle of Cazacu, Flynn and Lam by show-ing that the difference of both sides of the inequality controls the distance to theset of extremal functions both in L norm of function and in L norm of gradient offunctions. The Heisenberg uncertainty principle in quantum mechanics states that the position andthe momentum of a given particle cannot both be determined exactly at the same time (see[30]). The rigorous mathematical formulation of this principle is established by Kennard[32] and Weyl [48] (who attributed it to Pauli) stating that the function itself and its Fourier ∗ Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy Street, CauGiay District, Ha Noi, Viet Nam. † Department of Mathematics, FPT University, Ha Noi, Viet Nam.Email: [email protected]; [email protected] and [email protected]
Mathematics Subject Classification : 26D10, 46E35, 26D15
Key words and phrases : Caffarelli–Kohn–Nirenberg inequalities, uncertainty principle, sharp constant,extremal functions, stability estimates Z R n |∇ u | dx Z R n | x | | u | dx ≥ n (cid:16) Z R n | u | dx (cid:17) (1.1)for any function u ∈ H ( R n ) (the first order Sobolev space in R n ) such that R R n | x | | u | dx < ∞ . It is well known that the constant n / n − tγt Z R n | u | t | x | tγ dx ≤ (cid:16) Z R n |∇ u | p | x | αp dx (cid:17) p (cid:16) Z R n | u | p ( t − p − | x | β dx (cid:17) p − p (1.2)where n ≥
2, 1 < p < t , and α, β, γ satisfying the following conditions n − αp > , n − β > , n − tγ > , and the balanced condition γ = 1 + αt + βtp . Moreover, when 1 + α − βr > n − β < (cid:16) α − βr (cid:17) p ( t − t − p then the inequality (1.2) is sharp and the extremal functions are given by u ( x ) = ( λ + | x | α − βr (cid:17) − pt − p , λ > . The inequality (1.2) is extended to the Riemannian manifolds in [41], the Finsler manifoldsin [31] and the stratified Lie groups in [38].Both the Heisenberg-Pauli-Weyl principle (1.1) and the Xia inequality (1.2) belongto a larger class of the first order interpolation inequalities which are called Caffarelli-Kohn-Nirenberg (CKN) inequalities established in [8] to study the Navier-Stokes equationand the regularity of particular solutions [7]. The class of CKN inequalities containsmany well-known inequalities such as the Sobolev inequality, the Hardy inequality, theHardy-Sobolev inequality, the Gagliardo-Nirenberg inequality, etc. They play an importantrole in theory of partial differential equations and have been extensively studied in manysettings. Concerning to the sharp version of CKN inequalities, we refer the reader to thepapers [2, 10, 11, 17, 18, 20, 21, 33, 45].The higher order CKN inequalities were established by Lin [34]. In contrast to thefirst order inequalities, much less is known on the sharp version of the higher order CKNinequalities except the Rellich inequality [43] and the sharp higher order Sobolev inequality219, 33]. In recent paper [13], Cazacu, Flynn and Lam proved the following sharp secondorder uncertainty principle which is a special case of the second order CKN inequalities Z R n | ∆ u | dx Z R n | x | |∇ u | dx ≥ (cid:16) n + 22 (cid:17) (cid:16) Z R n |∇ u | dx (cid:17) . (1.3)The constant ( n + 2) / u by a divergence-free vector field U . In particular, the inequality (1.3) answers affirmativelythe question of Maz’ya in the case n = 2.The first aim in this paper is to extend the inequality (1.3) to a larger class of parametersin spirit of (1.2). For α, β satisfying the conditions n − α > n − β >
0, we denote by H α,β ( R n ) the second order Sobolev space which is completion of C ∞ ( R n ) under the norm k u k H α,β ( R n ) = (cid:16) Z R n | ∆ u | | x | − α dx + Z R n |∇ u | | x | − β dx (cid:17) , u ∈ C ∞ ( R n ) . Then the first main result in this paper reads as follows.
Theorem 1.1.
Let n ≥ and α ∈ R satisfy n − α > , n + 2 α > and n + 2 + 4 α > .Then the following inequality Z R n | ∆ u | | x | α dx Z R n | x | α |∇ u · x | dx ≥ (cid:16) n + 4 α + 22 (cid:17) (cid:16) Z R n |∇ u | dx (cid:17) (1.4) holds true for any function u ∈ H α, − − α ( R n ) . Furthermore, if α > then the inequality (1.4) is sharp and is attained by function U ( x ) = exp (cid:18) − | x | α ) α ) (cid:19) . In particular, when α = 0 , Theorem 1.1 implies the following.
Corollary 1.2.
Let n ≥ . Then there holds Z R n | ∆ u | dx Z R n |∇ u · x | dx ≥ (cid:16) n + 22 (cid:17) (cid:16) Z R n |∇ u | dx (cid:17) (1.5) for any u ∈ H , − ( R n ) . This inequality is sharp and is attained by the Gaussian function exp( −| x | / . Evidently, we have |∇ u · x | ≤ |∇ u || x | . Hence, our inequality (1.5) is still stronger than(1.3).Recall that the proof of (1.3) in [13] is quite long and complicated. The authors haveused the decomposition of function u into spherical harmonic and integral estimates forradial functions. 3n order to prove (1.4), we develop a new approach which is completely different withthe one of Cazacu, Flynn and Lam. Indeed, our approach is based on establishing a newidentity Z R n | ∆ u + ∇ u · x | x | α | | x | α dx = Z R n | ∆ u | | x | α dx + Z R n | x | α |∇ u · x | dx − ( n − Z R n |∇ u | dx. Then, using a factorization of u as u = vU , we are able to show that Z R n | ∆ u + ∇ u · x | x | α | | x | α dx ≥ n + 2 α ) Z R n |∇ u | dx. Combining two estimates above and a simple minimizing argument, we obtain (1.4). Moredetails in the proof are given in Section § Theorem 1.3.
Let n ≥ , t ≥ and α, β, γ be such that n − α > , n − β > , n − tγ > and γ = 1 + αt + β t . (1.7) Then the following inequality Z R n | ∆ u | | x | α dx Z R n |∇ u | t − | x | β dx ≥ (cid:18) n + t (1 + 2 α − γ ) t (cid:19) (cid:18)Z R n |∇ u | t | x | tγ dx (cid:19) (1.8) holds true for any radial function u ∈ H α,β ( R n ) . Moreover, under the following conditions (1 + 2 α )( t −
2) + 1 + α − β > , t < α − β , (1.9) n + 2 α > , (1.10) and n − β < t − t − α − β , (1.11) then the constant ( n + t (1 + 2 α − γ )) /t is sharp and is attained only up to a dilation anda multiplicative constant by the function the form U ( x ) = Z ∞| x | s α exp (cid:16) − s α − β α − β (cid:17) ds, if t = 2 , and U ( x ) = Z ∞| x | r α (cid:16) t − r (1+2 α )( t − α − β (1 + 2 α )( t −
2) + 1 + α − β (cid:17) − t dr, if t > .
4n the special case where α = β = 0, t = 2 and γ = 1 /
2, we recover the secondresult of Cazacu, Flynn and Lam in [13] for radial functions in H , ( R n ) which is the sharpsecond order Hydrogen uncertainty principle Z R n | ∆ u | dx Z R n |∇ u | dx ≥ ( n + 1) (cid:16) Z R n |∇ u | | x | dx (cid:17) . (1.12)In fact, in that paper, Cazacu, Flynn and Lam proved the inequality (1.12) for any function u (without radiality assumption) for any n ≥
5. The condition n ≥ n = 2 , ,
4. The inequality (1.12) issharp and an extremal function is given by u ( x ) = c (1 + a | x | ) e − a | x | with c ∈ R and a > n − γt ) /t ) to (( n + t (1 + 2 α − γ )) /t ) . Moreover, toobtain the sharpness and the attainability of constant, we need some more conditions onthe parameters (see (1.9) and (1.10)). Indeed, these conditions ensure that the function U (and U ) is well-defined, and ∆ U (and ∆ U ) exists in the distributional sense and belongsto H α,β ( R n ).In the non-radial case, following the approach of Cazacu, Flynn, and Lam by usingspherical harmonics, we are able to establish the sharp second order CKN inequalities(see Theorem 1.4 below) which extends (1.8) to any function in H α,β ( R n ) but with somerestrictions on the dimension n (as the case of the second order hydrogen uncertaintyprinciple (1.12)). However, this approach works only for t = 2.To state our next result, let us define for n, α, β, γ as in Theorem 1.3 with t = 2 and k ≥ A n,α,k ( g ) = Z ∞ ( g ′′ ) r n +2 k − α − dr + (1 + 2 α )( n + 2 k − Z ∞ ( g ′ ) r n +2 k − α − dr (1.13) B n,β,k ( g ) = Z ∞ ( g ′ ) r n +2 k − β − dr + βk Z ∞ g r n +2 k − β − dr (1.14) C n,γ,k ( g ) = Z ∞ ( g ′ ) r n +2 k − γ − dr + 2 γk Z ∞ g r n +2 k − γ − dr (1.15)and A k ( n, α, β ) = inf g A n,α,k ( g ) B n,β,k ( g ) C n,γ,k ( g ) (1.16)where the infimum is taken on all function g ∈ C ([0 , ∞ )) such that A n,α,β ( g ) and B n,β,k ( g )are finite. Our next result is given in the following theorem. Theorem 1.4.
Let n ≥ and α, β, γ satisfy the conditions of Theorem 1.3 with t = 2 ,then we have Z R n | ∆ f | | x | α dx Z R n |∇ f | | x | β dx ≥ min k ∈ N A k ( n, α, β ) (cid:16) Z R n |∇ f | | x | γ dx (cid:17) , (1.17)5 or any function u ∈ H α,β ( R n ) . Moreover, the constant min k ∈ N A k ( n, α, β ) in (1.17) issharp and satisfies the estimate min k ∈ N A k ( n, α, β ) ≥ min k ∈ N n , βk ( n +2 k − β − o n , γk ( n +2 k − γ − o (cid:16) n + 2 k + 4 α − γ + 22 (cid:17) . (1.18)The inequality (1.18) was proved by Cazacu, Flynn and Lam in [13] corresponding tothe case α = γ = 0 , β = − α = β = 0 , γ = . It plays an important role in their proofof the sharp second uncertainty principle (1.3) and the second order hydrogen uncertaintyprinciple (1.12). Indeed, by considering the right-hand side as a function of k , they showthat the infimum of the right-hand side is attained at k = 0 in the case α = γ = 0 , β = − n ≥ α = β = 0 , γ = for any n ≥
5. This implies (1.3) and(1.12). We believe that this argument together with (1.18) provides the same conclusionfor general α, β, γ when n large enough. Nevertheless, we do not pursue this direction inthis paper.The last aim of this paper is to establish the stability estimates for the sharp secondorder uncertainty principle (1.3) of Cazacu, Flynn and Lam. Let us define δ ( u ) = (cid:16) R R n | ∆ u | dx (cid:17) (cid:16) R R n | x | u dx (cid:17) n +22 R R n |∇ u | dx − u ∈ H , − ( R n ) \ { } and δ (0) = 0. Evidently, we always have δ ( u ) ≥
0. Establishingthe stability estimates for the sharp second order uncertainty principle (1.3) means thatwe use δ ( u ) to control the distance from u to the set of extremal functions, i.e., the set E = n c e − a | x | : c ∈ R , a > o . In fact, we shall prove two stability estimates for (1.3). The first stability result uses thedistance concerning to the L norm of gradient and reads as follows. Theorem 1.5.
Given n ≥ . Then the following inequality δ ( u ) ≥ C n + 2) inf n R R n |∇ u − ∇ v | dx R R n |∇ u | dx : v ∈ E, Z R n |∇ u | dx = Z R n |∇ v | dx o , (1.19) holds true for any u ∈ H , − ( R n ) \ { } with C = min { n/ , } . The second stability result concerns to the L norm of function and is stated as follows Theorem 1.6.
Given n ≥ . Then the following inequality δ ( u ) ≥ C n ( n + 2) inf n R R n | u − v | dx R R n | u | dx : v ∈ E, Z R n | u | dx = Z R n | v | dx o , (1.20) holds true for any u ∈ H , − ( R n ) \ { } with C = min { n/ , } .
6n recent years, there has been an enourmous attention on establishing the stabilityversion of the sharp inequalities in analysis and geometry, especially the Sobolev typeinequality. The question on the stability estimate for the sharp Sobolev inequality wasposed by Br´ezis and Lieb [6]. This question was affirmatively answered by Bianchi andEgnell [4] for functions in H ( R n ) by exploiting the Hilbert structure of this space. Forthe case of W ,p ( R n ) with p = 2, the stability estimates for the Sobolev inequality wereestablished in [14–16,25,26,28,37,44]. We also refer the readers to the papers [5,9,22–24,39]for the stability version of the Gagliardo-Nirenberg inequality and logarithmic Sobolevinequality. In recent paper [36], McCurdy and Venkatraman exploit the approach ofBianchi and Egnell [4] to prove a stability version of the classical Heisenberg-Pauli-Weylinequality (1.1) Z R n |∇ u | dx Z R n | x | | u | dx − n (cid:16) Z R n | u | dx (cid:17) ≥ ˜ C inf v ∈ E Z R n | u − v | dx for any u ∈ H ( R n ) with R R n | u | dx = 1 where ˜ C > n . In [42], the author gives a new and simple proof of the above inequality by using Poincar´einequality for Gaussian measure. In that paper, the author also proves the stability versionof (1.2).Let us finish this introduction by some comment on the proofs of Theorem 1.5 andTheorem 1.6. To prove our stability estimates, we shall provide a new proof of the inequalityof Cazacu, Flynn and Lam (the inequality (1.3)). This proof is based on the identity Z R n k∇ u + ∇ u ⊗ x k HS dx = Z R n | ∆ u | − n Z R n |∇ u | dx + Z R n |∇ u | | x | dx where ∇ u denotes the Hessian matrix of u , ∇ u ⊗ x denotes matrix ( ∂ i ux j ) ni,j =1 and k A k HS denotes the Hilbert-Schmidt norm of an n × n matrix A , i.e., k A k HS = (cid:16) Tr( A t A ) (cid:17) / with A t being the transpose of A . Using a factorization of u given by u = ve −| x | / , we obtain Z R n k∇ u + ∇ u ⊗ x k HS dx = Z R n k∇ v − x ⊗ ∇ v k HS e −| x | dx + 2 Z R n |∇ u | dx. Combining the two equalities, we have Z R n | ∆ u | − ( n + 2) Z R n |∇ u | dx + Z R n |∇ u | | x | dx = Z R n k∇ v − x ⊗ ∇ v k HS e −| x | dx. The right-hand side is an integral related to the Gaussian type measure. Then we can usespectral analysis of the Ornstein-Uhlenbeck type operator associated with the Gaussianmeasure and Hermite polynomials to obtain the following estimate Z R n k∇ v − x ⊗ ∇ v k HS e −| x | dx ≥ C Z R n |∇ v | e −| x | dx, C = min { n/ , } . This estimate is the key in the proofs of Theorem 1.5 and Theorem1.6. The detailed proof will be given in Section § §
2, we prove the second orderCKN inequalities given in Theorem 1.1, Theorem 1.3, and Theorem 1.4. Section § In this section, we provide the proof of the second order CKN inequalities in Theorems1.1, 1.3 and 1.4. We also show that under the conditions of parameters in these theorems,the obtained inequalities are sharp and we exhibit a class of extremal functions. We beginwith the proof of Theorem 1.1.
Proof of Theorem 1.1.
For any function u ∈ C ∞ ( R n ), we have Z R n | ∆ u + ∇ u · x | x | α | | x | − α dx = Z R n | ∆ u | | x | − α dx + |∇ u · x | | x | α dx + 2 Z R n ∆ u ∇ u · xdx. (2.1)Using integration by parts, we have Z R n ∆ u ∇ u · xdx = − Z R n ∇ u ( ∇ u ) · xdx − Z R n |∇ u | dx = − Z R n ∇ ( |∇ u | ) · xdx − Z R n |∇ u | dx = n − Z R n |∇ u | dx. Inserting this equality in (2.1), we get Z R n | ∆ u + ∇ u · x | x | α | | x | − α dx = Z R n | ∆ u | | x | − α dx + Z R n (cid:12)(cid:12)(cid:12)(cid:12) ∇ u · x | x | (cid:12)(cid:12)(cid:12)(cid:12) | x | α dx + ( n − Z R n |∇ u | dx. (2.2)We first consider the case α = −
1. Setting u = vU ( x ) with U ( x ) = exp (cid:16) − | x | α α (cid:17) , we have ∇ u = ∇ vU ( x ) + v ( x ) ∇ U ( x ) , ∆ u = ∆ vU + 2 ∇ v ∇ U + v ∆ U ∇ U ( x ) = − x | x | α e − | x | α α , ∆ U ( x ) = − ( n + 2 α ) | x | α e − | x | α α + | x | α e − | x | α α . Plugging this expression into ∆ u + ∇ u · x | x | α and using simple computations imply∆ u + ∇ u · x | x | α = (cid:16) ∆ v − ∇ v · x | x | α − ( n + 2 α ) v | x | α (cid:17) e − | x | α α . Hence, it holds Z R n | ∆ u + ∇ u · x | x | α | | x | − α dx = Z R n | ∆ v − ∇ v · x | x | α | | x | − α e − | x | α α dx + ( n + 2 α ) Z R n v | x | α e − | x | α α dx − n + 2 α ) Z R n (∆ v − ∇ v · x | x | α ) ve − | x | α α dx. We next compute the last integral in the right-hand side of the preceding equality. Noticethat the function v = ue | x | α α is not C at origin in general. So we can not use integrationby parts directly. To overcome this difficulty, we first notice that under the assumption u ∈ C ∞ ( R n ), n + 2 α > n + 2 + 4 α >
0, we have v ∆ ve − | x | α α = u (∆ u + 2 ∇ u · x | x | α + ( n + 2 α ) | x | α u + | x | α u ) ∈ L ( R n )and v ∇ v · x | x | α e − | x | α α = u ( ∇ u · x | x | α − u | x | α ) ∈ L ( R n ) . Therefore it holds Z R n (∆ v − ∇ v · x | x | α ) ve − | x | α α dx = lim ǫ → + Z B cǫ (∆ v − ∇ v · x | x | α ) ve − | x | α α dx, where B cǫ = { x ∈ R n : | x | ≥ ǫ } . Using integration by parts we have Z B cǫ (∆ v − ∇ v · x | x | α ) ve − | x | α α dx = Z B cǫ div (cid:0) ∇ ve − | x | α α (cid:1) ve − | x | α α dx = − Z B cǫ ∇ ve − | x | α α ∇ ( ve − | x | α α ) dx + Z {| x | = ǫ } ∇ v · x | x | ve − ǫ α α ds = − Z B cǫ |∇ v | e − | x | α α dx + 12 Z B cǫ ∇ v · x | x | α e − | x | α α dx + Z {| x | = ǫ } ∇ v · x | x | ve − ǫ α α ds = − Z B cǫ |∇ v | e − | x | α α dx − n + 2 α Z B cǫ v | x | α e − | x | α α dx + Z B cǫ v | x | α e − | x | α α dx + ǫ α e − ǫ α α Z {| x | = ǫ } v ds + Z {| x | = ǫ } ∇ v · x | x | ve − ǫ α α ds. ǫ → + and using the assumptions n + 2 α > n + 2 + 4 α >
0, we obtainlim ǫ → + Z B cǫ (∆ v − ∇ v · x | x | α ) ve − | x | α α dx = − Z R n |∇ v | e − | x | α α dx − n + 2 α Z R n v | x | α e − | x | α α dx + Z R n v | x | α e − | x | α α dx. Consequently, we arrive at Z R n | ∆ u + ∇ u · x | x | α | | x | − α dx = Z R n | ∆ v − ∇ v · x | x | α | | x | − α e − | x | α α dx + 2( n + 2 α ) Z R n |∇ v | e − | x | α α dx + 2( n + 2 α ) Z R n v | x | α e − | x | α α dx − n + 2 α ) Z R n v | x | α e − | x | α α dx. (2.3)Again by using integration by parts on B cǫ and letting ǫ → + , we have Z R n |∇ u | dx = Z R n |∇ v | U dx + Z R n v |∇ U | dx + Z R n ∇ v · U ∇ U dx = Z R n |∇ v | U dx − Z R n v U ∆ U dx = Z R n |∇ v | e − | x | α α dx + ( n + 2 α ) Z R n v | x | α e − | x | α α dx − Z R n v | x | α e − | x | α α dx. (2.4)Combining (2.2), (2.3) and (2.4), we obtain Z R n | ∆ u | | x | − α dx + Z R n (cid:12)(cid:12)(cid:12)(cid:12) ∇ u · x | x | (cid:12)(cid:12)(cid:12)(cid:12) | x | α dx + ( n − Z R n |∇ u | dx = Z R n | ∆ v − ∇ v · x | x | α | | x | − α e − | x | α α dx + 2( n + 2 α ) Z R n |∇ u | dx which is equivalent to Z R n | ∆ u | | x | − α dx + Z R n (cid:12)(cid:12)(cid:12)(cid:12) ∇ u · x | x | (cid:12)(cid:12)(cid:12)(cid:12) | x | α dx = Z R n | ∆ v − ∇ v · x | x | α | | x | − α e − | x | α α dx + ( n + 4 α + 2) Z R n |∇ u | dx. (2.5)10y density argument, (2.5) still holds for any functions u ∈ H α, − − α ( R n ). It follows from(2.5) that Z R n | ∆ u | | x | − α dx + Z R n (cid:12)(cid:12)(cid:12)(cid:12) ∇ u · x | x | (cid:12)(cid:12)(cid:12)(cid:12) | x | α dx ≥ ( n + 4 α + 2) Z R n |∇ u | dx for any u ∈ H α, − − β ( R n ). Replacing u by function u λ ( x ) = λ n − u ( λx ) with λ >
0, we get λ α Z R n | ∆ u | | x | − α dx + λ − − α Z R n (cid:12)(cid:12)(cid:12)(cid:12) ∇ u · x | x | (cid:12)(cid:12)(cid:12)(cid:12) | x | α dx ≥ ( n + 4 α + 2) Z R n |∇ u | dx. The left-hand side of the preceding inequality is minimized by λ = R R n | ∆ u | | x | − α dx R R n (cid:12)(cid:12)(cid:12) ∇ u · x | x | (cid:12)(cid:12)(cid:12) | x | α dx α . Hence, by taking λ = λ , we obtain (1.4) for α = −
1. The case α = − α → − α >
0. Taking u = U implies v ≡
1. Hence, (2.5) becomes Z R n | ∆ U | | x | − α dx + Z R n (cid:12)(cid:12)(cid:12)(cid:12) ∇ U · x | x | (cid:12)(cid:12)(cid:12)(cid:12) | x | α dx = ( n + 4 α + 2) Z R n |∇ U | dx. Furthermore, by the direct computations and integration by parts, we have Z R n | ∆ U | | x | − α dx = ( n + 2 α ) Z R n | x | α U dx − n + 2 α ) Z R n | x | α U dx + Z R n | x | α U dx = ( n + 2 α ) Z R n | x | α U dx + ( n + 2 α ) Z R n ∇ U · x | x | α dx + Z R n (cid:12)(cid:12)(cid:12)(cid:12) ∇ U · x | x | (cid:12)(cid:12)(cid:12)(cid:12) | x | α dx = Z R n (cid:12)(cid:12)(cid:12)(cid:12) ∇ U · x | x | (cid:12)(cid:12)(cid:12)(cid:12) | x | α dx. This implies that the equality occurs in (1.4) with u = U . Hence, the inequality (1.4) issharp and U is an extremal function. This completes the proof of Theorem 1.1.We next prove Theorem 1.3. 11 roof of Theorem 1.3. By density argument, it is enough to prove (1.8) for radial functions u ∈ C ∞ ( R n ). Let u ∈ C ∞ ( R n ) be a radial function, by using integration by parts, we have Z R n |∇ u | t | x | tγ dx = | S n − | Z ∞ | u ′ | t r n − tγ − dr = 1 n − tγ | S n − | Z ∞ | u ′ | t ( r n − tγ ) ′ dr = − tn − tγ | S n − | Z ∞ | u ′ | t − u ′ u ′′ r n − tγ dr = − tn − tγ | S n − | Z ∞ | u ′ | t − u ′ (cid:16) u ′′ + n − r u ′ − n + 2 αr u ′ (cid:17) r n − tγ dr − t (1 + 2 α ) n − tγ | S n − | Z ∞ | u ′ | t r n − tγ − dr. This gives n + t (1 + 2 α − γ ) t Z R n |∇ u | t | x | tγ dx = −| S n − | Z ∞ | u ′ | t − u ′ (cid:16) u ′′ + n − r u ′ − n + 2 αr u ′ (cid:17) r n − γ dr = −| S n − | Z ∞ | u ′ | t − u ′ r − β (cid:16) u ′′ + n − r u ′ − n + 2 αr u ′ (cid:17) r − α r n − dr, (2.6)here we use (1.7). By density argument, (2.6) still holds for radial function u ∈ H α,β ( R n ).Using H¨older inequality, we arrive at (cid:12)(cid:12)(cid:12)(cid:12) n + t (1 + 2 α − γ ) t (cid:12)(cid:12)(cid:12)(cid:12) Z R n |∇ u | t | x | tγ dx ≤ (cid:18) | S n − | Z ∞ (cid:16) u ′′ + n − r u ′ − n + 2 αr u ′ (cid:17) r n − α − dr (cid:19) × (cid:18) | S n − | Z ∞ | u ′ | t − r n − β − dr (cid:19) = (cid:18) | S n − | Z ∞ (cid:16) u ′′ + n − r u ′ − n + 2 αr u ′ (cid:17) r n − α − dr (cid:19) (cid:18)Z R n |∇ u | t − | x | β dx (cid:19) . (2.7)12urthermore, using integration by parts, we have Z ∞ (cid:16) u ′′ + n − r u ′ − n + 2 αr u ′ (cid:17) r n − α − dr = Z ∞ (∆ u ( r )) r n − α − dr + ( n + 2 α ) Z ∞ ( u ′ ) r n − α − dr − n + 2 α ) Z ∞ (cid:16) u ′′ + n − r u ′ (cid:17) u ′ r r n − α − dr = Z ∞ (∆ u ( r )) r n − α − dr − ( n + 2 α )( n − α − Z ∞ ( u ′ ) r n − α − dr − ( n + 2 α ) Z ∞ (( u ′ ) ) ′ r n − α − dr = Z ∞ (∆ u ( r )) r n − α − dr. Consequently, it holds | S n − | Z ∞ (cid:16) u ′′ + n − r u ′ − n + 2 αr u ′ (cid:17) r n − α − dr = Z R n (∆ u ) | x | α dx. Inserting this equality into (2.7), we obtain (1.8).Suppose that a nonzero radial function u ∈ H α,β ( R n ) is an extremal function for (1.8).Notice that under the conditions (1.9) and (1.10), we have n + t (1 + 2 α − γ ) = n + 2 α + (1 + 2 α )( t −
2) + 1 + α − β > . Hence, the equation holds when applying the H¨older inequality to (2.6) if and only if u ′′ + n − r u ′ − n + 2 αr u ′ = − λ | u ′ | t − u ′ r α − β for some λ >
0, which is equivalent to u ′′ − αr u ′ + λ | u ′ | t − u ′ r α − β = 0 . Denote u ′ = r α w , then w satisfies the equation w ′ + λr (1+2 α )( t − α − β | w | t − w = 0 . We have following two cases:
Case 1: t = 2 . In this case, we have w ′ + λr α − β w = 0 which implies w ( r ) = c exp( − λr α − β / (1 + α − β )) for some c ∈ R . Hence, u ′ ( r ) = cr α exp (cid:16) − λ r α − β/ α − β/ (cid:17) u ( x ) = c Z ∞| x | r α exp (cid:16) − λ r α − β/ α − β/ (cid:17) dr. Case 2: t > . In this case, we have ( | w | − t ) ′ = λ ( t − λr (1+2 α )( t − α − β which implies | w ( r ) | = (cid:16) c + λ ( t − r (1+2 α )( t − α − β (1 + 2 α )( t −
2) + 1 + α − β (cid:17) − t , for some c >
0, here we use (1.9). From this expression, up to a multiplicative constant 1or −
1, we can assume that w ( r ) = (cid:16) c + λ ( t − r (1+2 α )( t − α − β (1 + 2 α )( t −
2) + 1 + α − β (cid:17) − t . Therefore, the extremal function has the form u ( x ) = Z ∞| x | r α (cid:16) c + λ ( t − r (1+2 α )( t − α − β (1 + 2 α )( t −
2) + 1 + α − β (cid:17) − t dr as desired.We finish this section by proving Theorem 1.4. Our proof follows the approach ofCazacu, Flynn and Lam to prove (1.3) by using the decomposition of u into sphericalharmonics. It is well known that the technique of decomposing a function into sphericalharmonics is a very useful method to prove the Hardy-Rellich type inequalities (see [12,29, 40, 46, 47] and references therein). Let us recall some facts on this method which weborrow from [46, Section 2 . f ∈ C ∞ ( R n ) can be decomposed into sphericalharmonics as f ( x ) = ∞ X k =0 f k ( r ) φ k ( ω ) , x = rω, | x | = r, ω ∈ S n − (2.8)where φ k are the orthogonal eigenfunctions of the Laplace-Beltrami operator on S n − withthe corresponding eigenvalue c k = k ( n + k − φ ≡ φ k is restriction of the k order homogeneous, harmonic polynomials in R n to S n − and f k ( r ) = | S n − | R S n − f φ k ds with k ≥
0. Hence f k ∈ C ∞ ( R n ) with f k ( r ) = O ( r k ) as r → f ( x ) = ∞ X k =0 (cid:16) f ′′ k ( r ) + n − r f ′ k ( r ) − c k f k ( r ) r (cid:17) φ k ( ω )and |∇ f ( x ) | = ∞ X k =0 (cid:16) |∇ f k | φ k + f k r |∇ S n − φ k | (cid:17) . f k ( r ) = r k g k ( r ). By the simple computations,we have f ′′ k ( r ) + n − r f ′ k ( r ) − c k f k ( r ) r = r k g ′′ k + 2 kr k − g ′ k + k ( k − r k − g k + ( n − r k − g ′ k + k ( n − r k − g k − c k r k − g k = r k g ′′ k + ( n + 2 k − r k − g ′ k and |∇ f k | = k r k − g k + r k ( g ′ k ) + 2 kr k − g k g ′ k . Therefore, using integration by parts and the definitions (1.13), (1.14) and (1.15), we get Z R n | ∆ f | | x | α dx = ∞ X k =0 Z R n (cid:16) f ′′ k ( r ) + n − r f ′ k ( r ) − c k f k ( r ) r (cid:17) | x | − α dx = ∞ X k =0 Z R n (cid:16) g ′′ k ( r ) + n + 2 k − r g ′ k ( r ) (cid:17) | x | − α +2 k dx = | S n − | ∞ X k =0 (cid:16) Z ∞ ( g ′′ k ) r n +2 k − α − dr + ( n + 2 k − Z ∞ ( g ′ k ) r n +2 k − α − dr + 2( n + 2 k − Z ∞ g k g ′ k r n +2 k − α − dr (cid:17) = | S n − | ∞ X k =0 (cid:16) Z ∞ ( g ′′ k ) r n +2 k − α − dr + (1 + 2 α )( n + 2 k − Z ∞ ( g ′ k ) r n +2 k − α − dr (cid:17) = | S n − | A n,α,k ( g k ) , (2.9) Z R n |∇ f | | x | β dx = ∞ X k =0 | S n − | Z ∞ (cid:16) ( f ′ k ) + c k f k r (cid:17) r n − β − dr = | S n − | ∞ X k =0 (cid:16) Z ∞ ( g ′ k ) r n +2 k − β − dr + 2 k Z ∞ g k g ′ k r n +2 k − β − dr + ( c k + k ) Z ∞ g k r n +2 k − β − dr = | S n − | ∞ X k =0 (cid:16) Z ∞ ( g ′ k ) r n +2 k − β − dr + βk Z ∞ g k r n +2 k − β − dr (cid:17) = | S n − | B n,β,k ( g k ) (2.10)15nd Z R n |∇ f | | x | γ dx = | S n − | ∞ X k =0 (cid:16) Z ∞ ( g ′ k ) r n +2 k − γ − dr + 2 γk Z ∞ g k r n +2 k − γ − dr (cid:17) = | S n − | C n,γ,k ( g k ) . (2.11)With these preparations, we are ready to prove Theorem 1.4 Proof of Theorem 1.4.
It follows from (2.9), (2.10) and (2.11) that (cid:16) Z R n | ∆ f | | x | α dx (cid:17)(cid:16) Z R n |∇ f | | x | β dx (cid:17) = | S n − | (cid:16) ∞ X k =0 A n,α,k ( g k ) (cid:17)(cid:16) ∞ X k =0 B n,β,k ( g k ) (cid:17) and Z R n |∇ f | | x | γ dx = | S n − | ∞ X k =0 C n,γ,k ( g k ) . By Minkowski inequality and the definition (1.16) of A k ( n, α, β ), we have (cid:16) ∞ X k =0 A n,α,k ( g k ) (cid:17)(cid:16) ∞ X k =0 B n,β,k ( g k ) (cid:17) ≥ (cid:16) ∞ X k =0 q A n,α,k ( g k ) B n,β,k ( g k ) (cid:17) ≥ inf k ∈ N A k ( n, α, β ) (cid:16) ∞ X k =0 C n,γ,k ( g k ) (cid:17) , which yields (cid:16) Z R n | ∆ f | | x | α dx (cid:17)(cid:16) Z R n |∇ f | | x | β dx (cid:17) ≥ inf k ∈ N A k ( n, α, β ) (cid:16) Z R n |∇ f | | x | γ dx (cid:17) . Furthermore, by using one dimensional Hardy inequality, we have Z ∞ ( g ′ ) r n +2 k − β − dr ≥ (cid:16) n + 2 k − β − (cid:17) Z ∞ g r n +2 k − β − dr, and Z ∞ ( g ′ ) r n +2 k − γ − dr ≥ (cid:16) n + 2 k − γ − (cid:17) Z ∞ g r n +2 k − γ − dr which imply B n,β,k ( g ) ≥ (cid:16) n , βk ( n + 2 k − β − o(cid:17) Z ∞ ( g ′ ) r n +2 k − β − dr and C n,γ,k ( g ) ≤ (cid:16) n , γk ( n + 2 k − γ − o(cid:17) Z ∞ ( g ′ ) r n +2 k − γ − dr. n , βk ( n + 2 k − β − o > A n,α,k ( g ) = Z ∞ (cid:16) g ′′ ( r ) + n + 2 k − r g ′ ( r ) (cid:17) r n +2 k − α − dr. Hence, applying the inequality (1.8) for radial functions in dimension n + 2 k we get A k ( n, α, β ) ≥ n , βk ( n +2 k − β − o n , γk ( n +2 k − γ − o (cid:16) n + 2 k + 4 α − γ + 22 (cid:17) . (2.12)This gives lim k →∞ A k ( n, α, β ) = ∞ and hence inf k A k ( n, α, β ) = min k A k ( n, α, β ) . Consequently, we get (cid:16) Z R n | ∆ f | | x | α dx (cid:17)(cid:16) Z R n |∇ f | | x | β dx (cid:17) ≥ min k A k ( n, α, β ) (cid:16) Z R n |∇ f | | x | γ dx (cid:17) as wanted (1.17).It is easy to see that the constant min k A k ( n, α, β ) is sharp in (1.17). Indeed, thereexists k and a sequence of function h k such that min k A k ( n, α, β ) = A k ( n, α, β ) andlim k →∞ A n,α,k ( h k ) B n,β,k ( h k ) C n,γ,k ( h k ) = A k ( n, α, β ) . Testing (1.17) by functions h k φ k implies the sharpness of min k A k ( n, α, β ).Finally, the estimate (1.18) immediately follows from (2.12). In this section, we prove the stability versions of the second order uncertainty principle(1.3) due to Cazacu, Flynn and Lam, that is, we give the proof of Theorems 1.5 and 1.6.We start by preparing some main ingredients in our proof.17irstly, for any function u ∈ C ∞ ( R n ) it is easy to check by using integration by partsthat Z R n k∇ u + ∇ u ⊗ x k HS dx = Z R n k∇ u k HS dx + 2 Z R n ∇ u ( ∇ u ) · xdx + Z R n |∇ u | | x | dx = Z R n | ∆ u | + Z R n ∇ ( |∇ u | ) · xdx + Z R n |∇ u | | x | dx = Z R n | ∆ u | − n Z R n |∇ u | dx + Z R n |∇ u | | x | dx. (3.1)Next, we set u = ve −| x | / . Then, v ∈ C ∞ ( R n ) and we have ∇ u = [ ∇ v − ∇ v ⊗ x − x ⊗ ∇ v − v ( I n − x ⊗ x )] e − | x | and ∇ u ⊗ x = ( ∇ v ⊗ x − x ⊗ xv ) e − | x | . Consequently, it holds ∇ u + ∇ u ⊗ x = [ ∇ v − x ⊗ ∇ v − vI n ] e − | x | which yields k∇ u + ∇ u ⊗ x k HS = k∇ v − x ⊗ ∇ v k HS e −| x | − v − ∇ v · x ) ve −| x | + nv e −| x | . (3.2)By using again integration by parts, we get Z R n (∆ v − ∇ v · x ) ve −| x | dx = − Z R n |∇ v | e −| x | dx + Z R n ∇ v · xve −| x | dx = − Z R n |∇ v | e −| x | dx + 12 Z R n ∇ v · xe −| x | dx = − Z R n |∇ v | e −| x | dx − n Z R n v e −| x | dx + Z R n v | x | e −| x | dx. Integrating both sides of (3.2) in R n and using the preceding equality, we arrive at Z R n k∇ u + ∇ u ⊗ x k HS dx = Z R n k∇ v − x ⊗ ∇ v k HS e −| x | dx + 2 Z R n |∇ v | e −| x | dx + 2 n Z R n v e −| x | dx − Z R n v | x | e −| x | dx. (3.3)18inally, by using integration by parts again, we have Z R n |∇ u | dx = Z R n |∇ v − xv | e −| x | dx = Z R n |∇ v | e −| x | dx − Z R n ∇ v · xve −| x | dx + Z R n | x | v e −| x | dx = Z R n |∇ v | e −| x | dx − Z R n ∇ v · xe −| x | dx + Z R n | x | v e −| x | dx = Z R n |∇ v | e −| x | dx + n Z R n v e −| x | dx − Z R n | x | v e −| x | dx. (3.4)It follows from (3.1), (3.3) and (3.4) that Z R n | ∆ u | dx + Z R n | x | u dx = ( n + 2) Z R n |∇ u | dx + Z R n k∇ v − x ⊗ ∇ v k HS e −| x | dx, (3.5)where we recall that u = ve −| x | / .Notice that (3.5) provides another proof of (1.3). Indeed, from (3.5) we have Z R n | ∆ u | dx + Z R n | x | u dx ≥ ( n + 2) Z R n |∇ u | dx, (3.6)for any function u ∈ C ∞ ( R n ). Applying (3.6) for function u λ ( x ) = λ n/ − u ( λx ) with λ > λ Z R n | ∆ u | dx + λ − Z R n | x | u dx ≥ ( n + 2) Z R n |∇ u | dx for any λ >
0. Choosing λ = ( R R n | ∆ u | dx/ R R n | x | u dx ) implies the result of Cazacu,Flynn and Lam.Our next task is to use (3.5) to prove the stability estimates for (1.3) given in Theorem1.5 and 1.6. We remark that Lemma 3.1.
We have Z R n k∇ v − x ⊗ ∇ v k HS e −| x | dx = Z R n k∇ v k HS e −| x | dx + n Z R n |∇ v | e −| x | dx − Z R n | x | |∇ v | e −| x | dx = n X i =1 (cid:16) Z R n |∇ ∂ i v | e −| x | dx + n Z R n | ∂ i v | e −| x | dx − Z R n | x | | ∂ i v | e −| x | dx (cid:17) . Proof.
This lemma follows from integration by parts.The integrals on the right-hand side of the expression in Lemma 3.1 concern with theGaussian type measure. This suggests us using the Hermite type polynomials to analysis19hose integrals. We continue by recalling several properties of the Hermite polynomial. Let γ n denote the standard n − dimensional Gaussian measure, i.e., dγ n ( x ) = e −| x | / (2 π ) n/ dx. Let L n denote the Ornstein-Uhlenbeck operator L n v = ∆ v − ∇ v · x . Note that Z R n L n u · vdγ n = − Z R n ∇ u · ∇ vdγ n . It is well-known that L ( γ ) has an orthogonal basis consisting of the Hermite polynomials H , H , H , . . . with H i ( t ) = ( − i e t / d i dt i ( e − t / ) , i = 0 , , , . . . . Note that H i is a polynomial of degree i . For examples, H ≡ , H ( t ) = t, H ( t ) = t − , H ( t ) = t − t, . . . . The Hermite polynomials have the following properties (see [1,formulas 22 . .
14 and 22 . . H i +1 ( t ) = tH i ( t ) − iH i − ( t ) , i ≥ , and H ′ i ( t ) = iH i − ( t ) , i ≥ . Using these two properties, we get the following result
Proposition 3.2.
There holds t H ( t ) = H ( t ) + H ( t ) , t H ( t ) = H ( t ) + 3 H ( t ) and t H i ( t ) = H i +2 ( t ) + (2 i + 1) H i ( t ) + ( i − H i − ( t ) , i ≥ . It is known that L H i = − iH i for any i ≥
0. The following fact is important in ourproof.
Lemma 3.3.
For i ≤ j , there holds Z t H i ( t ) H j ( t ) dγ ( t ) = if j = i + 1 or j > i + 22 i + 1 if i = j if j = i + 2 .Proof. This lemma follows from Proposition 3.2.20or each I ∈ Z n + , I = ( i , . . . , i n ) , i j ≥ , j = 1 , . . . , n , we denote H I ( x ) = H i ( x ) · · · H i n ( x n ) . So, { H I } I ∈ Z n + forms an orthogonal basis of L ( γ n ) and L n H I = −| I | H I , with | I | = i + i + · · · + i n . Making the change of variable x → x/ √
2, we see that the set of functions H I ( x ) = H I ( √ x ) , I ∈ Z n + forms an orthonormal basis of L ( µ n ) with dµ n ( x ) = e −| x | dx/π n/ and∆ H I − ∇H I · x = − | I |H I . The next lemma plays an important role in our analysis.
Lemma 3.4.
The following inequality Z R n |∇ w | dµ n + n Z R n w dµ n − Z R n | x | dµ n ≥ C Z R n w dµ n holds for any w ∈ C ∞ ( R n ) with C = min { , n/ } .Proof. We decompose w as w = X I ∈ Z n + a I H I . Then, we have Z R n |∇ w | dµ n = − Z R n (∆ w − ∇ w · x ) wdµ n = X I ∈ Z n + | I | a I (3.7)and Z R n | w | dµ n = X I ∈ Z n + a I . (3.8)We next compute R R n | x | | w | dx . We have w = X I a I H I + X I = J a I a J H I H J . Using Lemma 3.3 and making the change of variable x → x/ √
2, we have Z R n | x | H I dµ n = 2 | I | + n . I, J ∈ Z n + , we say that I is a neighborhood of J (denote by I ∼ J ) if and only if thereexists uniquely a 1 ≤ k ≤ n such that | j k − i k | = 2 and j l = i l for any l = k . Using Lemma3.3 and making the change of variable x → x/ √
2, we have X I = J a I a J Z R n | x | H I H J dµ n = 12 X I X J ∼ I a I a J . Using Cauchy-Schwartz inequality, we have X I X J ∼ I a I a J = X J ∼ a a J + X | I | =1 ,J ∼ I a I a J + X | I | =2 ,J ∼ I a I a J + X | I |≥ ,J ∼ I a I a J = X J ∼ a a J + X | I | =1 ,J ∼ I a I a J + X | I | =2 ,I ∼ a I a + X | I | =2 ,J ∼ I,J =0 a I a J + X | I |≥ ,J ∼ I a I a J ≤ X J ∼ (cid:16) a + a J (cid:17) + X | I | =1 ,J ∼ I (cid:16) a I a J (cid:17) + X | I | =2 ,I ∼ ( a I + 14 a )+ X | I | =2 ,J ∼ I,J =0 (cid:16) a I a J (cid:17) + X | I |≥ ,J ∼ I (cid:16) a I a J (cid:17) . For each I ∈ Z n + we denote by N I the number of J which is neighborhood of I . We remarkthat the number of a I which appears in the sum above is equal to twice number of J whichis a neighborhood of I (and hence is equal to 2 N I ). Evidently, we have N = n , N I = n if | I | = 1, and if | I | ≥ I has k coordinates which are greater or equal to 2 then N I = n + k . Hence, we have X I X J ∼ I a I a J ≤ n a + n X | I | =1 a I + ( n + 2) X | I | =2 a I + X | I |≥ N I a I . (3.9)Combining (3.7), (3.8) and (3.9), we get Z R n |∇ w | dµ n + n Z R n w dµ n − Z R n | x | dµ n ≥ n a + X | I | =1 a I + X | I | =2 a I + X | I |≥ (cid:16) | I | + n − N I (cid:17) a I . (3.10)For | I | ≥
3, if all coordinates of I are smaller than 2 then N I = n and hence | I | + n − N I ≥ I has k coordinates which are at least 2 then N I = n + k and | I | ≥ k , hence | I | + n − N I ≥ k ≥ . So, in any case, we have | I | + n − N I ≥ if | I | ≥
3. This togetherwith (3.10) implies Z R n |∇ w | dµ n + n Z R n w dµ n − Z R n | x | dµ n ≥ n a + X | I | =1 a I + X | I | =2 a I + 32 X | I |≥ a I ≥ min { , n/ } X I a I = min { , n/ } Z R n w dµ n . ∂ i v and using Lemma 3.1, we obtain Z R n k∇ v − x ⊗ ∇ v k HS dµ n ≥ C Z R n |∇ v | dµ n . (3.11)To proceed, we need the following result. Lemma 3.5.
It holds inf c ∈ R Z R n |∇ [( v − c ) e −| x | / ] | dx ≤ n + 22 Z R n |∇ v | e −| x | dx. Proof.
We begin by recall the Gaussian Poincar´e inequality (see [3, Introduction]) Z R n |∇ w | dγ n ≥ Z R n (cid:12)(cid:12)(cid:12) w − Z R n wdγ n (cid:12)(cid:12)(cid:12) dγ n = inf c ∈ R Z R n | w − c | dγ n . Making the change of function w ( x ) = w ( √ x ), we obtain the following Poincar´e inequalityfor µ n , Z R n |∇ w | dµ n ≥ Z R n (cid:12)(cid:12)(cid:12) w − Z R n wdµ n (cid:12)(cid:12)(cid:12) dµ n = 2 inf c ∈ R Z R n | w − c | dµ n . (3.12)Using integration by parts, we have Z R n |∇ [( v − c ) e −| x | / ] | dx = Z R n |∇ v | e −| x | dx − Z R n ∇ ( v − c ) · x ( v − c ) e −| x | dx + Z R n | x | ( v − c ) e −| x | dx = Z R n |∇ v | e −| x | dx + n Z R n | v − c | dx − Z R n | x | | v − c | dx ≤ Z R n |∇ v | e −| x | dx + n Z R n | v − c | e −| x | dx. Applying the Poincar´e inequality (3.12) for function v , we getinf c ∈ R Z R n |∇ [( v − c ) e −| x | / ] | dx ≤ Z R n |∇ v | e −| x | dx + nπ n/ inf c ∈ R Z R n | v − c | dµ n ≤ n + 22 Z R n |∇ v | e −| x | dx. With Lemma 3.5 and the estimate (3.11) at hand, we are ready to prove Theorem 1.5.23 roof of Theorem 1.5.
By the density argument, it is enough to prove Theorem 1.5 forfunction u ∈ C ∞ ( R n ). Moreover, since both sides of the inequality (1.19) are invariantunder the dilation and multiplication by a nonzero constant, then without loss of generalitywe can assume that R R n |∇ u | dx = 1 and Z R n | ∆ u | dx = Z R n | x | |∇ u | dx. Set u = ve −| x | / , we have by (3.11) and Lemma 3.5 δ ( u ) = R R n | ∆ u | dx + R R n | x | |∇ u | dxn + 2 −
1= 1 n + 2 Z R n k∇ v − x ⊗ ∇ v k HS e −| x | dx ≥ Cn + 2 Z R n |∇ v | e −| x | dx ≥ C ( n + 2) inf c ∈ R Z R n |∇ [( v − c ) e −| x | / ] | dx ≥ C ( n + 2) inf v ∈ E Z R n |∇ u − ∇ v | dx. In the next, we divide the proof into two cases according to the range of δ ( u ). Case 1:
We suppose that δ ( u ) ≤ C n +2) . From the previous estimate, we getinf v ∈ E Z R n |∇ u − ∇ v | dx ≤ . We show that the infimum above is attained by a function v ∈ E . Indeed, let v i = c i e − λ i | x | / be a sequence in E such thatlim i →∞ Z R n |∇ u − ∇ v i | dx = inf v ∈ E Z R n |∇ u − ∇ v | dx ≤ . Using triangle inequality, we have (cid:16) Z R n |∇ u − ∇ v i | dx (cid:17) ≥ (cid:12)(cid:12)(cid:12)(cid:16) Z R n |∇ v i | dx (cid:17) − (cid:12)(cid:12)(cid:12) . Thus for i large enough, we have 14 ≤ Z R n |∇ v i | dx ≤ . By a simple computation, we have Z R n |∇ v i | dx = c i λ − n +1 i Z R n | x | e −| x | dx.
24o there are a, A > a ≤ c i λ − n +1 i ≤ A . We note that Z R n |∇ u −∇ v i | dx = 1+ c i λ − n +1 i Z R n | x | e −| x | dx +2 c i λ i Z R n ∇ u · ( λ i x ) e −| λ i x | / dx. (3.13)We claim that λ i is bounded from above and below by positive constants. Indeed, supposethat lim i →∞ λ i = ∞ . For any ǫ >
0, there exists
R > R B R |∇ u | dx < ǫ . Hence, itholds (cid:12)(cid:12)(cid:12) Z R n ∇ u · ( λ i x ) e −| λ i x | / dx (cid:12)(cid:12)(cid:12) ≤ Z B R |∇ u || λ i x | e −| λ i x | / dx + Z B cR |∇ u | | λ i x | e −| λ i x | / dx ≤ (cid:16) Z B cR |∇ u | dx (cid:17) (cid:16) Z B cR | λ i x | e −| λ i x | dx (cid:17) + ǫ (cid:16) Z B R | λ i x | e −| λ i x | dx (cid:17) ≤ λ − n i (cid:16) Z B cR √ λi | x | e −| x | dx (cid:17) + λ − n i ǫ (cid:16) Z B R √ λi | x | e −| x | dx (cid:17) which implies (cid:12)(cid:12)(cid:12) c i λ i Z R n ∇ u · ( λ i x ) e −| λ i x | / dx (cid:12)(cid:12)(cid:12) ≤ A (cid:16) Z B cR √ λi | x | e −| x | dx (cid:17) + Aǫ (cid:16) Z B R √ λi | x | e −| x | dx (cid:17) . Let i → ∞ , we obtainlim sup i →∞ (cid:12)(cid:12)(cid:12) c i λ i Z R n ∇ u · ( λ i x ) e −| λ i x | / dx (cid:12)(cid:12)(cid:12) ≤ A (cid:16) Z R n | x | e −| x | dx (cid:17) ǫ. Since ǫ > i →∞ (cid:12)(cid:12)(cid:12) c i λ i Z R n ∇ u · ( λ i x ) e −| λ i x | / dx (cid:12)(cid:12)(cid:12) = 0 . This together with (3.13) implies18 ≥ lim i →∞ Z R n |∇ u − ∇ v i | dx > λ i , suppose thatlim i →∞ λ i = 0 . ǫ >
0, there exists
R > R B cR |∇ u | dx < ǫ . Hence, it holds (cid:12)(cid:12)(cid:12) Z R n ∇ u · ( λ i x ) e −| λ i x | / dx (cid:12)(cid:12)(cid:12) ≤ Z B R |∇ u || λ i x | e −| λ i x | / dx + Z B cR |∇ u | | λ i x | e −| λ i x | / dx ≤ (cid:16) Z B R |∇ u | dx (cid:17) (cid:16) Z B R | λ i x | e −| λ i x | dx (cid:17) + ǫ (cid:16) Z B cR | λ i x | e −| λ i x | dx (cid:17) ≤ λ − n i (cid:16) Z B R √ λi | x | e −| x | dx (cid:17) + λ − n i ǫ (cid:16) Z B cR √ λi | x | e −| x | dx (cid:17) which implies (cid:12)(cid:12)(cid:12) c i λ i Z R n ∇ u · ( λ i x ) e −| λ i x | / dx (cid:12)(cid:12)(cid:12) ≤ A (cid:16) Z B R √ λi | x | e −| x | dx (cid:17) + Aǫ (cid:16) Z B cR √ λi | x | e −| x | dx (cid:17) . Let i → ∞ , we obtainlim sup i →∞ (cid:12)(cid:12)(cid:12) c i λ i Z R n ∇ u · ( λ i x ) e −| λ i x | / dx (cid:12)(cid:12)(cid:12) ≤ A (cid:16) Z R n | x | e −| x | dx (cid:17) ǫ. Since ǫ > i →∞ (cid:12)(cid:12)(cid:12) c i λ i Z R n ∇ u · ( λ i x ) e −| λ i x | / dx (cid:12)(cid:12)(cid:12) = 0 . This together with (3.13) implies18 ≥ lim i →∞ Z R n |∇ u − ∇ v i | dx > a , a , a > | c i | ≤ a , < a ≤ λ i ≤ a for any i . Extracting a subsequence, we can assume that c i → c and λ i → λ ∈ [ a , a ].Denote v = ce − λ | x | / ∈ E . We then have ∇ v i → ∇ v in L and hence Z R n |∇ u − ∇ v | dx = lim i →∞ Z R n |∇ u − ∇ v i | dx = inf v ∈ E Z R n |∇ u − ∇ v | dx. Note that Z R n |∇ u | dx = 1 , Z R n |∇ u − ∇ v | dx = inf v ∈ E Z R n |∇ u − ∇ v | dx ≤ v
0. So we can choose a positive constant a such that Z R n |∇ ( av ) | dx = Z R n |∇ u | dx = 1or, equivalently a = ( R R n |∇ v | dx ) − . By triangle inequality, we have (cid:12)(cid:12)(cid:12)(cid:16) Z R n |∇ v | dx (cid:17) − (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:16) Z R n |∇ v | dx (cid:17) − (cid:16) Z R n |∇ u | dx (cid:17) (cid:12)(cid:12)(cid:12) ≤ (cid:16) Z R n |∇ u − ∇ v | dx (cid:17) ≤ r ( n + 2) C δ ( u ) . Using this estimate and Cauchy-Schwartz inequality, we have Z R n |∇ u − ∇ ( av ) | dx = Z R n |∇ u − ∇ v + (1 − a ) ∇ v | dx ≤ Z R n |∇ u − ∇ v | dx + 2( a − Z R n |∇ v | dx ≤ ( n + 2) C δ ( u ) + 2 (cid:12)(cid:12)(cid:12)(cid:16) Z R n |∇ v | dx (cid:17) − (cid:12)(cid:12)(cid:12) ≤ n + 2) C δ ( u ) . This gives δ ( u ) ≥ C n + 2) inf n R R n |∇ u − ∇ v | dx R R n |∇ u | dx : v ∈ E, Z R n |∇ u | dx = Z R n |∇ v | dx o , (3.14)when δ ( u ) ≤ C n +2) . Case 2:
Suppose that δ ( u ) > C n +2) . Since we always haveinf n R R n |∇ u − ∇ v | dx R R n |∇ u | dx : v ∈ E, Z R n |∇ u | dx = Z R n |∇ v | dx o ≤ δ ( u ) ≥ C n + 2) inf n R R n |∇ u − ∇ v | dx R R n |∇ u | dx : v ∈ E, Z R n |∇ u | dx = Z R n |∇ v | dx o . (3.15)The inequality (1.19) follows immediately from (3.14) and (3.15).We next prove Theorem 1.6. The proof follows the same way in the proof of Theorem1.5. Hence, we will only outline the different points.27 roof of Theorem 1.6. As in the proof of Theorem 1.5, we can assume that u ∈ C ∞ ( R n ), R R n |∇ u | dx = 1 and Z R n | ∆ u | dx = Z R n | x | |∇ u | dx. Denote u = ve −| x | / , we get δ ( u ) ≥ Cn + 2 Z R n |∇ v | e −| x | dx, (3.16)or equivalently R R n |∇ v | e −| x | dx ≤ n +2 C δ ( u ). We have1 = Z R n |∇ u | dx = Z R n |∇ v | e −| x | dx + n Z R n v e −| x | dx − Z R n | x | v e −| x | dx = Z R n |∇ v | e −| x | dx + n Z R n u dx − Z R n | x | u dx. The Heisenberg-Pauli-Weyl uncertainty principle (1.1) implies1 ≤ n + 2 C δ ( u ) + n Z R n u dx − n (cid:16) Z R n | u | dx (cid:17) . We also divide the rest of the proof into two cases.
Case 1:
We suppose that δ ( u ) ≤ C n ( n +2) . The estimate above gives2 n (cid:16) − r n + 2 C δ ( u ) (cid:17) ≤ Z R n u dx ≤ n (cid:16) n + 2 C δ ( u ) (cid:17) . Using the Poincar´e inequality (3.12) and the estimate (3.16), we arrive at δ ( u ) ≥ Cn + 2 Z R n (cid:12)(cid:12)(cid:12) v − Z R n vdµ n (cid:12)(cid:12)(cid:12) e −| x | dx = 2 Cn + 2 Z R n (cid:12)(cid:12)(cid:12) u − (cid:16) Z R n vdµ n (cid:17) e −| x | / (cid:12)(cid:12)(cid:12) dx ≥ Cn + 2 inf v ∈ E Z R n | u − v | dx ≥ Cn + 2 Z R n | u | dx inf v ∈ E R R n | u − v | dx R R n | u | dx ≥ Cn ( n + 2) (cid:16) − r n + 2 C δ ( u ) (cid:17) inf v ∈ E R R n | u − v | dx R R n | u | dx ≥ Cn ( n + 2) inf v ∈ E R R n | u − v | dx R R n | u | dx , by the choice of δ ( u ). This impliesinf v ∈ E R R n | u − v | dx R R n | u | dx ≤ n ( n + 2)2 C δ ( u ) ≤ .
28y repeating the proof of Theorem 1.5, the infimum above is attained by a function v ∈ E, v
0. Furthermore, mimicking the last argument in the proof of Theorem 1.5, we have δ ( u ) ≥ C n ( n + 2) inf n R R n | u − v | dx R R n | u | dx : v ∈ E, Z R n | u | dx = Z R n | v | dx o (3.17)when δ ( u ) ≤ C n ( n +2) . Case 2: If δ ( u ) > C n ( n +2) then obviously we have δ ( u ) ≥ C n ( n + 2) inf n R R n | u − v | dx R R n | u | dx : v ∈ E, Z R n | u | dx = Z R n | v | dx o . (3.18)The inequality (1.20) follows directly from (3.17) and (3.18). Acknowledgments
This work was initiated and done when the second author visit Vietnam Institute forAdvanced Study in Mathematics (VIASM) in 2020. He would like to thank the institutefor hospitality and support during the visit.
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