Communications in Applied Analysis 14 (2010), no. 2, SHARP WEIGHTED RELLICH AND UNCERTAINTY PRINCIPLE INEQUALITIES ON CARNOT GROUPS

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1 Communications in Alied Analysis 1 (010), no., 51-7 SHARP WEIHTED RELLICH AD UCERTAITY PRICIPLE IEQUALITIES O CAROT ROUPS ISMAIL KOMBE 1 Deartment of Mathematics, Oklahoma City University 501 orth Blackwelder Oklahoma City, OK , USA ikombe@okcu.edu ABSTRACT. In this work we rove shar weighted Rellich-tye inequalities and their imroved versions for general Carnot grous. To derive the imroved Rellich-tye inequalities we have established new weighted Hardy-tye inequalities with remainder terms. We also rove new shar forms of the weighted Hardy-Poincaré and uncertainty rincile inequalities for olarizable Carnot grous. AMS (MOS) E30, 3A80, 6D ITRODUCTIO The classical Rellich inequality [6] states that for n 5, for all φ C 0 (R n ), Rn φ(x) n (n ) Rn φ(x) 16 x. (1.1) It is well-known that the constant n (n ) in inequality (1.1) is shar. In a recent aer, Tertikas and Zograhooulos [8] obtained the following Rellich-tye 16 inequality that connects first to second-order derivatives: Rn Rn φ(x) n φ(x), (1.) x where φ C0 (R n ), n 5 and the constant n is shar. There has been considerable amount work on the Rellich-tye inequalities in Euclidean saces and Riemannian manifolds, e.g., [1], [5], [8], [], [0] and the references therein. However, Rellichtye inequalities have not been established for general Carnot grous. Our main contribution in this direction is to find shar weighted Rellich-tye inequalities and their imroved versions for general Carnot grous. The Rellich inequality (1.1) is the first generalization of Hardy s inequality Rn φ(x) (n ) Rn φ(x) x (1.3) Received August 31, $15.00 c Dynamic Publishers, Inc.

2 ISMAIL KOMBE to the higher order derivatives and they are intimately related. For examle, the Rellich inequality (1.1) is an easy consequence of (1.) and the weighted Hardy inequality: Rn φ(x) x (n ) Rn φ(x) where φ C 0 (R n ), n 5 and the constant (n ) is shar. x, (1.) It is well-known that Hardy and Rellich inequalities as well as their imroved versions lay imortant roles in many questions from sectral theory, harmonic analysis and analysis of linear and nonlinear artial differential equations. A striking examle where the shar Hardy inequality (1.3) lays a major role is the following linear heat equation: u u t x in R n (0, T ), u(x, 0) = u 0 (x) 0 in R n. (1.5) In their classical aer Baras and oldstein [] roved that the initial value roblem (1.5) has no nonnegative solutions excet u 0 if c > C (n) = ( n ). Moreover, all ositive solutions blow u instantaneously in the sense that if u n is the solution of the same roblem with the otential c/ x relaced by V n = min{c/ x, n}, then lim n u n (x, t) = for all x R n and t > 0. If c C (n) = ( n ), ositive weak solutions do exist. ote that the above inequalities are strict unless φ is identically equal to 0. Therefore it is natural to exect some extra term might be added on the right hand side of the inequalities (1.1), (1.), (1.3) and (1.). A remarkable result in this direction has been obtained by Brezis and Vázquez [8]. They have discovered the following shar imroved Hardy inequalities for a bounded domain R n ( n ) φ(x) φ(x) + µ ( ω n ) /n φ, (1.6) x ( n ) φ(x) φ(x) ( + C φ q x ) q, (1.7) where φ H 1 0(), C = C(, n) > 0, ω n and denote the n-dimensional Lebesgue measure of the unit ball B R n and the domain resectively. Here µ is the first eigenvalue of the Lalace oerator in the two dimensional unit disk, and it is otimal when is a ball centered at the origin. In (1.7) we assume that q < n n and the critical Sobolev exonent q = = n is not included. The work of Brezis and n Vazquez [8] has been a continuous source of insiration and a lot of rogress has been made to find further imrovement of the inequalities (1.1), (1.), (1.3), (1.), (1.6) and (1.7) in the various settings e.g., [30], [], [6], [31], [1], [5], [19], [8], [], [0], [3] and the references therein.

3 SHARP RELLICH AD UCERTAITY PRICIPLEIEQUALITIES 3 The connection between weighted Hardy and Rellich inequalities, and the imortance of Hardy s inequality in analysis and artial differential equations motivates us to establish weighted Hardy-tye inequalities and their imroved versions on Carnot grous. Indeed, in our earlier aer, we found some shar weighted Hardy-tye inequalities and their imroved versions for L -norms of gradients on Carnot grous [3]. In the resent aer we first rove a new (non standard) form of weighted L -Hardy-tye inequality with a shar constant and then derive new weighted L - Hardy-tye inequalities with remainder terms for bounded domains in Carnot grous. We stress here that weighted Hardy-tye inequalities and their imroved versions are the main tools for establishing weighted Rellich-tye inequalities and their imroved versions, resectively. We should mention that Hardy-tye inequalities have been the target of investigation in Carnot-Carathéodory saces since the work of arofalo and Lanconelli [18], and there has been a continuously growing literature in this direction. We refer to the recent aers by Danielli, arofalo and Phuc [13], and oldstein and Kombe [1], and the monograh by Caogna et al. [11] and the references therein. It is known that Hardy and Sobolev inequalities are closely related to the Heisenberg uncertainty rincile in quantum mechanics. The Heisenberg uncertainty rincile says that the osition and momentum of a article cannot be determined exactly at the same time but only with an uncertainty. More recisely, the uncertainty rincile on the Euclidean sace R n can be stated in the following way: ( )( ) ( ) x f(x) f(x) n f(x) (1.8) R n R n R n for all f L (R n ). It is well known that equality is attained in the above if and only if f is a aussian function (i.e. f(x) = Ae α x for some A R, α > 0). There exists large literature devoted to deriving various uncertainty rincile tye inequalities in the Euclidean and other settings (see [16], [3] and the references therein). However, much less is known about shar uncertainty rincile inequalities on Carnot grous. In [3] we obtained the following uncertainty rincile-tye inequality: ( )( ) ( Q ) (, φ φ φ ) (1.9) where = u 1/( Q) is the homogeneous norm associated to Folland s fundamental solution u for the sub-lalacian and Q is the homogeneous dimension of. It is clear that this inequality is not shar. In this aer, motivated by a result of Balogh and Tyson [3], we rove a shar analog of the uncertainty rincile inequality (1.8) for olarizable Carnot grous. In order to state and rove our theorems, we first recall the basic roerties of Carnot grou and some well-known results that will be used in the sequel. Further information can be found in [3], [7], [10], [1], [15], [17], [5], [7], [9].

4 ISMAIL KOMBE. PRELIMIARIES A Carnot grou is a connected, simly connected, nilotent Lie grou whose Lie algebra admits a stratification. That is, there exist linear subsaces V 1,..., V k of such that = V 1... V k, [V 1, V i ] = V i+1, for i = 1,,..., k 1 and [V 1, V k ] = 0 (.1) where [V 1, V i ] is the subsace of generated by the elements [X, Y ] with X V 1 and Y V i. This defines a k-ste Carnot grou and integer k 1 is called the ste of. Via the exonential ma, it is ossible to induce on a family of automorhisms of the grou, called dilations, δ λ : R n R n (λ > 0) such that δ λ (x 1,..., x n ) = (λ α 1 x 1,..., λ αn x n ) where 1 = α 1 =... = α m < α m+1... α n are integers and m = dim(v 1 ). The grou law can be written in the following form x y = x + y + P (x, y), x, y R n (.) where P : R n R n R n has olynomial comonents and P 1 =... = P m = 0. ote that the inverse x 1 of an element x has the form x 1 = x = ( x 1,..., x n ). Let X 1,..., X m be a family of left invariant vector fields which form an orthonormal basis of V 1 R m at the origin, that is, X 1 (0) = x1,..., X m (0) = xm. The vector fields X j have olynomial coefficients and can be assumed to be of the form n X j (x) = j + a ij (x) i, X j (0) = j, j = 1,..., m, i=j+1 where each olynomial a ij is homogeneous with resect to the dilations of the grou, that is a ij (δ λ (x)) = λ α i α j a ij (x). The horizontal gradient on Carnot grou is the vector valued oerator = (X 1,..., X m ) where X 1,..., X m are the generators of. The sub-lalacian is the second-order artial differential oerator on given by m = Xj. j=1 The fundamental solution u for is defined to be a weak solution to the equation u = δ (.3) where δ denotes the Dirac distribution with singularity at the neutral element 0 of. In [15], Folland roved that in any Carnot grou, there exists a homogeneous norm such that u = Q

5 SHARP RELLICH AD UCERTAITY PRICIPLEIEQUALITIES 5 is harmonic in \{0}. Furthermore there exists a constant c > 0 so that c u satisfies (.3) in the sense of distributions. The number Q, called the homogeneous dimension of, is defined by k Q = j(dimv j ) j=1 and lays an imortant role in in the analysis of Carnot grous. We now set u 1 Q if x 0, (x) := (.) 0 if x = 0. We recall that a homogeneous norm on is a continuous function : [0, ) smooth away from the origin which satisfies the conditions : (δ λ (x)) = λ(x), (x 1 ) = (x) and (x) = 0 iff x = 0. A Carnot grou is said to be olarizable if the homogeneous norm = u 1/( Q) satisfies the following - sub-lalace equation,, := 1 ( ), = 0, in \ {0}. (.5) This class of grous were introduced by Balogh and Tyson [3] and admit the analogue of olar coordinates. It is known that Euclidean sace, the Heisenberg grou and the Kalan s H-tye grou [] are olarizable Carnot grous (see [1], [3]). In [3], Balogh and Tyson roved that the homogeneous norm = u 1/( Q), associated to Folland s solution u for the sub-lalacian, enters also in the exression of the fundamental solution of the sub-ellitic -Lalacian: m, u = X i ( Xu X i u), 1 < <, (.6) i=1 on olarizable Carnot grous. More recisely, they roved that for every 1 < <, = Q Q 1, if Q, u = log, if = Q. (.7) is -harmonic in \ {0}. Furthermore there exists a constant c so that c u satisfies, (c u ) = δ in the sense of distributions. In the setting of H-tye grous, exlicit formulas for the fundamental solutions of the sub-ellitic -Lalacian has been found by Caogna, Danielli and arofalo [10]. The following formula: ( ) = Q in \ Z (.8)

6 6 ISMAIL KOMBE was roved by Balogh and Tyson [3]. Here Z := {0} {x \ {0} : (x) = 0} has Haar measure zero and 0 for a.e. x. The curve γ : [a, b] R is called horizontal if its tangents lie in V 1, i.e, γ (t) san{x 1,..., X m } for all t. Then, the Carnot-Carathéodory distance d CC (x, y) between two oints x, y is defined to be the infimum of all horizontal lengths b a γ (t), γ (t) 1/ dt over all horizontal curves γ : [a, b] such that γ(a) = x and γ(b) = y. otice that d cc is a homogeneous distance and satisfies the invariance roerty d cc (z x, z y) = d cc (x, y), for all x, y, z, and is homogeneous of degree one with resect to the dilation δ λ, i.e. d cc (δ λ (x), δ λ (y)) = λd cc (x, y), for all x, y, z, for all λ > 0. The Carnot-Carathéodory balls are defined by B(y, R) = {x d cc (y, x) < R}. By left-translation and dilation, it is easy to see that the Haar measure of B(y, R) is roortional by R Q. More recisely B(y, R) = R Q B(y, 1) = R Q B(0, 1). We now set B ϱ := B(0, R) = {x : ϱ(x) < R} where ϱ := d cc (0, x) is the Carnot-Carathéodory distance of x from the origin. ote that ϱ is a homogeneous norm and equivalent to other homogeneous norm on. At this oint we remark that is uniformly bounded and : (, d cc ) R is Lischitz (see [3]). We now recall the following integration formula in olar coordinates on f(x) = f(δ λ u)λ Q 1 dσ(u)dλ 0 S which is valid for all f L 1 (). Here S = { = 1} is the unit shere with resect to the homogeneous norm and dσ is a Radon measure on S (see [17], [3], [5], [7]). ow it is clear the radial function ρ α ( ϱ is any homogeneous norm on ) is locally integrable if α > Q. 3. SHARP WEIHTED HARDY TYPE IEQUALITIES In this section we rove various weighted Hardy-tye inequalities and their imroved versions. We begin this section by roving a new form of the weighted Hardy- Poincaré-tye inequality with a shar constant.

7 SHARP RELLICH AD UCERTAITY PRICIPLEIEQUALITIES 7 Theorem 3.1. Let be a olarizable Carnot grou with homogeneous dimension Q 3 and let φ C 0 (), 1 < < Q and α > Q. Then the following inequality is valid : α+ φ Furthermore, the constant ( Q+α ) is shar. ( Q + α ) α φ. (3.1) Proof. Using the volume growth condition formula (.8) and integration by arts, we get (Q + α) α φ φ φ α+1 = φ. An alication of Hölder s and Young s inequality yields (Q + α) ( α φ for any ɛ > 0. Therefore ) ( 1)/ ( α φ ( 1)ɛ /( 1) α φ + ɛ α+ φ ) 1/ α+ φ α+ φ ) ɛ (Q + α ( 1)ɛ /( 1) α φ. (3.) ( ) ote that the function ɛ ɛ Q + α ( 1)ɛ /( 1) attains the maximum for ( ) ɛ /( 1) =, and this maximum is equal to Q+α. Q+α ow we obtain the desired inequality α+ φ ( Q + α ) α φ. ext we claim that ( ) Q+α is the best constant in (3.1): It is clear that C H : = inf 0 φ C 0 () ( Q + α ). = α+ φ α φ, (Q + α) α+ φ (3.3) α φ holds for all φ C0 (). If we ass to the inf in (3.3) we get that ( ) Q+α CH. We only need to show that C H ( ) Q+α and for this we use the following family of radial functions

8 8 ISMAIL KOMBE where ɛ > 0. comact suort in. Q+α +ɛ if [0, 1], φ ɛ () = Q+α ( +ɛ) if > 1, (3.) otice that φ ɛ () can be aroximated by smooth functions with A direct comutation shows that α+ φ ɛ ( Q+α + ɛ ) Q+α+ɛ if [0, 1], = ( Q+α + ɛ ) Q ɛ if > 1. Let us denote by B 1 = {x : (x) 1} the unit ball with resect to the homogeneous norm. Hence α φ ɛ = Q+α+ɛ + B 1 Q ɛ. \B 1 ote that, for every ɛ > 0, the weights Q+α+ɛ and Q ɛ are integrable at 0 and, resectively. This imlies that α φ ɛ is finite. Thus we have (Q + α + ɛ ) α φ ɛ = ( Q + α + ɛ ) [ ] Q+α+ɛ + Q ɛ B 1 \B 1 = α+ φ ɛ. On the other hand ( Q+α C H + ɛ ) α+ φ ɛ ( Q + α = + ɛ ) α φ ɛ α+ φ ɛ. It is clear that ( Q+α + ɛ ) CH and letting ɛ 0 we obtain ( ) Q+α CH. Therefore C H = ( ) Q+α. The following L -Hardy-tye inequality is the weighted extension of Theorem 3.1 in [1] and lays imortant roles in the roof of Theorem 3.6 and Section. Theorem 3.. Let be a olarizable Carnot grou with homogeneous dimension Q 3 and let φ C 0 (), α R, 1 < < Q and Q + α > 0. Then the following inequality is valid : ( Q + α ) α φ Furthermore, the constant ( Q+α ) is shar. α φ. (3.5)

9 SHARP RELLICH AD UCERTAITY PRICIPLEIEQUALITIES 9 Proof. Let φ C 0 () and define ψ = γ φ where γ < 0. A direct calculation shows that We now use the following convexity inequality φ = γ γ 1 ψ + γ ψ. (3.6) a + b a c() b + a a b, (3.7) where a, b R n, and c() > 0 (see [6]). In view of (3.7) we have that α φ γ α+γ ψ + γ γ α+γ +1 ( ψ ) + c() α+γ ψ. Clearly α φ γ α+γ ψ + γ γ α+γ +1 ( ψ ), and integration by arts gives α φ γ α+γ ψ γ γ ( α+γ +1 ) ψ. We now choose γ = Q α ; then we get ( α+γ +1 ) ψ = ( 1 Q ) ψ. Since u is the fundamental solution of sub--lalacian,, we then have ( 1 Q ) ψ = c(, ) φ(0) (Q+α ) (0) = 0 (3.8) (3.9) where c(, ) is a ositive constant (see [3]). Hence we obtain the desired inequality ( Q + α ) α φ α φ. (3.10) ( ) To show that the constant is shar, we use the following family of radial functions Q+α φ ɛ () = Q+α +ɛ Q+α ( +ɛ) if [0, 1], if > 1,

10 10 ISMAIL KOMBE and ass to the limit as ɛ 0. ote that the Theorem (3.) also holds for 1 < < and in this case we use the following inequality: a + b a b c() ( a + b ) + a a b (3.11) where a R n, b R n and c() > 0 (see [6]). Remark 3.3. We remark that if = then we remove the olarizability condition and the inequality (3.5) holds in any Carnot grou (see [3]). IMPROVED HARDY TYPE IEQUALITIES. We now rove imroved weighted Hardy-tye inequalities and also collect other known imroved weighted Hardy-tye inequalities that will be used in Section. To motivate our discussion, let us recall the the following shar imroved Hardy inequality from the Euclidean setting: ( n + α ) x α φ x α φ x + 1 x α φ, (3.1) x (ln R ) x where is abounded domain with smooth boundary, 0, φ C 0 (), n 1, α R, R e su x and n + α > 0. Furthermore, the constant 1 is shar and this inequality has immediate alications in artial differential equations (see [], [31], [1] [19]). Motivated by the above results our first goal is to obtain the analog of (3.1) for bounded domains in Carnot grous. Theorem 3.. Let be a Carnot grou with homogeneous norm = u 1/( Q) and let be a bounded domain with smooth boundary, 0, R e su, α R, Q 3, Q + α > 0. Then the following inequality holds: ( Q + α ) α φ α φ + 1 for all comactly suorted smooth function φ C 0 (). α φ (ln R ) (3.13) Proof. Let φ C 0 () and define ψ = β φ where β < 0. A direct calculation shows that φ = β β ψ + β β 1 ψ ψ + β ψ. (3.1) Multilying both sides of (3.1) by the α and alying integration by arts over gives α φ = β + α+β ψ α+β ψ. β ( α+β )ψ α + β (3.15)

11 SHARP RELLICH AD UCERTAITY PRICIPLEIEQUALITIES 11 We can easily show that β α + β ( α+β ) = β(α + β + Q ) α+β β Q α+β+q u. (3.16) Substituting (3.16) into (3.15) and using the fact that ψ = β φ yield α φ = ( β β(α + Q ) α φ β ( u) α+q φ Q + α+β ψ. The middle integral vanishes since u is the fundamental solution of sub-lalacian, therefore we have α φ = ( β β(α + Q ) ) α φ + α+β ψ. ote that the quadratic function β β(α+q ) attains maximum for β = Q α and this maximum equal to ( Q+α ). Therefore ( Q + α ) α φ = α φ + Q ψ. (3.17) Let us define ϕ(x) = (ln R ) 1 ψ(x) where is the homogeneous norm which is defined as in (.). A direct calculation shows that Q ψ = 1 Q (ln( R ) 1 ϕ + 1 ( Q )ϕ. ( Q) It is clear that the last integral term vanishes. Therefore we have Q ψ 1 Q (ln( R ) 1 ϕ = 1 Q ψ (ln R ) = 1 α φ. (ln R ) Q ln( R ) ϕ Substituting (3.18) into (3.17) which yields the desired inequality (3.13). (3.18) One of the advantages of our aroach is that it automatically yields a remainder term and then using a suitable functional change lead us to obtain an exlicit remainder term as in the Theorem 3.3. On the other hand, there are other techniques that we can use to obtain exlicit remainder term. In our earlier aer [3] we have used weighted Sobolev-Poincare inequalities and obtained the following imroved weighted Hardy-tye inequalities.

12 1 ISMAIL KOMBE Theorem 3.5. ([3]) Let be a Carnot grou with homogeneous norm = u 1/( Q), and let φ C 0 (B ϱ ), α R, Q 3 and Q + α > 0. Then the following inequality is valid: ( Q + α α φ B ϱ ) Bϱ α φ + 1 α φ, (3.19) C R B ϱ where C is a ositive constant and R is the radius of the ball B ϱ. Theorem 3.6. ([3]) Let be a Carnot grou with homogeneous norm = u 1/( Q) and let φ C 0 (B ϱ ), α R, Q 3, Q + α > 0 and q >. Then the following inequality is valid: ( Q + α α φ B ϱ ) Bϱ α φ + K C R ( ) /q, σ φ q B ϱ (3.0) where C > 0, R is the radius of the ball B ϱ, σ = ( Q)( q)+αq and K = ( B ϱ Q otice that the remainder terms in Theorems 3. and 3.5 contain functions of the homogeneous norm and φ. Motivated by the recent work of Abdellaoui, Colorado and Peral [1] we have following inequality (which is a weighted version of the inequality (3.) in [3]) so that remainder term contains functions of and φ. Theorem 3.7. Let be a olarizable Carnot grou with homogeneous dimension Q 3 and let be a bounded domain with smooth boundary which contains the origin, α R, Q + α > 0, and 1 < q <. Then there exists a ositive constant C = C(Q, q, ) such that the following inequality holds: ( Q + α ) α φ α ( ) φ + C αq /q φ q (3.1) for all comactly suorted smooth function φ C 0 (). Proof. The roof is similar to the roof Theorem in [3] ( see also []). We only need to use the weighted L -Hardy-tye inequality (3.5). ) q q.. SHARP WEIHTED RELLICH TYPE IEQUALITIES AD THEIR IMPROVED VERSIOS Our main goal in this section is to obtain weighted analogues of the Rellich inequality (1.1) and (1.) for general Carnot grous. Furthermore, we shall also obtain their imroved versions for bounded domains. The following is the first result of this section.

13 SHARP RELLICH AD UCERTAITY PRICIPLEIEQUALITIES 13 Theorem.1. Let be a Carnot grou with homogeneous norm = u 1/( Q) and let φ C0 (), α R, Q 3, Q + α > 0. Then the following inequality is valid: α φ (Q + α ) (Q α) α φ. (.1) 16 Furthermore, the constant (Q+α ) (Q α) 16 is shar. Proof. A straightforward comutation shows that α = (Q + α )(α ) α + α Q Q+α u. (.) Multilying both sides of (.) by φ and integrating over the domain, we obtain φ α = α (φ φ + φ ). Since u is the fundamental solution of and Q + α > 0 we obtain φ α = (Q + α )(α ) α φ. Therefore (Q + α )(α ) α φ α φ φ = α φ. (.3) Alying the weighted Hardy inequality (3.5) on the right hand side of (.3), we get (Q + α )(α ) α φ α φ φ ( Q + α ) α φ. ow it is clear that, α φ φ ( Q + α )( Q α ) α φ. (.) ext, we aly the Cauchy-Schwarz inequality to the integrand α φ φ and we obtain ( α φ φ ) 1/ ( α φ φ ) 1/. α (.5) Combining (.5) and (.), we obtain the inequality (.1). ow we rove that the constant C(Q, α) = (Q+α ) (Q α) 16 is the best constant for the Rellich-tye inequality (.1), that is C R : = inf 0 f C 0 α f () α = (Q + α ) (Q α). 16 f

14 1 ISMAIL KOMBE It is clear that (Q + α ) (Q α) 16 α f α f. (.6) If we ass to the infimum in (.6) we get that (Q+α ) (Q α) 16 C R. We only need to show that C R (Q+α ) (Q α) 16. iven ɛ > 0, we define the function φ ɛ () by ( Q+α + ɛ) ( 1 ) + 1 if [0, 1], φ ɛ () = Q+α ( +ɛ) if > 1. (.7) otice that φ ɛ () can be well aroximated by smooth functions with comact suort in. By direct comutation we get φ ɛ = ( Q+α + ɛ) (Q 1) if 1, ( Q+α + ɛ) ( Q α ɛ) Q α ɛ if > 1. Let us denote by B 1 = {x : 1} the unit ball with resect to the homogeneous norm. Hence α φ ɛ = A(Q, α, ɛ) α +B(Q, α, ɛ) Q ɛ B 1 \B 1 where A(Q, α, ɛ) = (Q 1) ( Q+α +ɛ) and B(Q, α, ɛ) = ( Q+α +ɛ) ( Q α ɛ). ote that the integrand B 1 α is finite because is uniformly bounded and Q + α > 0. Therefore ext, α φ ɛ = B(Q, α, ɛ) Q ɛ + O(1). (.8) \B 1 α φ ɛ = B1 α φ ɛ + \B1 α φ ɛ. It is clear that the first integrand B 1 α φ ɛ is finite and we get α φ ɛ = Q ɛ + O(1). (.9) \B 1 Taking the limit as ɛ 0 and noting that we get \B 1 Q ɛ

15 SHARP RELLICH AD UCERTAITY PRICIPLEIEQUALITIES 15 Therefore C R = (Q+α ) (Q α) 16. α φ ɛ α φ ɛ (Q + α ) (Q α). 16 Remark.. In the Abelian case, when = R n with the ordinary dilations, one has = V 1 = R n so that Q = n. It is clear that the inequality (.1) with the homogeneous norm (x) = x and α = 0 reduces the Rellich inequality (1.1). IMPROVED RELLICH TYPE IEQUALITIES. In this subsection we obtain various imroved versions of the weighted Rellich-tye inequality (.1) for smooth bounded domains. One virtue of our aroach is that, one can obtain as many as imroved weighted Rellich-tye inequalities as one can construct imroved weighted L -Hardy-tye inequalities. The following theorem is the first result in this direction. Theorem.3. Let be a Carnot grou with homogeneous norm = u 1/( Q) and let be a bounded domain with smooth boundary, 0, Q 3, Q < α < Q and R e su. Then the following inequality holds: α φ (Q + α ) (Q α) α φ 16 + (Q + α )(Q α) 8 for all comactly suorted functions φ C 0 (). α φ (ln R ) (.10) Proof. The roof of Theorem. is similar to that of Theorem.1. Let φ C 0 () and using the same argument as in Theorem.1, we have the following identity: (Q + α )(α ) α φ α φ φ = α φ. (.11) We now aly imroved weighted Hardy-tye inequality (3.13) to the right hand side of (.11): (Q + α )(α ) α φ [ ( Q + α ) α φ + 1 ow it is clear that α φ φ ( Q + α + 1 α φ α φ φ α φ (ln R ). ] )(Q α) α φ. (ln R ) (.1)

16 16 ISMAIL KOMBE On the other hand we have, by the Young s inequality, α φ φ ɛ α φ + 1 B B ɛ B α φ, (.13) where ɛ > 0 and will be chosen later. Substituting (.13) into (.1) we obtain α B φ ( ɛ + (Q + α )(Q α)ɛ ) α φ B + ɛ α φ. (ln R ) It is clear that the quadratic function ɛ +(Q+α )(Q α)ɛ attains the maximum for ɛ = (Q+α )(Q α) and this maximum is equal to (Q+α ) (Q α). Hence we obtain 8 16 the desired inequality: B α φ (Q + α ) (Q α) 16 (Q + α )(Q α) + 8 B α φ α φ. (ln R ) Using the same arguments as in Theorem. and imroved Hardy-tye inequalities (3.19) and (3.0) we obtain the following imroved Rellich-tye inequalities on a metric ball, resectively. Theorem.. Let be a Carnot grou with homogeneous norm = u 1/( Q) and let B ϱ be a ϱ-ball in, Q 3, α R and Q < α < Q. Then the following inequality holds: φ (Q + α ) (Q α) 16 Bϱ α φ (Q + α )(Q α) + α φ c r B ϱ Bϱ α for all comactly suorted smooth functions φ C 0 (B ϱ ). (.1) Theorem.5. Let be a Carnot grou with homogeneous norm = u 1/( Q) and let B ϱ be a ϱ-ball in, φ C 0 (B ϱ ), Q 3, α R, Q < α < Q and q >. Then the following inequality is valid: Bϱ α φ (Q + α ) (Q α) 16 + (Q + α )(Q α) K c r Bϱ α φ ( B ϱ σ ) (.15) /q, φ q

17 SHARP RELLICH AD UCERTAITY PRICIPLEIEQUALITIES 17 where c is a ositive constant, σ = ( Q)( q)+(α )q and K = ( B ϱ Q ) q q. The following imroved Rellich-tye inequality holds for bounded domains in olarizable Carnot grous. Theorem.6. Let be a olarizable Carnot grou and let be a bounded domain with smooth boundary, 0, α R, Q 3 and Q < α < Q. Then the following inequality holds: α φ (Q + α ) (Q α) 16 + C(Q + α )(Q α) for all comactly suorted smooth functions φ C 0 (). α φ ( φ q (α )q ) /q (.16) Proof. The roof is similar to the roof of Theorem.. We only need to use the imroved Hardy-tye inequality (3.1). WEIHTED RELLICH TYPE IEQUALITY II. We now turn our attention to another Rellich-tye inequality that connects first to second order derivatives. The following theorem is first result in this direction. Theorem.7. (Weighted Rellich-tye inequality II) Let be a Carnot grou with homogeneous norm = u 1/( Q) and let φ C0 (), Q 3 and 8 Q < α < Q. 3 Then the following inequality is valid: α φ (Q α) Furthermore, the constant C(Q, α) = ( Q α Proof. Our starting oint is the identity ) is shar. α φ (Q + α )(α ) = α φ φ valid for all φ C 0 () and Q + α > 0 (see (.3)). By alying Cauchy s inequality we obtain α φ φ ɛ α φ + 1 ɛ α φ. (.17) α φ (.18) α φ, (.19)

18 18 ISMAIL KOMBE where ɛ > 0 and will be chosen later. Combining (.19) and (.18), we get ( (Q + α )(α ) ) α φ + ɛ α φ + 1 α φ ɛ. (.0) We only consider the case (Q+α )(α ) + ɛ > 0 because other cases do not allow us to obtain shar weighted Rellich-tye inequality that connects first to second-order derivatives. We now aly the Rellich-tye inequality (.1) to the first integral term on the right hand side of (.0) and get α φ [ 16ɛ (Q + α ) (Q α) + 8(α ) (Q + α )(Q α) + 1 ] ɛ ote that the function ɛ 16ɛ + (Q+α ) (Q α) 8(α ) (Q+α )(Q α) + 1 ɛ φ. attains the minimum for ɛ = (Q+α )(Q α), and this minimum is equal to. Therefore we obtain the 8 (Q α) desired inequality: α φ (Q α) α φ. (.1) ) is shar, we again use the same sequence of func- To show that constant ( Q α tions (.7) and we get as ɛ 0. α φ ɛ α φ ɛ (Q α) Remark.8. ote that one can also aly the weighted Hardy-tye inequality (3.5) with = to the first integral on the right hand side of (.0) and reach the same inequality (.1). IMPROVED RELLICH TYPE IEQUALITY II. We now resent imroved versions of the Rellich-tye inequality (.17) for bounded domains. Their roofs are very similar to that of Theorem.6, excet instead of using lain weighted Hardy-tye inequality, we use imroved weighted Hardy-tye inequalities, (3.13), (3.19), (3.0) and (3.1), resectively. Theorem.9. Let be a Carnot grou with homogeneous norm = u 1/( Q) and let be a bounded domain with smooth boundary, 0, Q 3, 8 Q < α < Q 3 and R e su. Then the following inequality holds: α φ (Q α) + C(Q, α) α φ α φ (ln R ) (.)

19 SHARP RELLICH AD UCERTAITY PRICIPLEIEQUALITIES 19 for all comactly suorted smooth functions φ C 0 (). Here C(Q, α) = (Q α)(q+3α 8) 16. Theorem.10. Let be a Carnot grou with homogeneous norm = u 1/( Q) and let B ϱ be a ϱ-ball in, φ C 0 (B ϱ ), Q 3 and 8 Q 3 < α < Q. Then the following inequality holds: Bϱ α φ (Q α) + Bϱ α φ (Q α)(q + 3α 8) C R where C > 0 and R is the radius of the ball B ϱ. Bϱ α φ, (.3) Theorem.11. Let be a Carnot grou with homogeneous norm = u 1/( Q) and let B ϱ be a ϱ-ball in, φ C 0 (B ϱ ), Q 3 and 8 Q 3 < α < Q. Then the following inequality is valid: Bϱ α φ (Q α) + Bϱ α φ (Q α)(q + 3α 8) K C R where R is the radius of the ball B ϱ, C > 0, σ ( q ) Q q. Bϱ B ϱ σ φ q, (.) = ( Q)( q)+(α )q and K = Theorem.1. Let be a olarizable Carnot grou with homogeneous norm = u 1/( Q) and let be a bounded domain with smooth boundary, 0, Q 3 and 8 Q < α < Q. Then the following inequality holds: 3 α φ (Q α) α φ + C ( φ q (α )q ) /q (.5) for all comactly suorted smooth functions φ C 0 (). Here C = C(Q α)(q+3α 8) and C > UCERTAITY PRICIPLE IEQUALITY In [3] we obtained the following uncertainty rincile-tye inequality for general Carnot grous: ( )( ) φ φ ( Q ) ( φ ), (5.1) where φ C 0 (). It is clear that this inequality does not recover the Euclidean uncertainty rincile inequality (1.8). As we ointed out before one of the main goal of this aer is to establish a shar uncertainty rincile inequality for Carnot grous and the following theorem is the main result of this section.

20 0 ISMAIL KOMBE Theorem 5.1. Let be a olarizable Carnot grou with homogeneous norm = u 1/( Q) and let Q 3 and φ C0 (). Then the following inequality is valid: ( )( φ φ ) (. Q φ ) (5.) Proof. By the volume growth formula (.8) and integration by arts, we get ( φ ) Qφ = φ. (5.3) Alying Cauchy-Schwarz inequality to the right hand-side of (5.3) gives the desired inequality: ( )( φ φ ) (. Q φ ) It is easy to verify that the equality is attained in Theorem 5.1 by the functions φ = Ae β for some A R, β > 0. Remark 5.. In the Abelian case, when = R n with the ordinary dilations, one has = V 1 = R n so that Q = n. It is clear that the inequality (5.) with the homogeneous norm (x) = x recover the uncertainty rincile inequality (1.8). In connection with uncertainty rincile inequality we now resent the following Caffarelli-Kohn-irenberg [9]-tye inequality for olarizable Carnot grous. It is clear that this inequality reduces to the uncertainty rincile inequality (5.) for α = 0 and = q =. Theorem 5.3. Let be a olarizable Carnot grou with homogeneous norm = u 1/( Q) and let Q 3, α > Q, > 1, q = and φ 1 C 0 (). Then the following inequality is valid: ( ) q q 1 φ ( 1 q 1 ( q 1 )q q φ q ) 1/q (Q αq + α) α φ. (5.) q Proof. The roof is similar to the roof of Theorem 3.1. We omit the details. REFERECES [1] B. Abdellaoui, D. Colorado, I. Peral, Some imroved Caffarelli-Kohn-irenberg inequalities, Calculus of Variations and Partial Differential Equations 3 (005), [] Adimurthi,. Chaudhuri and M. Ramaswamy, An imroved Hardy-Sobolev inequality and its alications, Proceedings of American Mathematical Society 130 (00), [3] Z. Balogh and J. Tyson, Polar coordinates on Carnot grous, Mathematische Zeitschrift 1 (00), [] P. Baras and J. A. oldstein, The heat equation with a singular otential, Transactions of American Mathematical Society 8 (198), [5]. Barbatis, Best constants for higher-order Rellich inequalities in L (), Mathematische Zeitschrift 55 (007), no.,

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