FACTORIZATIONS AND HARDY-RELLICH INEQUALITIES ON STRATIFIED GROUPS
arXiv:1706.05108v1 [math.FA] 15 Jun 2017
MICHAEL RUZHANSKY AND NURGISSA YESSIRKEGENOV Abstract. In this paper, we obtain Hardy, Hardy-Rellich and refined Hardy inequalities on general stratified groups and weighted Hardy inequalities on general homogeneous groups using the factorization method of differential operators, inspired by the recent work of Gesztesy and Littlejohn [GL17]. We note that some of the obtained inequalities are new also in the usual Euclidean setting. We also obtain analogues of Gesztesy and Littlejohn’s 2-parameter version of the Rellich inequality on stratified groups and on the Heisenberg group, and a new two-parameter estimate on Rn which can be regarded as a counterpart to the Gesztesy and Littlejohn’s estimate.
Contents 1. Introduction 2. Preliminaries 3. Factorizations and weighted Hardy inequalities on homogeneous groups 4. Factorizations and Hardy-Rellich inequalities on stratified groups 5. Factorizations on the Heisenberg group References
1 7 10 16 24 33
1. Introduction Consider the Hardy inequality 2 Z Z n−2 |f (x)|2 2 |(∇f )(x)| dx ≥ dx, n ≥ 3, 2 |x|2 Rn Rn
(1.1)
for all functions f ∈ C0∞ (Rn \{0}), where ∇ is the standard gradient in Rn . The purpose of this paper is to obtain Hardy, weighted Hardy, improved Hardy and Hardy-Rellich inequalities on stratified, on Heisenberg and on homogeneous groups by factorizing differential expressions. Therefore, let us first recall known results in this direction. In the recent paper + Tα,β [GL17], inspiring our work, Gesztesy and Littlejohn used the nonnegativity of Tα,β 2010 Mathematics Subject Classification. 22E30, 43A80. Key words and phrases. Hardy inequality, Hardy-Rellich inequality, factorization, homogeneous Lie group, stratified group, Heisenberg group. The authors were supported in parts by the EPSRC grant EP/K039407/1 and by the Leverhulme Grant RPG-2014-02, as well as by the MESRK grant 0825/GF4. No new data was collected or generated during the course of research. 1
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M. RUZHANSKY AND N. YESSIRKEGENOV
on C0∞ (Rn \{0}), for two-parameter differential expressions Tα,β := −△ + α|x|−2 x · ∇ + β|x|−2 and its formal adjoint + Tα,β := −△ − α|x|−2 x · ∇ + (β − α(n − 2))|x|−2 ,
for α, β ∈ R, x ∈ Rn \{0} and n ≥ 2. As a result, they obtained the following inequality for all f ∈ C0∞ (Rn \{0}): Z Z 2 |(△f )(x)| dx ≥((n − 4)α − 2β) |x|−2 |(∇f )(x)|2dx n n R R Z − α(α − 4) |x|−4 |x · (∇f )(x)|2 dx (1.2) n R Z + β((n − 4)(α − 2) − β) |x|−4 |f (x)|2 dx, Rn
which implies classical Rellich and Hardy-Rellich-type inequalities after choosing special values of parameters α and β. We refer to [GL17] for a thorough discussion of the factorization method, its history and different features. We also refer to [GP80] for obtaining the Hardy inequality and to [Ges84] for logarithmic refinements by this factorization method. We note that among other things, in Corollary 5.2 we show the counterpart of the Gesztesy and Littlejohn estimate (1.2) which follows from the nonnegativity of the + operator Tα,β Tα,β , namely, for α, β ∈ R and f ∈ C0∞ (Rn \{0}) with n ≥ 2, we have Z Z 2 |(△f )(x)| dx ≥ (nα − 2β) |x|−2 |(∇f )(x)|2 dx n n R Z R − α(α + 4) |x|−4 |x · (∇f )(x)|2 dx n (1.3) R 2 2 + (2(n − 4)(α(n − 2) − β) − 2α (n − 2) + αβn − β )× Z × |x|−4 |f (x)|2 dx. Rn
For the convenience of the reader let us now shortly recapture the main results of this paper, first for general homogeneous groups, then for stratified groups, and finally for the Heisenberg group, where more precise expressions are possible due to the possibility of using explicit formulae for its commutators. The following result is new already in the usual setting of Rn , however, we may also formulate it in the general setting of homogeneous groups in the sense of Folland and Stein [FS82]. We recall the details of such construction in Section 2. • Let G be a homogeneous group of homogeneous dimension Q ≥ 3 and let d | · | be an arbitrary homogeneous quasi-norm. Let R := d|x| be the radial 2 derivative. Let φ, ψ ∈ Lloc (G\{0}) be any real-valued functions such that Rφ, Rψ ∈ L2loc (G\{0}). Let α ∈ R. Then for all complex-valued functions f ∈ C0∞ (G\{0}) we have Z (φ(x))2 |Rf (x)|2 dx G
FACTORIZATIONS ON STRATIFIED GROUPS
3
Z φ(x)ψ(x) |f (x)|2 dx ≥α φ(x)Rψ(x) + ψ(x)Rφ(x) + (Q − 1) |x| G Z 2 (ψ(x))2 |f (x)|2dx, −α G
and also
Z
(φ(x))2 |Rf (x)|2 dx
G
Z φ(x)ψ(x) ≥α ψ(x)Rφ(x) − φ(x)Rψ(x) + (Q − 1) |f (x)|2dx |x| G Z 2 (ψ(x))2 |f (x)|2 dx −α Z
G
φ(x)Rφ(x) (φ(x))2 − (Q − 1) + φ(x)R2 φ(x))|f (x)|2 dx. 2 |x| |x| G A number of consequences of these estimates for different choices of functions φ and ψ are given in Section 3. In particular, the above estimates are new already in the usual Euclidean setting, in which case we can take G = Rn , Q = n, |x| is the usual Euclidean norm of x, and x · ∇f (x), x ∈ Rn , Rf (x) = |x| +
((Q − 1)
is the usual radial derivative. For general homogeneous groups the radial derivative operator R is explained in detail in formulae (2.4)-(2.6). We can also refer to [RSY17] for another type of weighted Hardy inequalities, the socalled Caffarelli-Kohn-Nirenberg inequalities on homogeneous groups, and to references therein. Let us now list the results on a stratified group G of homogeneous dimension Q ≥ 3. Let X1 , ..., XN be left-invariant vector fields giving the first stratum of the Lie algebra of G, ∇H = (X1 , ..., XN ) the horizontal gradient, (x′ , x(2) , . . . , x(r) ) ∈ RN × . . . × RNr with N + N2 + . . . + Nr = n, | · | the Euclidean norm on RN , r being the step of G, and L the sub-Laplacian operator L=
N X
Xk2 .
k=1
• Let G be a stratified group with N ≥ 2 being the dimension of the first stratum, and let α, β ∈ R. Then for all complex-valued functions f ∈ C0∞ (G\{x′ = 0}) we have kLf k2L2 (G)
′
x · ∇H f 2
∇H f 2 2
≥ (α(N − 2) − 2β)
|x′ | 2 − α |x′ |2 2 L (G) L (G)
f 2
+ (α(N − 4)(N − 2) − α (N − 2) + 2β(4 − N) − β + αβ(N − 2))
|x′ |2 2 2
2
.
L (G)
(1.4)
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M. RUZHANSKY AND N. YESSIRKEGENOV
• Let G be a stratified group with N ≥ 3 being the dimension of the first stratum. Then for all complex-valued functions f ∈ C0∞ (G\{x′ = 0}) we have the refined Hardy inequality
′
f
x · ∇H f N − 2
(1.5)
|x′ | 2 ≥ 2 |x′ | 2 , L (G) L (G) where the constant
N −2 2
is sharp.
In particular, by the Cauchy-Schwartz inequality (1.5) implies the Hardy inequality
N −2 f
k∇H f kL2 (G) ≥ , (1.6) 2 |x′ | L2 (G)
where the constant N 2−2 is also sharp. In fact, the sharpness of this constant was the nature of the Badiale-Tarantello conjecture [BT02, Remark 2.3]. This conjecture was solved in [SSW03] and further improvements were obtained in [RS17a]. In Theorem 4.4 we obtain yet another proof of (1.6) showing that the factorization method gives an elementary half-page proof for it. On stratified
we note
that the weighted Hardy-Rellich type inequalities,
group,
∇H f 2
Lf 2 with a, b ∈ R instead of kLf k2L2(G) in (1.4), were namely for |x′ |a
|x′ |b L2 (G)
L2 (G)
obtained in [SS17]. If we use the L-gauge in (1.6) instead of the norm |x′ |, different versions of the Hardy inequality have been actively investigated on stratified groups, we refer to [DGP11], [GL90], [GK08], [Gri03], [JS13], [KO13], [KS16], [Lia13], [NZW01] for a partial list of references for this and related problems. For Hardy and Rellich inequalities for H¨ormander’s sums of squares of vector fields we refer to [RS17c] and references therein. We finally state the analogue of the inequality (1.2) on the Heisenberg group. We + note that compared to (1.2), the analogous expression of Tα,β Tα,β contains additional terms involving commutators of invariant vector fields – such terms do not appear in the Abelian Euclidean setting. In the case of general stratified groups, to get inequality (1.4), we cancel these commutators by considering the non-negative ex+ + pression Tα,β Tα,β + Tα,β Tα,β instead. However, in the case of the Heisenberg group, using explicit knowledge of left-invariant vector fields, we can calculate the inequal+ + ities corresponding to both Tα,β Tα,β and Tα,β Tα,β , which will be given in (1.12) and (1.13), respectively. We briefly recall that the Heisenberg group Hn is a manifold R2n+1 with the group law given by 1 (x(1) , y (1) , t(1) )(x(2) , y (2) , t(2) ) := (x(1) + x(2) , y (1) + y (2) , t(1) + t(2) + (x(1) y (2) −x(2) y (1) )), 2 for (x(1) , y (1) , t(1) ), (x(2) , y (2) , t(2) ) ∈ Rn × Rn × R ∼ Hn , where we denote by x(1) y (2) and x(2) y (1) their usual scalar products on Rn . The canonical basis of its Lie algebra hn is given by the left-invariant vector fields yj xj Xj = ∂xj − ∂t , Yj = ∂yj + ∂t , j = 1, . . . , n, T = ∂t . (1.7) 2 2
FACTORIZATIONS ON STRATIFIED GROUPS
5
The canonical commutation relations of the basis {Xj , Yj , T } for j = 1, ..., n are given by [Xj , Yj ] = T,
j = 1, . . . , n,
with all the other commutators being zero. We also note that the Heisenberg Lie algebra hn is stratified via hn = V1 ⊕ V2 , where V1 is linearly spanned by the Xj ’s and Yj ’s, and V2 = RT . Therefore, the natural dilations on hn are given by Dr (Xj ) = rXj , Dr (Yj ) = rYj , Dr (T ) = r 2 T, and on Hn by Dr (x, y, t) = r(x, y, t) = (rx, ry, r 2t), (x, y, t) ∈ Hn , r > 0. Consequently, Q = 2n + 2 is the homogeneous degree of the Lebesgue measure dxdydt and the homogeneous dimension of the Heisenberg group Hn as well. The sub-Laplacian is given by L :=
n X
(Xj2
+
Yj2 )
j=1
n X yj 2 xj 2 = ∂xj − ∂t + ∂yj + ∂t . 2 2 j=1
To simplify and unify the following formulations, we denote by x e′ the variables of the first stratum: ( xj , 1 ≤ j ≤ n; x e′j = (1.8) yj−n , n + 1 ≤ j ≤ 2n, with its dimension being N = 2n, the norm |e x′ | =
N q X j=1
v uX u n 2 ′ 2 (e xj ) = t (xj + yj2 ),
(1.9)
j=1
and the horizontal gradient ∇H := (X1 , . . . , Xn , Y1 , . . . , Yn ).
(1.10)
We can also write |e x′ |2 2 L = ∆xe′ + ∂ + Z∂t , 4 t
with Z =
n X
(xj ∂yj − yj ∂xj ),
(1.11)
j=1
e′ , and Z is the tangential where ∆xe′ is the Euclidean Laplacian with respect to x derivative in the x e′ -variables.
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M. RUZHANSKY AND N. YESSIRKEGENOV
• Let α, β ∈ R. Then for all complex-valued functions f ∈ C0∞ (Hn \{e x′ = 0}), we have
∇H f 2 2
kLf kL2 (Hn ) ≥ ((N − 4)α − 2β) ′ |e x | L2 (Hn )
′
2
x
e · ∇ f H
− α(α − 4)
|e ′ 2 x | L2 (Hn ) (1.12)
f 2
+ β((N − 4)(α − 2) − β)
|e x′ |2 L2 (Hn ) Zf T f + αkT f k2L2 (Hn ) , ′ + 2α ′ |e x | |e x | L2 (Hn ) and kLf k2L2 (Hn )
∇H f 2
≥ (Nα − 2β) ′ |e x | L2 (Hn )
2
′
x e · ∇ f H
− α(α + 4)
|e x′ |2 2
L (Hn )
f 2
+ (2(N − 4)(α(N − 2) − β) − 2α (N − 2) + αβN − β ) ′ 2 |e x | L2 (Hn ) Zf T f − αkT f k2L2(Hn ) . , ′ − 2α |e x′ | |e x | L2 (Hn ) (1.13) 2
2
Moreover, we have kLf k2L2 (Hn )
∇H f 2
≥ ((N − α)α − 2β)
|e x′ | L2 (Hn )
f 2
+ β((N − 4)(α − 2) − β) ′ 2 |e x | L2 (Hn ) Zf T f + αkT f k2L2 (Hn ) , ′ + 2α |e x′ | |e x | L2 (Hn )
(1.14)
for α(α − 4) ≥ 0, and kLf k2L2(Hn )
∇H f 2
≥ (−2β + α(N − α) − 4α) ′ |e x | L2 (Hn )
f 2
+ (2(N − 4)(α(N − 2) − β) − 2α (N − 2) + αβN − β )
|e x′ |2 L2 (Hn ) Zf T f − 2α − αkT f k2L2(Hn ) , ′ |e x′ | |e x | L2 (Hn ) 2
2
(1.15)
FACTORIZATIONS ON STRATIFIED GROUPS
7
for α(α + 4) ≥ 0. Comparing (1.14) with Gesztesy and Littlejohn’s estimate (1.2) we note the appearing two last terms involving the commutators T = [Xj , Yj ] due to the non-commutative structure of the Heisenberg group. There are also other ways of writing these inequalities, see (5.12) and (5.13). Zf T f The term |ex′ | , |ex′ | , although appearing to be complex-valued, is acL2 (Hn )
tually real-valued for any complex-valued function f , see Remark 5.3, and we have n Z X Zf T f log |e x′ |2 Re(∂yj (f (e x′ , t))∂txj f (e =− , ′ x′ , t))de x′ dt. (1.16) |e x′ | |e x | L2 (Hn ) H n j=1
For functions f (e x′ , t) = f (e x′ ) the last two terms in the above inequalities vanish, while for functions f (e x′ , t) = f (|e x′ |, t) we have Zf = 0, so that the inner product term involving Zf vanishes. The estimate (1.14) for α > 0 and (1.15) for α < 0 show that kT f kL2 (Hn ) is also controlled by kLf kL2 (Hn ) , although without the weight, which is natural in view of the homogeneity degrees. In Section 2 we briefly recall the main concepts of general homogeneous groups and stratified groups, and fix the notation. In Section 3 the Hardy inequalities with more general weights on homogeneous groups are proved by factorization of differential expressions. In Section 4 we investigate the Hardy and Hardy-Rellich inequalities on stratified Lie groups using this factorization method. Finally, the Hardy-Rellich inequalities on the Heisenberg group are discussed in Section 5. 2. Preliminaries In this section we briefly recall the necessary notation concerning the setting of homogeneous groups following Folland and Stein [FS82] as well as a recent treatise [FR16]. A connected simply connected Lie group G is called a homogeneous group if its Lie algebra g is equipped with a family of the following dilations: ∞ X 1 Dλ = Exp(A lnλ) = (ln(λ)A)k . k! k=0 Here A is a diagonalisable positive linear operator on Lie algebra g, and every dilation Dλ satisfies the following ∀X, Y ∈ g, λ > 0, [Dλ X, Dλ Y ] = Dλ [X, Y ], that is, every Dλ is a morphism of the Lie algebra g. Then, in particular, we have Z Z Q −Q f (x)dx, (2.1) |Dλ (S)| = λ |S| and f (λx)dx = λ G
G
where Q := Tr A is a homogeneous dimension of G. Here dx is the Haar measure on homogeneous group G and |S| is the volume of a measurable set S ⊂ G. We recall that the Haar measure on a homogeneous group G is the standard Lebesgue measure for Rn (see, for example [FR16, Proposition 1.6.6]). Let | · | be a homogeneous quasi-norm on homogeneous groups G: it satisfies the usual properties of the norm except that the triangle inequality may hold with a
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M. RUZHANSKY AND N. YESSIRKEGENOV
constant ≥ 1, see [FR16, Section 3.1.6] for a detailed discussion. Let us now introduce the polar decomposition on homogeneous Lie group, which can be found in [FS82] and [FR16, Section 3.1.7]: there is a positive Borel measure σ on the unit quasi-sphere S := {x ∈ G : |x| = 1}, so that for all f ∈ L1 (G) we have Z Z f (x)dx = G
∞
0
Z
f (ry)r Q−1dσ(y)dr.
(2.2)
(2.3)
S
If we fix a basis {X1 , . . . , Xn } of a Lie algebra g such that AXk = νk Xk for every k, then the matrix A can be taken to be A = diag(ν1 , . . . , νn ). Then each Xk is homogeneous of degree νk . By a decomposition of exp−1 G (x) in g, we define the vector e(x) = (e1 (x), . . . , en (x)) by the formula n X −1 ej (x)Xj , expG (x) = e(x) · ∇ ≡ j=1
where ∇ = (X1 , . . . , Xn ). It gives the following equality
x = expG (e1 (x)X1 + . . . + en (x)Xn ) . By homogeneity and denoting x = ry, y ∈ S, we get e(x) = e(ry) = (r ν1 e1 (y), . . . , r νn en (y)). So one obtains d d d (f (x)) = (f (ry)) = (f (expG (r ν1 e1 (y)X1 + . . . + r νn en (y)Xn ))). d|x| dr dr
(2.4)
Throughout this paper, we use the notation R :=
d , dr
(2.5)
that is, d (f (x)) = Rf (x), ∀x ∈ G, (2.6) d|x| for a homogeneous quasi-norm |x| on the homogeneous group G. Now we very briefly recall the necessary notation concerning the setting of stratified groups (or homogeneous Carnot groups), as a special case of homogeneous groups. The triple G = (Rn , ◦, Dλ) is called a stratified group if it satisfies the conditions: • For some natural numbers N = N1 , N2 , ..., Nr with N + N2 + . . . + Nr = n, the following decomposition Rn = RN × . . . × RNr is valid, and for each λ > 0 the dilation Dλ : Rn → Rn is defined by Dλ (x) = Dλ (x′ , x(2) , . . . , x(r) ) := (λx′ , λ2 x(2) , . . . , λr x(r) ) is an automorphism of the stratified group G. Here x′ ≡ x(1) ∈ RN and x(k) ∈ RNk for k = 2, . . . , r.
FACTORIZATIONS ON STRATIFIED GROUPS
9
• Let N be as in above and let X1 , . . . , XN be the left invariant vector fields on stratified group G such that Xk (0) = ∂x∂ k |0 for k = 1, . . . , N. Then rank(Lie{X1 , . . . , XN }) = n, for each x ∈ Rn , that is, the iterated commutators of X1 , . . . , XN span the Lie algebra of stratified group G. Note that the left invariant vector fields X1 , . . . , XN are called the (Jacobian) generators of the stratified group G and r is called a step of this stratified group G. For the expressions for left invariant vector fields on G in terms of the usual (Euclidean) derivatives and further properties see e.g. [FR16, Section 3.1.5]. The homogeneous dimension Q of the stratified group G is then given by Q=
r X
kNk , N1 = N.
k=1
The differential operator
L=
N X
Xk2 ,
(2.7)
k=1
is called the (canonical) sub-Laplacian on the stratified group G. This sub-Laplacian L is a left invariant homogeneous of order 2 hypoelliptic differential operator and it is known that L is elliptic if and only if r = 1. The left invariant vector fields Xj have an explicit form and satisfy the divergence theorem, which can be found in [FR16, Section 3.1.5] and [RS17b]: N
r
1 XX ∂ ∂ (ℓ) ak,m(x′ , . . . , xℓ−1 ) (ℓ) . Xk = ′ + ∂xk ℓ=2 m=1 ∂xm
(2.8)
We will also use the following notations:
∇H := (X1 , . . . , XN ) for the horizontal gradient, divH υ := ∇H · υ for the horizontal divergence, and |x′ | =
q ′ ′ x12 + . . . + xN2
for the Euclidean norm on RN . Since we have the explicit representation of the left invariant vector fields Xj in (2.8), one can readily obtain the identities |∇H |x′ |γ | = γ|x′ |γ−1 , and divH
x′ |x′ |γ
=
for all γ ∈ R, x′ 6= 0.
PN
′ γ ′ j=1 |x | Xj xj
P ′ ′ γ−1 Xj |x′ | − N N −γ j=1 xj γ|x | = , ′ 2γ |x | |x′ |γ
(2.9)
(2.10)
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M. RUZHANSKY AND N. YESSIRKEGENOV
3. Factorizations and weighted Hardy inequalities on homogeneous groups In this section, using the factorization of differential expressions, we obtain Hardy type inequalities with general weights φ(x) and ψ(x), which are real-valued functions in L2loc (G\{0}), with their radial derivatives also in L2loc (G\{0}). The obtained results are new already in the standard setting of Rn , but the proof below works for general homogeneous groups, in particular including Rn with both isotropic and anisotropic structures, as well as general stratified and graded Lie groups. Theorem 3.1. Let G be a homogeneous group of homogeneous dimension Q ≥ 3 and let | · | be an arbitrary homogeneous quasi-norm. Let φ, ψ ∈ L2loc (G\{0}) be any real-valued functions such that Rφ, Rψ ∈ L2loc (G\{0}). Let α ∈ R. Then for all complex-valued functions f ∈ C0∞ (G\{0}) we have Z Z 2 2 (φ(x)) |Rf (x)| dx ≥ α (φ(x)Rψ(x) + ψ(x)Rφ(x)) |f (x)|2 dx G
and
G
Z φ(x)ψ(x) 2 2 (ψ(x))2 |f (x)|2 dx + α(Q − 1) |f (x)| dx − α |x| G G Z Z 2 2 (φ(x)) |Rf (x)| dx ≥ α (ψ(x)Rφ(x) − φ(x)Rψ(x)) |f (x)|2 dx G G Z Z φ(x)ψ(x) 2 2 +α(Q − 1) (ψ(x))2 |f (x)|2 dx |f (x)| dx − α |x| G G Z Z φ(x)Rφ(x) (φ(x))2 2 |f (x)| dx + (Q − 1) |f (x)|2 dx −(Q − 1) 2 |x| |x| G G Z + φ(x)R2 φ(x)|f (x)|2 dx. Z
(3.1)
(3.2)
G
From these we can get different weighted Hardy inequalities. For example, in the most physical situation with φ(x) ≡ 1 we obtain for α ∈ R and for any f ∈ C0∞ (G\{0}) the inequalities Z Z ψ(x) 2 2 2 |Rf (x)| dx ≥ αRφ(x) + α(Q − 1) − α (ψ(x)) |f (x)|2 dx |x| G G and Z
Z ψ(x) Q−1 2 2 |Rf (x)| dx ≥ −αRφ(x) + α(Q − 1) − α (ψ(x)) − |f (x)|2 dx. 2 |x| |x| G G 2
Remark 3.2. If φ(x) = |x|−a and ψ(x) = |x|−b for a, b ∈ R, then (3.1) implies that Z Z Z |Rf (x)|2 |f (x)|2 |f (x)|2 2 dx ≥ α(Q − a − b − 1) dx − α dx. (3.3) a+b+1 2b |x|2a G G |x| G |x| In the case b = a + 1, we have Z Z |Rf (x)|2 |f (x)|2 2 dx ≥ (α(Q − 2a − 2) − α ) dx. 2a+2 |x|2a G G |x|
FACTORIZATIONS ON STRATIFIED GROUPS
11
Then, by maximising the constant (α(Q − 2a − 2) − α2 ) with respect to α we obtain the weighted Hardy inequality on homogeneous groups Z Z |f (x)|2 (Q − 2a − 2)2 |Rf (x)|2 dx ≥ dx. (3.4) 2a+2 |x|2a 4 G |x| G It is known that the constant in (3.4) is sharp (see [RS16, Corollary 4.2] and also [RSY16, Theorem 3.4]). Thus, in the case b = a + 1 and α = Q−2a−2 , we see that (3.3) gives (3.4). 2 Remark 3.3. If φ(x) = |x|−a (log |x|)c and ψ(x) = |x|−b (log |x|)d for a, b, c, d ∈ R, then we obtain from (3.1) the inequality Z (log |x|)2c |Rf (x)|2 dx 2a |x| G Z c+d−1 (c + d)(log |x|) + (Q − 1 − a − b)(log |x|)c+d ≥α |f (x)|2 dx a+b+1 |x| G Z (log |x|)2d |f (x)|2dx. −α2 2b |x| G Q Q−2 When a = 2 , b = 2 , c = 1 and d = 0, it follows that Z Z (log |x|)2 |f (x)|2 2 2 |Rf (x)| dx ≥ (α − α ) dx, (3.5) Q−2 Q G |x| G |x| which after maximising the above constant with respect to α again, we obtain the critical Hardy inequality Z Z 1 |f (x)|2 (log |x|)2 2 |Rf (x)| dx ≥ dx. (3.6) Q−2 4 G |x|Q G |x| The sharpness of the constant in (3.6) is proved in [RSY16, Theorem 3.4]). So, we note that (3.5) gives (3.6) when α = 21 . Remark 3.4. On Carnot groups, we can refer to the recent work [YKG17] for the Hardy inequalities with a pair of nonnegative weight functions. However, we note that in Theorem 3.1 there are no restrictions on the weights while in [YKG17] the weights have to satisfy certain relations for the weighted estimate to hold true. Proof of Theorem 3.1. Let us introduce one-parameter differential expression Tα := φ(x)R + αψ(x). Let us calculate a formal adjoint operator of Tα on C0∞ (G\{0}): Z Z φ(x)Rf (x)g(x)dx + α ψ(x)f (x)g(x)dx G G Z Z ∞Z d Q−1 dσ(y)dr + α ψ(x)f (x)g(x)dx = φ(ry) (f (ry))g(ry)r dr G 0 S Z ∞Z d =− φ(ry)f (ry) (g(ry))r Q−1 dσ(y)dr dr 0 ZS ∞ Z −(Q − 1) φ(ry)f (ry)g(ry)r Q−2dσ(y)dr 0
S
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M. RUZHANSKY AND N. YESSIRKEGENOV
Z
∞
Z
d (φ(ry))f (ry)g(ry)r Q−1 dσ(y)dr dr 0 S Z +α ψ(x)f (x)g(x)dx G Z Z φ(x) = f (x) −φ(x)Rg(x) dx − (Q − 1) f (x) g(x) dx |x| G G Z Z − f (x) Rφ(x)g(x) dx + α f (x) ψ(x)g(x) dx. −
G
G
Thus, the formal adjoint operator of Tα has the following form Q−1 Tα+ := −φ(x)R − φ(x) − Rφ(x) + αψ(x), |x| where x 6= 0. Then we have (Tα+ Tα f )(x) = −φ(x)R(φ(x)Rf (x)) − αφ(x)R(f (x)ψ(x)) − −
Q−1 (φ(x))2 Rf (x) |x|
α(Q − 1) φ(x)ψ(x)f (x) − φ(x)Rφ(x)Rf (x) − αψ(x)f (x)Rφ(x) |x|
+αφ(x)ψ(x)Rf (x) + α2 (ψ(x))2 f (x). x ) ∈ By the nonnegativity of Tα+ Tα , introducing polar coordinates (r, y) = (|x|, |x| (0, ∞) × S on G, where S is the quasi-sphere as in (2.2), and using (2.3), one calculates Z Z 2 0≤ |(Tα f )(x)| dx = f (x)(Tα+ Tα f )(x)dx G G Z = Re f (x)(Tα+ Tα f )(x)dx = I1 + I2 + I3 + I4 + I5 + I6 , (3.7) G
where
d d φ(ry) (f (ry)) r Q−1 dσ(y)dr, I1 = −Re f (ry)φ(ry) dr dr 0 S Z ∞Z d I2 = −αRe f (ry)φ(ry) (f (ry)ψ(ry))r Q−1dσ(y)dr, dr 0 S Z ∞Z d (f (ry)) Q−1 (φ(ry))2 dr r dσ(y)dr, I3 = −(Q − 1)Re f (ry) r 0 S Z Z φ(x)ψ(x) 2 I4 = −α(Q − 1) |f (x)| dx − α Rφ(x)ψ(x)|f (x)|2 dx |x| G G Z + α2 (ψ(x))2 |f (x)|2 dx, Z
I5 = −Re and
Z
∞
Z
∞
0
I6 = αRe
Z
0
Z
∞
G
f (ry) S
Z
S
d d (φ(ry))φ(ry) (f (ry))r Q−1 dσ(y)dr dr dr
f (ry)φ(ry)ψ(ry)
d (f (ry))r Q−1 dσ(y)dr. dr
(3.8)
FACTORIZATIONS ON STRATIFIED GROUPS
Now we calculate I1 , I2 , I3 , I5 and I6 . By a direct calculation we obtain Z ∞Z d I1 = (Q − 1)Re (φ(ry))2f (ry) (f (ry))r Q−2 dσ(y)dr dr 0 S Z ∞Z d d +Re (φ(ry))2 (f (ry)) (f (ry))r Q−1 dσ(y)dr dr dr 0 S Z ∞Z d d +Re φ(ry) (φ(ry)) f (ry)f (ry)r Q−1dσ(y)dr dr dr 0 SZ Z Q−1 ∞ d = (φ(ry))2 |f (ry)|2r Q−2 dσ(y)dr 2 dr 0 S 2 Z ∞Z 2 d + (φ(ry)) (f (ry)) r Q−1dσ(y)dr dr Z 0Z S d d 1 ∞ φ(ry) (φ(ry)) |f (ry)|2r Q−1dσ(y)dr + 2 0 dr dr S 2 Z ∞Z 2 d = (φ(ry)) (f (ry)) r Q−1 dσ(y)dr dr 0 S Z ∞Z (Q − 1)(Q − 2) − (φ(ry))2|f (ry)|2r Q−3 dσ(y)dr 2 S Z ∞Z 0 d −(Q − 1) φ(ry) (φ(ry))|f (ry)|2r Q−2 dσ(y)dr dr 0 S 2 Z ∞Z 1 d − (φ(ry)) |f (ry)|2r Q−1dσ(y)dr 2 0 dr Z ZS d2 1 ∞ φ(ry) 2 (φ(ry))|f (ry)|2r Q−1dσ(y)dr − 2 0 dr Z ∞S Z Q−1 d − φ(ry) (φ(ry))|f (ry)|2r Q−2 dσ(y)dr 2 dr 0 S Z Z (Q − 1)(Q − 2) (φ(x))2 2 2 = (φ(x)) |Rf (x)| dx − |f (x)|2 dx 2 2 |x| G Z G Z 1 φ(x)Rφ(x) (Rφ(x))2 |f (x)|2dx −(Q − 1) |f (x)|2dx − |x| 2 G Z Z G φ(x)Rφ(x) Q − 1 1 R2 φ(x)φ(x)|f (x)|2 dx − |f (x)|2 dx. − 2 G 2 |x| G Now let us calculate I2 : Z ∞Z d I2 = −αRe φ(ry)ψ(ry)f (ry) (f (ry))r Q−1dσ(y)dr dr 0 S Z ∞Z d −α φ(ry) (ψ(ry))|f (ry)|2r Q−1 dσ(y)dr dr 0 Z SZ α ∞ d =− φ(ry)ψ(ry) |f (ry)|2r Q−1 dσ(y)dr 2 dr Z ∞0Z S d −α φ(ry) (ψ(ry))|f (ry)|2r Q−1 dσ(y)dr dr 0 S
13
(3.9)
14
M. RUZHANSKY AND N. YESSIRKEGENOV
Z
∞
Z
d = −α φ(ry) (ψ(ry))|f (ry)|2r Q−1 dσ(y)dr dr 0 S Z ∞Z α d + ψ(ry) (φ(ry))|f (ry)|2r Q−1dσ(y)dr 2 0 dr S Z ∞Z d α φ(ry) (ψ(ry))|f (ry)|2r Q−1dσ(y)dr + 2 0 dr S Z ∞Z α(Q − 1) + φ(ry)ψ(ry)|f (ry)|2r Q−2dσ(y)dr 2 0 S Z Z α α 2 = ψ(x)Rφ(x)|f (x)| dx + φ(x)Rψ(x)|f (x)|2 dx 2 G 2 G Z Z φ(x)ψ(x) α(Q − 1) 2 |f (x)| dx − α φ(x)Rψ(x)|f (x)|2 dx. (3.10) + 2 |x| G G For I3 , one has Z ∞Z d I3 = −(Q − 1)Re (φ(ry))2f (ry) (f (ry))r Q−2 dσ(y)dr dr 0 S Z ∞Z d Q−1 (φ(ry))2 |f (ry)|2r Q−2dσ(y)dr =− 2 dr 0 S Z ∞Z d = (Q − 1) φ(ry) (φ(ry))|f (ry)|2r Q−2 dσ(y)dr dr 0 S Z ∞Z (Q − 1)(Q − 2) + (φ(ry))2|f (ry)|2r Q−3 dσ(y)dr 2 0 S Z Z (φ(x))2 (Q − 1)(Q − 2) φ(x)Rφ(x) 2 |f (x)| dx + |f (x)|2 dx. (3.11) = (Q − 1) 2 |x| 2 |x| G G For I5 , we have Z ∞Z d d I5 = −Re f (ry) (φ(ry))φ(ry) (f (ry))r Q−1 dσ(y)dr dr dr 0 S Z ∞Z d d 1 (φ(ry))φ(ry) |f (ry)|2r Q−1 dσ(y)dr =− 2 0 dr S dr Z ∞Z 2 1 d = φ(ry) 2 (φ(ry))|f (ry)|2r Q−1dσ(y)dr 2 0 dr S 2 Z ∞Z 1 d + (φ(ry)) |f (ry)|2r Q−1dσ(y)dr 2 0 dr S Z ∞Z d φ(ry) dr (φ(ry)) Q−1 |f (ry)|2r Q−1dσ(y)dr + 2 r 0 S Z Z 1 1 2 2 = R φ(x)φ(x)|f (x)| dx + (Rφ(x))2 |f (x)|2 dx 2 G 2 G Z Q−1 Rφ(x)φ(x) + |f (x)|2 dx. (3.12) 2 |x| G
FACTORIZATIONS ON STRATIFIED GROUPS
Finally, for I6 we obtain Z I6 = αRe
∞
0
Z
Z
f (ry)φ(ry)ψ(ry)
S
∞
Z
d (f (ry))r Q−1dσ(y)dr dr
α d φ(ry)ψ(ry) |f (ry)|2r Q−1dσ(y)dr 2 0 dr Z ∞ ZS α d =− (φ(ry))ψ(ry)|f (ry)|2r Q−1 dσ(y)dr 2 0 dr Z Z S α ∞ d − (ψ(ry))φ(ry)|f (ry)|2r Q−1 dσ(y)dr 2 0 dr S Z Z |f (ry)|2 Q−1 α(Q − 1) ∞ φ(ry)ψ(ry) r dσ(y)dr − 2 r 0 S Z Z α α 2 =− Rφ(x)ψ(x)|f (x)| dx − φ(x)Rψ(x)|f (x)|2 dx 2 G 2 G Z φ(x)ψ(x) α(Q − 1) |f (x)|2 dx. − 2 |x| G Putting (3.8)-(3.13) in (3.7), we obtain that Z (φ(x))2 |Rf (x)|2 dx =
15
(3.13)
G
Z φ(x)ψ(x) |f (x)|2 dx −α φ(x)Rψ(x) + ψ(x)Rφ(x) + (Q − 1) |x| G Z +α2 (ψ(x))2 |f (x)|2 dx ≥ 0, G
which implies (3.1). Thus, we have obtained (3.1) using the nonnegativity of Tα+ Tα . Now we obtain (3.2) using the nonnegativity of Tα Tα+ . So, we calculate Q−1 (Tα Tα+ f )(x) = −φ(x)R(φ(x)Rf (x)) + αφ(x)R(f (x)ψ(x)) − (φ(x))2 Rf (x) |x| −
α(Q − 1) φ(x)ψ(x)f (x) − φ(x)Rφ(x)Rf (x) − αψ(x)f (x)Rφ(x) |x|
−αφ(x)ψ(x)Rf (x) + α2 (ψ(x))2 f (x) φ(x) −(Q − 1)φ(x)f (x)R − φ(x)f (x)R2 φ(x). |x| Using the nonnegativity of Tα Tα+ , we get Z Z 2 0≤ |(Tα f )(x)| dx = f (x)(Tα Tα+ f )(x)dx = Re
Z
G
where
G
G
f (x)(Tα Tα+ f )(x)dx = Ie1 + Ie2 + Ie3 + Ie4 + Ie5 + Ie6 ,
Ie1 = I1 , Ie2 = −I2 , Ie3 = I3 , Ie4 = I4 + A, Ie5 = I5 , Ie6 = −I6
(3.14)
16
M. RUZHANSKY AND N. YESSIRKEGENOV
with
Z
φ(x) A = −(Q − 1) φ(x)|f (x)| R |x| G 2
dx −
Z
φ(x)R2 φ(x)|f (x)|2 dx.
G
Taking into account these and (3.8)-(3.13), we obtain (3.2).
4. Factorizations and Hardy-Rellich inequalities on stratified groups In this section, we obtain Hardy-Rellich and improved Hardy inequalities on stratified groups by factorization. We refer to [Ges84], [GP80] and to the recent paper [GL17] for obtaining Hardy and Hardy-Rellich inequalities using this factorization method in the isotropic abelian case. In this section L is the sub-Laplacian operator defined by (2.7). Theorem 4.1. Let G be a stratified group with N ≥ 2 being the dimension of the first stratum, and let α, β ∈ R. Then for all complex-valued functions f ∈ C0∞ (G\{x′ = 0}) we have kLf k2L2 (G)
∇H f 2
≥ (α(N − 2) − 2β) ′ |x | L2 (G)
′
2
2 x · ∇H f −α |x′ |2 L2 (G)
f 2
+ (α(N − 4)(N − 2) − α (N − 2) + 2β(4 − N) − β + αβ(N − 2)) ′ 2 |x | 2 2
2
.
L (G)
(4.1)
We note that the term x′ · ∇H f can be eliminated.
Corollary 4.2. Using the Cauchy-Schwarz inequality in (4.1) Z Z |(∇H f )(x)|2 |x′ · (∇H f )(x)|2 dx ≤ dx, |x′ |4 |x′ |2 G G
(4.2)
we obtain kLf k2L2 (G)
∇H f 2
≥ (α(N − 2) − 2β − α )
|x′ | 2 L (G) 2
f 2
+ (α(N − 4)(N − 2) − α (N − 2) + 2β(4 − N) − β + αβ(N − 2))
|x′ |2 2 2
2
.
L (G)
(4.3)
We also record the special case of Theorem 4.1 in the abelian estting, to contrast it later in Corollary 5.2 with the Gesztesy-Littlejohn inequality (1.2).
FACTORIZATIONS ON STRATIFIED GROUPS
17
Corollary 4.3. In the abelian case G = (Rn , +), we have N = n, ∇H = ∇ = (∂x1 , ..., ∂xn ) is the usual (full) gradient, so (4.1) implies for α, β ∈ R and for any f ∈ C0∞ (Rn \{0}) with n ≥ 2 the inequality k△f k2L2(Rn )
∇f 2
≥ (α(n − 2) − 2β)
|x| 2 n L (R )
2
x · ∇f
− α2
|x|2 2 n L (R )
(4.4)
f 2
+ (α(n − 4)(n − 2) − α (n − 2) + 2β(4 − n) − β + αβ(n − 2))
|x|2 2 2
2
.
L (Rn )
Proof of Theorem 4.1. We will be applying the factorization method with the following differential expressions for two real parameters α, β ∈ R,
Tα,β := −L + α
x′ · ∇H β + ′2 ′ 2 |x | |x |
(4.5)
and its formal adjoint
+ Tα,β
α(N − 2) − β x′ · ∇H − , := −L − α ′ 2 |x | |x′ |2
(4.6)
where x′ 6= 0. Then, by a direct calculation for any function f ∈ C0∞ (G\{x′ = 0}) we have + (Tα,β Tα,β f )(x) x′ · ∇H x′ · (∇H f ) βf (x) α(N − 2) − β = −L − α −(Lf )(x) + α − + |x′ |2 |x′ |2 |x′ |2 |x′ |2 ′ x · (∇H f ) x′ · ∇H N −2 2 = (L f )(x) + α −L (x) + (Lf )(x) + (Lf )(x) |x′ |2 |x′ |2 |x′ |2 (4.7) (Lf )(x) f (x) − + β −L |x′ |2 |x′ |2 ′ x · ∇H x′ · (∇H f )(x) (N − 2)f (x) f + αβ − (x) + − |x′ |2 |x′ |2 |x′ |4 |x′ |4 ′ ′ x · ∇H x · (∇H f ) x′ · (∇H f )(x) f (x) 2 +α − (x) − (N − 2) + β2 ′ 4 . ′ 2 ′ 2 ′ 4 |x | |x | |x | |x |
18
M. RUZHANSKY AND N. YESSIRKEGENOV
Now we calculate
+ (Tα,β Tα,β f )(x) x′ · (∇H f )(x) (α(N − 2) − β)f (x) β x′ · ∇H −(Lf )(x) − α + ′2 − = −L + α |x′ |2 |x | |x′ |2 |x′ |2 ′ x · (∇H f ) x′ · ∇H f 2 = (L f )(x) + α L (x) − (Lf )(x) + (N − 2)L (x) |x′ |2 |x′ |2 |x′ |2 (Lf )(x) f (x) − + β −L |x′ |2 |x′ |2 ′ x · ∇H f x′ · (∇H f )(x) (N − 2)f (x) + αβ (x) − − |x′ |2 |x′ |2 |x′ |4 |x′ |4 ′ ′ x · ∇H x · (∇H f ) x′ · ∇H f 2 2 f (x) +α − (x) − (N − 2) (x) + β . |x′ |2 |x′ |2 |x′ |2 |x′ |2 |x′ |4 (4.8) Using
f L |x′ |2
(x) =
N X
Xj2
j=1
=
N X
Xj
j=1
=
f |x′ |2
2x′j f Xj f − |x′ |2 |x′ |4
N X (Xj2 f )(x)
|x′ |2
j=1
(x)
−
(x)
4x′j (Xj f )(x) |x′ |4
+
8(x′j )2 f (x) |x′ |6
−
2f (x) |x′ |4
(4.9)
(Lf )(x) 4x′ · (∇H f )(x) f (x) = − − (2N − 8) ′ 4 ′ 2 ′ 4 |x | |x | |x |
and
x′ · ∇H |x′ |2
f |x′ |2
N
f (x) X ′ x′ · (∇H f )(x) (x) = ′ 2 xj Xj (|x′ |−2 ) + |x | j=1 |x′ |4 = −2
N X (x′j )2 f (x) j=1
= −2
|x′ |6
x′ · (∇H f )(x) + |x′ |4
f (x) x′ · (∇H f )(x) + , |x′ |4 |x′ |4
(4.10)
FACTORIZATIONS ON STRATIFIED GROUPS
19
+ + in (4.8), then we have for (Tα,β Tα,β f )(x) + (Tα,β Tα,β f )(x) that
+ + (Tα,β Tα,β f )(x) + (Tα,β Tα,β f )(x) x′ · (∇H f )(x) f (x) (Lf )(x) 2 −2 + (4 − N) ′ 4 = 2(L f )(x) + 2α(N − 2) |x′ |2 |x′ |4 |x | ′ 2(Lf )(x) 4x · (∇H f )(x) f (x) f (x) + 2β − + + (2N − 8) − 2αβ(N − 2) |x′ |2 |x′ |4 |x′ |4 |x′ |4 ′ ′ x · ∇H x′ · (∇H f )(x) f (x) x · (∇H f ) 2 + 2α − (x) − (N − 2) + (N − 2) ′ 4 |x′ |2 |x′ |2 |x′ |4 |x | f (x) + 2β 2 ′ 4 |x | (Lf )(x) x′ · (∇H f )(x) f (x) 2 =: 2(L f )(x) + 2α(N − 2) −2 + (4 − N) ′ 4 |x′ |2 |x′ |4 |x | f (x) f (x) − 2αβ(N − 2) ′ 4 + 2β 2 ′ 4 + βJ1 + α2 J2 . |x | |x | (4.11)
In order to simplify this, let us calculate the following
′ x′ · ∇H x′ · (∇H f )(x) f (x) x · (∇H f ) −2 (x) − 2(N − 2) + 2(N − 2) ′ 4 ′ 2 ′ 2 ′ 4 |x | |x | |x | |x | PN N ′ ′ ′ X 2 j,k=1(xj Xj )(xk (Xk f ))(x) ′ ′ −3 ′ xk (Xk f )(x) − 2 x (−2)|x | X |x | =− j j |x′ |4 |x′ |2 j,k=1 x′ · (∇H f )(x) f (x) − 2(N − 2) + 2(N − 2) ′ 4 ′ 4 |x | |x | PN PN ′ P ′ ′ ′ 2 k=1 xk (Xk f )(x) 2 j,k=1 xj xk Xj (Xk f )(x) 4 N k=1 xk (Xk f )(x) =− − + |x′ |4 |x′ |4 |x′ |4 x′ · (∇H f )(x) f (x) − 2(N − 2) + 2(N − 2) ′ 4 ′ 4 |x | |x | PN x′ · (∇H f )(x) 2 j,k=1 x′j x′k (Xj Xk f )(x) f (x) = −2(N − 3) − + 2(N − 2) ′ 4 . ′ 4 ′ 4 |x | |x | |x |
(4.12)
20
M. RUZHANSKY AND N. YESSIRKEGENOV
Now putting this in (4.11), we obtain (Lf )(x) |x′ |2 x′ · (∇H f )(x) + (−4α(N − 2) − 2α2 (N − 3) + 8β) |x′ |4 + (2α(N − 2)(4 − N) + 2α2 (N − 2) − 2αβ(N − 2)
+ + (Tα,β Tα,β f )(x) + (Tα,β Tα,β f )(x) =2(L2 f )(x) + (2α(N − 2) − 4β)
+ (4N − 16)β + 2β 2 ) − 2α
2
PN
f (x) |x′ |4
′ ′ j,k=1 xj xk (Xj Xk f )(x) . |x′ |4
(4.13)
+ + Now using the nonnegativity of Tα,β Tα,β + Tα,β Tα,β and integrating by parts, we get Z Z 2 + |(Tα,β f )(x)| dx + |(Tα,β f )(x)|2 dx G
=
Z
G
+ + f (x)((Tα,β Tα,β + Tα,β Tα,β )f )(x)dx ≥ 0.
G
Putting (4.13) into this inequality, one calculates Z Z f (x)(Lf )(x) 2 dx 2 |(Lf )(x)| dx + (2α(N − 2) − 4β) |x′ |2 G G Z f (x)(x′ · (∇H f )(x)) 2 dx + (−4α(N − 2) − 2α (N − 3) + 8β) |x′ |4 G + (2α(N − 2)(4 − N) + 2α2 (N − 2) − 2αβ(N − 2) Z |f (x)|2 2 + (4N − 16)β + 2β ) dx ′ 4 G |x | N Z X f (x)x′j x′k (Xj Xk f )(x) 2 − 2α dx ≥ 0. |x′ |4 G j,k=1
(4.14)
Using the identities Z
G
N Z X f (x)(Lf )(x) (Xj f )(x)(Xj f )(x) dx = − ′ 2 |x | |x′ |2 j=1 G
−
N Z X j=1
=2
Z
G
f (x)(−2)|x′ |−3 Xj |x′ |(Xj f )(x)dx
G
f (x)(x′ · (∇H f )(x)) dx − |x′ |4
Z
G
|(∇H f )(x)|2 dx |x′ |2
(4.15)
FACTORIZATIONS ON STRATIFIED GROUPS
and
21
N Z X f (x)x′j x′k (Xj Xk f )(x) dx |x′ |4 j,k=1 G
N Z N Z X X f (x)x′k (Xk f )(x) f (x)x′k (Xk f )(x) = −(N − 1) dx − 2 dx |x′ |4 |x′ |4 G G k=1
k=1
N Z N Z X X x′j x′k (Xj f )(x)(Xk f )(x) − dx − f (x)x′j x′k (Xk f )(x)(−4)|x′ |−5 Xj |x′ |dx ′ |4 |x j,k=1 G j,k=1 G N Z N Z X X f (x)(x′j )2 x′k (Xk f )(x) f (x)x′k (Xk f )(x) = −(N + 1) dx + 4 dx |x′ |4 |x′ |6 G G k=1
−
Z
G
j,k=1
Z f (x)(x′ · (∇H f )(x)) |x′ · (∇H f )(x)|2 dx = −(N − 3) dx |x′ |4 |x′ |4 G Z |x′ · (∇H f )(x)|2 dx − |x′ |4 G
(4.16)
in (4.14), we obtain Z 2 |(Lf )(x)|2dx + (4α(N − 2) − 8β − 4α(N − 2) + 8β − 2α2(N − 3) + 2α2 (N − 3)) G
×
Z
f (x)(x′ · (∇H f )(x)) dx |x′ |4
G
2
2
+(2α(N − 2)(4 − N) + 2α (N − 2) − 2αβ(N − 2) + (4N − 16)β + 2β )
Z
G
−(2α(N − 2) − 4β)
Z
G
+2α2
Z
G
which implies (4.1).
′
|(∇H f )(x)|2 dx |x′ |2
|f (x)|2 dx |x′ |4
|x · (∇H f )(x)|2 dx ≥ 0, |x′ |4
Now let us give a very elementary proof of a version of the Hardy inequality on stratified groups using the factorization method. We note that this inequality on stratified groups was obtained in [RS17a] by a different method. Theorem 4.4. Let G be a stratified group with N ≥ 3 being the dimension of the first stratum. Let α ∈ R. Then for all complex-valued functions f ∈ C0∞ (G\{x′ = 0}) we have
N −2 f
k∇H f kL2 (G) ≥ , (4.17)
2 |x′ | L2 (G) where the constant
N −2 2
is sharp.
22
M. RUZHANSKY AND N. YESSIRKEGENOV
Proof of Theorem 4.4. Here we use the following one-parameter differential expression x′ Teα := ∇H + α ′ 2 , |x | and its formal adjoint x′ · Teα+ := −divH (·) + α ′ 2 , |x | ′ where x 6= 0. Taking into account (2.10) we have ′ x′ · (∇H f )(x) f (x) x +e e f (x) + α + α2 ′ 2 Tα Tα f = −(Lf )(x) − αdivH ′ 2 ′ 2 |x | |x | |x | ′ x 2 f (x) = −(Lf )(x) − αdivH f (x) + α |x′ |2 |x′ |2 α(α + 2 − N) f (x). |x′ |2 By integrating by parts and using the nonnegativity of Teα+ Teα one calculates Z Z 2 f (x)(Teα+ Teα f )(x)dx 0≤ |Teα f | dx = G ZG Z |f (x)|2 2 = |∇H f | dx + α(α + 2 − N) dx. ′ 2 G G |x | = −(Lf )(x) +
It follows that
Z
Z
|f (x)|2 dx, ′ 2 G G |x | which after maximising the constant in the above inequality with respect to α, we obtain (4.17). The sharpness of the constant in the obtained inequality was shown in [RS17a]. 2
|∇H f | dx ≥ α(N − 2 − α)
The interesting result here is that by modifying the differential expression Teα , the factorization method gives a refinement of the Hardy inequality (4.17):
Theorem 4.5. Let G be a stratified group with N ≥ 3 being the dimension of the first stratum. Let α ∈ R. Then for all complex-valued functions f ∈ C0∞ (G\{x′ = 0}) we have
′
f
x · ∇H f N − 2
(4.18)
|x′ | 2 ≥ 2 |x′ | 2 , L (G) L (G) where the constant
N −2 2
is sharp.
Remark 4.6. We note that the estimate (4.18) implies (4.17) by the Cauchy-Schwarz inequality. Consequently, the sharpness of the constant in (4.18) follows from the sharpness of the constant in (4.17). Proof of Theorem 4.5. Here we take the one parameter differential expressions in the form x′ · ∇H α Tbα := + ′ , ′ |x | |x |
FACTORIZATIONS ON STRATIFIED GROUPS
23
and
x′ · ∇H α − N + 1 + , Tbα+ := − |x′ | |x′ | where x′ 6= 0. By a direct calculation and using (2.9) we get ′ ′ x · ∇H x · (∇H f )(x) |x′ | |x′ | PN N ′ ′ ′ X j,k=1 (xj Xj )(xk (Xk f )(x)) ′ ′ −2 ′ xk (Xk f )(x) + x (−1)|x | X |x | = j j |x′ |2 |x′ | j,k=1 PN PN ′ PN ′ ′ ′ j,k=1 xj xk (Xj Xk f )(x) k=1 xk (Xk f )(x) k=1 xk (Xk f )(x) = + − |x′ |2 |x′ |2 |x′ |2 PN ′ ′ j,k=1 xj xk (Xj Xk f )(x) = |x′ |2 and ′
(x · ∇H )
f |x′ |
=
PN
′ k=1 xk (Xk f )(x) |x′ |
+
N X
x′k (−1)|x′ |−2 Xk |x′ |f (x)
k=1
x′ · (∇H f )(x) f (x) = − ′ . |x′ | |x | Taking into account these identities, we obtain Tbα+ Tbα f (x) =−
PN
′ ′ j,k=1 xj xk (Xj Xk f )(x) |x′ |2
− (N − 1)
x′ · (∇H f )(x) α(α + 2 − N) + f (x). |x′ |2 |x′ |2
Using (2.9), we calculate N Z X f (x)x′j x′k (Xj Xk f )(x) dx ′ |2 |x G j,k=1
N Z N Z X X f (x)x′k (Xk f )(x) f (x)x′k (Xk f )(x) = −(N − 1) dx − 2 dx |x′ |2 |x′ |2 k=1 G k=1 G
N Z N Z X X x′j x′k (Xj f )(x)(Xk f )(x) dx − − f (x)x′j x′k (Xk f )(x)(−2)|x′ |−3 Xj |x′ |dx ′ 2 |x | j,k=1 G j,k=1 G N Z N Z X X f (x)(x′j )2 x′k (Xk f )(x) f (x)x′k (Xk f )(x) = −(N + 1) dx + 2 dx |x′ |2 |x′ |4 k=1 G j,k=1 G
−
Z
G
Z f (x)(x′ · (∇H f )(x)) |x′ · (∇H f )(x)|2 dx = −(N − 1) dx |x′ |2 |x′ |2 G Z |x′ · (∇H f )(x)|2 − dx. |x′ |2 G
(4.19)
24
M. RUZHANSKY AND N. YESSIRKEGENOV
Taking into account this, integrating by parts, and using the nonnegativity of the operator Tbα+ Tbα , we get Z Z 2 0≤ |Tbα f | dx = f (x)(Tbα+ Tbα f )(x)dx G G ! PN Z ′ ′ ′ f (x)(X X f )(x) x x (x)(x · (∇ f )(x)) (N − 1)f j k H j,k=1 j k + dx =− ′ |2 ′ |2 |x |x G +α(α − N + 2)
Z
G
|f (x)|2 dx. |x′ |2
Consequently, using (4.19) one obtains Z ′ |x · (∇H f )(x)|2 |f (x)|2 + α(α − N + 2) ′ 2 dx ≥ 0. |x′ |2 |x | G It now follows that Z Z |x′ · (∇H f )(x)|2 |f (x)|2 dx ≥ α((N − 2) − α) dx. ′ 2 |x′ |2 G G |x | As usual, by maximising α((N − 2) − α) with respect to α, we obtain (4.18).
5. Factorizations on the Heisenberg group Now we discuss the Hardy-Rellich inequalities on the Heisenberg group. In this section we adopt all the notation concerning the Heisenberg group from the introduction as well as some new notation that will be useful in the computations: ( ( xj , 1 ≤ j ≤ n; ′ ej = Xj , 1 ≤ j ≤ n; x ej = , X (5.1) yj−n , n + 1 ≤ j ≤ 2n, Yj−n , n + 1 ≤ j ≤ 2n, and
v v u N uX uX u n ′ ′ 2 t (e xj ) = t (x2j + yj2). |e x|= j=1
(5.2)
j=1
Taking into account the above notations, ∇H is defined by e1 , . . . , X eN ), ∇H := (X
and the sub-Laplacian L :=
N X j=1
ej2 X
2 2 n X x e′j+n x e′j ∂t + ∂xe′j+n + ∂t . = ∂xe′j − 2 2 j=1
(5.3)
(5.4)
Therefore, we see that N = 2n and the formulae (2.9) and (2.10) also hold on the Heisenberg group. the following formulation we recall the tangential derivative operator Z = PFor n j=1 (xj ∂yj − yj ∂xj ) given in (1.11).
FACTORIZATIONS ON STRATIFIED GROUPS
25
Theorem 5.1. Let α, β ∈ R. Then for all complex-valued functions f ∈ C0∞ (Hn \{e x′ = 0}) we have
∇H f 2 2
kLf kL2 (Hn ) ≥ ((N − 4)α − 2β)
|e x′ | L2 (Hn )
2
′
x e · ∇H f
− α(α − 4) |e x′ |2 L2 (Hn ) (5.5)
f 2
+ β((N − 4)(α − 2) − β)
|e x′ |2 L2 (Hn ) Z f (e x′ , t)ZT f (e x′ , t) ′ − 2α de x dt + αkT f k2L2(Hn ) . ′ |2 |e x Hn and kLf k2L2 (Hn )
∇H f 2
≥ (Nα − 2β) ′ |e x | L2 (Hn )
′
2
x
e · ∇ f H
− α(α + 4)
|e x′ |2 L2 (Hn )
2
f
+ (2(N − 4)(α(N − 2) − β) − 2α2 (N − 2) + αβN − β 2 )
|e x′ |2 L2 (Hn ) Z f (e x′ , t)ZT f (e x′ , t) ′ + 2α de x dt − αkT f k2L2 (Hn ) . ′ |2 |e x Hn (5.6) Moreover, kLf k2L2(Hn )
∇H f 2
≥ ((N − α)α − 2β)
|e x′ | L2 (Hn )
f 2
+ β((N − 4)(α − 2) − β) ′ 2 |e x | L2 (Hn ) Z f (e x′ , t)ZT f (e x′ , t) ′ − 2α de x dt + αkT f k2L2(Hn ) ′ |2 |e x Hn
(5.7)
for α(α − 4) ≥ 0, and kLf k2L2 (Hn )
∇H f 2
≥ (−2β + α(N − α) − 4α)
|e x′ | L2 (Hn )
2
f
+ (2(N − 4)(α(N − 2) − β) − 2α2 (N − 2) + αβN − β 2 )
|e x′ |2 L2 (Hn ) Z f (e x′ , t)ZT f (e x′ , t) ′ + 2α de x dt − αkT f k2L2 (Hn ) ′ |2 |e x Hn (5.8)
26
M. RUZHANSKY AND N. YESSIRKEGENOV
for α(α + 4) ≥ 0. Corollary 5.2. In the abelian case G = (Rn , +), we have ∇H = ∇ = (∂x1 , ..., ∂xn ) and we can note that if we argue as in the proof of Theorem 5.1, the last two terms in (5.5) and (5.6) vanish. Therefore, in this case, (5.5) would coincide with the Gesztesy and Littlejohn’s estimate (1.2), while (5.6) would give the following new estimate for α, β ∈ R and f ∈ C0∞ (Rn \{0}) with n ≥ 2: k△f k2L2 (Rn )
∇f 2
≥ (nα − 2β)
|x| 2 n L (R )
x · ∇f 2
− α(α + 4) |x|2 L2 (Rn )
(5.9)
f 2
+ (2(n − 4)(α(n − 2) − β) − 2α (n − 2) + αβn − β )
|x|2 2 2
2
.
L (Rn )
Remark 5.3. Let us show that the following summand in the Theorem 5.1 is actually real-valued. Indeed, using integration by parts, we calculate Z n Z X f (e x′ , t)(xj ∂yj − yj ∂xj )(∂t f (e x′ , t)) ′ f (e x′ , t)ZT f (e x′ , t) ′ de x dt = de x dt |e x′ |2 |e x′ |2 Hn j=1 Hn n
1X =− 2 j=1
Z
Hn
n
1X = 2 j=1 n
1X = 2 j=1
Z
′ ∂t f (e x′ , t) ∂yj (f (e x′ , t))∂xj (log |e x′ |2 ) − ∂xj (f (e x′ , t))∂yj (log |e x′ |2 ) de x dt x′ , t) − ∂xj (f (e x′ , t))∂tyj f (e x′ , t))de x′ dt log |e x′ |2 (∂yj (f (e x′ , t))∂txj f (e
Hn
Z
log |e x′ |2 (∂yj (f (e x′ , t))∂txj f (e x′ , t) + ∂yj (f (e x′ , t))∂txj f (e x′ , t))de x′ dt Hn
=
n Z X j=1
log |e x′ |2 Re(∂yj (f (e x′ , t))∂txj f (e x′ , t))de x′ dt.
Hn
Remark 5.4. We note that we can use that ZT = T Z and integrate by parts in t, so that the term Z f (e x′ , t)ZT f (e x′ , t) ′ Zf T f de x dt = − , ′ |e x′ |2 |e x′ | |e x | L2 (Hn ) Hn can be seen as an interaction between the central and the tangential derivatives. This formula also implies (1.12)-(1.15) using (5.5)-(5.8), respectively. Writing (1.11) in the form ZT = L − ∆xe′ −
|e x′ |2 2 T , 4
(5.10)
FACTORIZATIONS ON STRATIFIED GROUPS
27
we have Z
Hn
f (x)ZT f (x) ′ de x dt |e x′ |2 ∆xe′ f f 1 Lf f − + kT f k2L2 (Hn ) . (5.11) = , ′ , ′ ′ ′ |e x | |e x | L2 (Hn ) |e x | |e x | L2 (Hn ) 4
Consequently, inequalities (5.5) and (5.6) can be also written as kLf k2L2 (Hn )
∇H f 2
≥ ((N − 4)α − 2β)
|e x′ | L2 (Hn )
2
′
x e · ∇H f
− α(α − 4) |e x′ |2 L2 (Hn ) (5.12)
f 2
+ β((N − 4)(α − 2) − β)
|e x′ |2 L2 (Hn ) Lf f α ∆xe′ f f − 2α + 2α + kT f k2L2(Hn ) , , ′ , ′ ′ ′ |e x | |e x | L2 (Hn ) |e x | |e x | L2 (Hn ) 2
and kLf k2L2 (Hn )
∇H f 2
≥ (Nα − 2β)
|e x′ | L2 (Hn )
′
2
x e · ∇H f
− α(α + 4) |e x′ |2 L2 (Hn )
f 2
+ (2(N − 4)(α(N − 2) − β) − 2α (N − 2) + αβN − β )
|e x′ |2 L2 (Hn ) Lf f ∆xe′ f f α + 2α − 2α − , , kT f k2L2(Hn ) , |e x′ | |e x′ | L2 (Hn ) |e x′ | |e x′ | L2 (Hn ) 2 (5.13) 2
2
respectively. Proof of Theorem 5.1. We will be applying the factorization method with the following differential expressions for two real parameters α, β ∈ R, Tα,β := −L + α and its formal adjoint + Tα,β := −L − α
β x e′ · ∇H + |e x′ |2 |e x′ |2
x e′ · ∇H α(N − 2) − β − , ′ 2 |e x| |e x′ |2
(5.14)
(5.15)
28
M. RUZHANSKY AND N. YESSIRKEGENOV
where x e′ 6= 0. Then, by a direct calculation for any function f ∈ C0∞ (Hn \{e x′ = 0}) we have + (Tα,β Tα,β f )(x) α(N − 2) − β x e′ · (∇H f ) βf (x) x e′ · ∇H − −(Lf )(x) + α + = −L − α |e x′ |2 |e x′ |2 |e x′ |2 |e x′ |2 ′ x e · (∇H f ) x e′ · ∇H N −2 2 = (L f )(x) + α −L (x) + (Lf )(x) + (Lf )(x) |e x′ |2 |e x′ |2 |e x′ |2 f (Lf )(x) + β −L (x) − |e x′ |2 |e x′ |2 ′ f x e′ · (∇H f )(x) (N − 2)f (x) x e · ∇H (x) + − + αβ − |e x′ |2 |e x′ |2 |e x′ |4 |e x′ |4 ′ ′ x e · ∇H x e′ · (∇H f )(x) f (x) x e · (∇H f ) 2 +α − (x) − (N − 2) + β2 ′ 4 ′ 2 ′ 2 ′ 4 |e x| |e x| |e x| |e x| =: (L2 f )(x) + αJ1 + βJ2 + αβJ3 + α2 J4 + β 2 J5 . (5.16) In order to simplify this, let us first calculate J1 : ′ x e · (∇H f ) x e′ · ∇H N −2 J1 = −L (x) + (Lf )(x) + (Lf )(x) ′ 2 ′ 2 |e x| |e x| |e x′ |2 ! PN ′ e N X x e′ · ∇H N −2 ek (Xk f ) 2 k=1 x e (x) + (Lf )(x) + (Lf )(x) =− Xj ′ 2 ′ 2 |e x| |e x| |e x′ |2 j=1 ! P N N ′ e e X ej f ) + N x 2e x′j X e ( X X f ) ( X j k k=1 k ej − ek f )(x) + =− X x e′k (X (x) ′ |4 |e x |e x′ |2 j=1 k=1 +
x e′ · ∇H N −2 (Lf )(x) + (Lf )(x) ′ 2 |e x| |e x′ |2
N N X 8(e x′j )2 X ek f )(x) + 2 ek f )(x) x e′k (X x e′k (X = − ′6 ′ |4 |e x | |e x j=1 k=1 k=1 ! N N X X 2e x′j ej f )(x) + ej X ek f )(x) + (X x e′k (X ′ |4 |e x j=1 k=1 PN ′ ′ e e e N ( X f )(x) + x e ( X X f )(x) 2e x X j j j k k=1 k + |e x′ |4 j=1 N X
−
N ′ e2 e e 2 f )(x) + PN x X (2X j k=1 ek (Xj Xk f )(x) j=1
+
|e x′ |2
x e′ · ∇H N −2 (Lf )(x) + (Lf )(x) ′ 2 |e x| |e x′ |2
!
FACTORIZATIONS ON STRATIFIED GROUPS
29
N 2N − 4 ′ N −4 4 X ′ ′ e e x e x e ( X X f )(x) + x e · (∇ f )(x) + (Lf )(x) j k H j k |e x′ |4 |e x′ |4 j,k=1 |e x′ |2 PN ′ ek f ) ek L(X x e′ · ∇H k=1 x (x) + (Lf )(x). (5.17) − |e x′ |2 |e x′ |2 Here using two times the formulae ej , X ek ] = T, [X ek , X ej ] = −T, k = n + j, j = 1, . . . , n, [X
=
for the last two summands in the above equality and noticing that all other commutators are zero, distinguishing between the cases k = j + n and j = k + n in the sums below we get PN ′ ek f ) x e L(X x e′ · ∇H − k=1 k′ 2 (x) + (Lf )(x) |e x| |e x′ |2 N X x e′k ek X ej X ej f )(x) − (X ej X ej X ek f )(x)) ((X = ′ |2 |e x j,k=1 =
n X x e′j+n j=1
+
n X k=1
=
|e x′ |2
ej+n X ej X ej f )(x) − (X ej T f )(x) − (X ej X ej+n X ej f )(x)) ((X
x e′k ek X ek+n X ek+n f )(x) + (X ek+n T f )(x) − (X ek+n X ek X ek+n f )(x)) ((X |e x′ |2
n X x e′j+n j=1
|e x′ |2
ej+n X ej X ej f )(x) − 2(X ej T f )(x) − (X ej+n X ej X ej f )(x)) ((X
n X x e′k ek X ek+n X ek+n f )(x) + 2(X ek+n T f )(x) − (X ek X ek+n X ek+n f )(x)) ((X + |e x′ |2 k=1
=2
n X (xj Yj − yj Xj )T f (x) j=1
|e x′ |2
,
where in the last line we use the usual notation (1.7) for variables and vector fields on the Heisenberg group. Putting this in (5.17), we obtain J1 =
N 2N − 4 ′ 4 X ′ ′ e e x ex e (Xj Xk f )(x) x e · (∇ f )(x) + H |e x′ |4 |e x′ |4 j,k=1 j k
n X N −4 (xj Yj − yj Xj )T f (x) + (Lf )(x) + 2 . ′ 2 |e x| |e x′ |2 j=1
Now for J2 we have N X f (Lf )(x) (Lf )(x) f 2 Xj (x) − =− (x) − J2 = −L ′ 2 ′ 2 ′ 2 |e x| |e x| |e x| |e x′ |2 j=1 ! N X ej f 2e x′j f X (Lf )(x) ej =− X − ′ 4 (x) − ′ 2 |e x| |e x| |e x′ |2 j=1
(5.18)
30
M. RUZHANSKY AND N. YESSIRKEGENOV
=−
N X j=1
e 2 f )(x) 4e ej f )(x) 8(e (X x′j (X x′j )2 f (x) 2f (x) j − + − ′4 |e x′ |2 |e x′ |4 |e x′ |6 |e x| =−
!
−
(Lf )(x) |e x′ |2
x′ · (∇H f )(x) f (x) 2(Lf )(x) 4e + + (2N − 8) . |e x′ |2 |e x′ |4 |e x′ |4
For J3 , one calculates x e′ · ∇H J3 = − |e x′ |2
f |e x′ |2
(x) +
(5.19)
x e′ · (∇H f )(x) (N − 2)f (x) − |e x′ |4 |e x′ |4 N
(N − 2)f (x) f (x) X ′ e (N − 2)f (x) f (x) ′ x · ∇H )(|e x′ |−2 ) − = − x ej Xj (|e x′ |−2 ) − = − ′ 2 (e ′ 4 ′ 2 |e x| |e x| |e x | j=1 |e x′ |4 =2
N X (e x′j )2 f (x) j=1
|e x′ |6
−
(N − 2)f (x) (N − 4)f (x) =− . ′ 4 |e x| |e x′ |4
(5.20)
By a direct calculation, we have for J4 ′ ′ x e · ∇H x e′ · (∇H f )(x) x e · (∇H f ) J4 = − (x) − (N − 2) |e x′ |2 |e x′ |2 |e x′ |4 PN N ej )(e ek f ))(x) X ek f )(x) x′j X x′k (X e′k (X j,k=1 (e ′ ′ −3 e ′ x − x e (−2)|e x | X |e x | =− j j ′ 4 |e x| |e x′ |2 j,k=1
x e′ · (∇H f )(x) |e x′ |4 (5.21) PN PN ′ ′ ′ e ek f )(x) 2 PN x ′ e ek f )(x) x e x e X ( X x e ( X e ( X f )(x) j j k k j,k=1 k=1 k = − k=1 k ′ 4 − + |e x| |e x′ |4 |e x′ |4 x e′ · (∇H f )(x) − (N − 2) |e x′ |4 PN ej X ek f )(x) e′j x e′k (X x e′ · (∇H f )(x) j,k=1 x − . = −(N − 3) |e x′ |4 |e x′ |4 − (N − 2)
Now putting (5.18)-(5.21) in (5.16), we obtain
(Lf )(x) |e x′ |2 x e′ · (∇H f )(x) + (2(N − 2)α + 4β − (N − 3)α2 ) |e x′ |4 f (x) + (β 2 + 2(N − 4)β − (N − 4)αβ) ′ 4 |e x| PN ′ ′ e e ej x ek (Xj Xk f )(x) j,k=1 x − α(α − 4) |e x′ |4 n X (xj Yj − yj Xj )T f (x) + 2α . |e x′ |2 j=1
+ (Tα,β Tα,β f )(x) =(L2 f )(x) − (2β − (N − 4)α)
(5.22)
FACTORIZATIONS ON STRATIFIED GROUPS
31
+ Now using the nonnegativity of Tα,β Tα,β and integrating by parts, we get Z Z + 2 f (x)(Tα,β Tα,β f )(x)dx ≥ 0, |(Tα,β f )(x)| dx = Hn
Hn
wherefor brevity we can write dx for the integral on the Heisenberg group Hn . Putting (5.22) into this inequality, one calculates Z Z f (x)(Lf )(x) 2 |(Lf )(x)| dx − (2β − (N − 4)α) dx |e x′ |2 Hn Hn Z f (x)(e x′ · (∇H f )(x)) 2 +(2(N − 2)α + 4β − (N − 3)α ) dx |e x′ |4 Hn Z |f (x)|2 2 dx +(β + 2(N − 4)β − (N − 4)αβ) (5.23) x′ |4 Hn |e N Z ej X ek f )(x) X f (x)e x′j x e′k (X −α(α − 4) dx ′ |4 |e x H n j,k=1 Z n X f (x)(xj Yj − yj Xj )T f (x) +2α dx ≥ 0. |e x′ |2 j=1 Hn
Let us analyse the last term in this inequality. Using formula (1.7), we have xj Yj − yj Xj = xj (∂yj +
x2j + yj2 xj yj ∂t ) − yj (∂xj − ∂t ) = xj ∂yj − yj ∂xj + ∂t . (5.24) 2 2 2
Consequently, we have n X
(xj Yj − yj Xj ) = Z +
j=1
|e x′ |2 T, 2
(5.25)
with the tangential derivative Z defined in (1.11) and T = ∂t . For the other terms in (5.23), using (2.9), we have Z N Z X ej f )(x)(X ej f )(x) f (x)(Lf )(x) (X dx = − |e x′ |2 |e x′ |2 Hn j=1 Hn −
N Z X
Hn
j=1
=2
Z
Hn
and
= −(N − 1)
N Z X k=1
f (x)(e x′ · (∇H f )(x)) dx − |e x′ |4
N Z X
j,k=1
Hn
ej |e ej f )(x)dx f (x)(−2)|e x′ |−3 X x′ |(X
Hn
Z
Hn
ej X ek f )(x) f (x)e x′j x e′k (X dx |e x′ |4
|(∇H f )(x)|2 dx |e x′ |2
N Z X ek f )(x) ek f )(x) f (x)e x′k (X f (x)e x′k (X dx − 2 dx ′ |4 |e x′ |4 |e x H n k=1
(5.26)
32
−
M. RUZHANSKY AND N. YESSIRKEGENOV N Z ej f )(x)(X ek f )(x) X x e′j x e′k (X ek f )(x)(−4)|e ej |e dx − f (x)e x′j x e′k (X x′ |−5 X x′ |dx |e x′ |4 Hn
N Z X
j,k=1
Hn
j,k=1
= −(N + 1)
N Z X k=1
−
Z
Hn
Hn
N Z ek f )(x) X ek f )(x) f (x)(e x′j )2 x e′k (X f (x)e x′k (X dx + 4 dx |e x′ |4 |e x′ |6 Hn j,k=1
Z |e x′ · (∇H f )(x)|2 f (x)(e x′ · (∇H f )(x)) dx = −(N − 3) dx |e x′ |4 |e x′ |4 Hn Z |e x′ · (∇H f )(x)|2 − dx. |e x′ |4 Hn
(5.27)
Putting (5.25), (5.26) and (5.27) in (5.23), we obtain Z |(Lf )(x)|2 dx + (2α(N − 4) − 4β + α(α − 4)(N − 3) − α2 (N − 3) + 2α(N − 2) + 4β) Hn
×
Z
Hn
f (x)(e x′ · (∇H f )(x)) dx + (β 2 + 2(N − 4)β − (N − 4)αβ) |e x′ |4 Z |(∇H f )(x)|2 −(α(N − 4) − 2β) dx |e x′ |2 Hn Z |e x′ · (∇H f )(x)|2 +α(α − 4) dx |e x′ |4 Hn ′ 2 Z f (x)(Z + |ex2| T )T f (x) dx ≥ 0, +2α |e x′ |2 Hn
Z
Hn
|f (x)|2 dx |e x′ |4
which implies (5.5). Then, we can also obtain (5.7) from (5.5) using the CauchySchwarz inequality (4.2) for α(α − 4) ≥ 0 in the last estimate. + Thus, we have obtained (5.5) and (5.7) using the nonnegativity of Tα,β Tα,β . Now + let us show (5.6) and (5.8) using the nonnegativity of Tα,β Tα,β . By writing (4.13) on + Heisenberg group and subtracting the expression Tα,β Tα,β in (5.22) from this, we get + (Tα,β Tα,β f )(x)
= (L2 f )(x) − (2β − Nα)
(Lf )(x) |e x′ |2
+ (6(2 − N)α + 4β − (N − 3)α2 )
x e′ · (∇H f )(x) |e x′ |4
+ (β 2 + 2(N − 4)β − Nαβ + 2α2 (N − 2) − 2α(N − 4)(N − 2)) − α(α + 4) − 2α
PN
ej X ek f )(x) e′j x e′k (X j,k=1 x |e x′ |4
n X (xj Yj − yj Xj )T f (x) j=1
|e x′ |2
.
f (x) |e x′ |4
(5.28)
FACTORIZATIONS ON STRATIFIED GROUPS
33
+ Then using the nonnegativity of Tα,β Tα,β and integrating by parts, one has Z Z + f (x)(Tα,β Tα,β f )(x)dx ≥ 0. |(Tα,β f )(x)|2 dx = Hn
Hn
Putting (5.28) into this inequality, one gets Z Z f (x)(Lf )(x) 2 dx |(Lf )(x)| dx − (2β − Nα) |e x′ |2 Hn Hn Z f (x)(e x′ · (∇H f )(x)) 2 + (6(2 − N)α + 4β − (N − 3)α ) dx |e x′ |4 Hn 2
2
+ (β + 2(N − 4)β − Nαβ + 2α (N − 2) − 2α(N − 4)(N − 2)) − α(α + 4) − 2α
n Z X j=1
N X
j,k=1
Hn
Z
Hn
Z
Hn
ej X ek f )(x) f (x)e x′j x e′k (X dx |e x′ |4
|f (x)|2 dx |e x′ |4
f (x)(xj Yj − yj Xj )T f (x) dx ≥ 0. |e x′ |2 (5.29)
+ Then, similarly as for Tα,β Tα,β , i.e. putting (5.25), (5.26) and (5.27) in (5.29), we obtain (5.6) and (5.8).
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Michael Ruzhansky: Department of Mathematics Imperial College London 180 Queen’s Gate, London SW7 2AZ United Kingdom E-mail address
[email protected] Nurgissa Yessirkegenov: Institute of Mathematics and Mathematical Modelling 125 Pushkin str. 050010 Almaty Kazakhstan and Department of Mathematics Imperial College London 180 Queen’s Gate, London SW7 2AZ United Kingdom E-mail address
[email protected]
