Hardy inequalities for the fractional powers of the Grushin operator

We establish uncertainty principles and Hardy inequalities for the fractional Grushin operator, which are reduced to those inequalities for the fractional generalized sublaplacian. The key ingredients to obtain them are an explicit integral representation and a ground state representation for the fractional powers of generalized sublaplacian.


1.
Introduction. The uncertainty principle is a fundamental attribute of quantum mechanics. In the case of the Laplacian ∆ = − d k=1 which can be deduced from Hardy inequality by Cauchy-Schwarz inequality. Here the fractional powers of Laplacian ∆ s = −(−∆) s is defined via spectral decomposition by is the Fourier transform. The well-known Hardy inequality of the Laplacian is given by

MANLI SONG AND JINGGANG TAN
where ∇ is the standard gradient operator on R d and the sharp constant C d is given by The inequality (1.1) can be proved equivalent to where the symbol ·, · stands for the inner product in L 2 (R d ). Moreover, Hardy inequality has also been generalized to fractional powers of Laplacian ∆ s . Now we state the Hardy type inequality for the fractional Laplacian ∆ s . For 0 < s < d/2, where the sharp constant C d,s (see [2]) is given as However, it is known the equality is never achieved. Later, Frank, Lieb and Seiringer [6] used a different approach called ground state representation to prove the inequality (1.2) when 0 < s < min{1, d/2}. Hardy's inequality for the fractional powers of Laplacian has been extensively studied, referring the reader to [2,7,19,14]. Recently, inspired by the ideas in Frank, Lieb and Seiringer [6], Hardy type inequalities concerning fractional powers have been developed to the Heisenberg group and the Grushin operator. Roncal and Thangavelu [10] proved analogues of Hardy type inequalities for fractional powers of the sublaplacian on the Heisenberg group and also obtained corresponding versions of Heisenberg uncertainty inequality.
Balhara [1] introduced the fractional powers of the Grushin operator and proved the non-homogeneous Hardy inequality for the fractional powers of the Grushin operator. They found an integral representation and a ground state representation for the fractional powers of generalized sublaplacian, which related to the Grushin operator by the Hecke-Bochner formula.
We are concerned with a Hardy inequality for the fractional powers of the Grushin operator involving a homogeneous weight. We consider the Grushin operator defined on R n+1 (n ≥ 2) by Instead of the fractional powers G s , we will study the related operators G s and Λ s = G −1 1−s G which behave like G s , see Section 2.1 for definitions, and prove the homogeneous Hardy inequality. Since the operators Λ s G −s and G s G −s are bounded on L 2 (R n+1 ), we can deduce the corresponding inequalities for G s from the inequalities for Λ s and G s . Given an appropriate function in the Grushin space, it can be expressed by a form of solid harmonics. By spectral decomposition and Hecke-Bochner formula, the action of the fractional powers of the Grushin operator Λ s can be transformed to the fractional powers of generalized sublaplacian. Then it suffices to prove the homogeneous Hardy inequality involving the fractional powers of generalized sublaplacian. We obtain its fundamental solution and a positive modified kernel that derived from the heat kernel. Here the modified heat kernel is adopted to establish an integral representation and the fundamental solution is used to compute a ground state representation of the fractional powers of generalized sublaplacian. Finally, as a consequence of the Hardy inequalities, we also obtain two versions of Heisenberg uncertainty for the fractional Grushin operator. Now we state our Hardy inequality for Λ s with a homogeneous weight.
Theorem 1.1. For 0 < s < 1, we have for all f ∈ C ∞ 0 (R n+1 ). Since we shall show that the operator V s = Λ s G −s is bounded on L 2 (R n+1 ) and its operator norm is given by the constant, we can immediately deduce a Hardy inequality for the pure fractional power G s .
. We denote by W s,2 (R n+1 ) the Sobolev space consisting of all L 2 functions f such that G s/2 f ∈ L 2 . Therefore, it is a Sobolev space naturally associated to G. Note that an L 2 function f belongs to W s,2 (R n+1 ) if and only if G s/2 f ∈ L 2 . We also give another version of Hardy inequality for G s involving a non-homogeneous weight.
for all f ∈ W s,2 (R n+1 ). Moreover, the inequality is optimal and the equality is It can be shown that the operator U s = G s G −s is bounded and its operar norm is given by for all f ∈ W s,2 (R n+1 ).

MANLI SONG AND JINGGANG TAN
Finally, Heisenberg type uncertainty inequalities for G s and Λ s follow from our Hardy inequalities.
provided 0 < s < n+2 4 and δ > 0. In the smaller range 0 < s < 1, we have where dx and dw are the standard Lebesgue measures. We denote by L p (X, dµ α ) (1 ≤ p ≤ ∞) the classical Lebesgue spaces with respect to the measure dµ α endowed with the norm · p . Let ·, · α stand for the inner product derived by L 2 (X, dµ α ). The generalized sublaplacian L α is defined on X by To establish the Hardy inequality in Theorem 1.1 and 1.3, we turn to demonstrate the reduced corresponding inequalities for the fractional powers of generalized sublaplacian L α , which are stated as follows. For the definition of the following fractional operators L s α , L α,s and Λ α,s , we see Section 2.3. Theorem 1.6. For α ≥ 0 and 0 < s < 1, we have . The desired result comes from the combination of an integral representation, the fundamental solution and a ground state representation for the fractional generalized sublaplacian.
Since the operator V α,s = Λ α,s L −s α is bounded on L 2 (X, dµ α ), we can immediately obtain a Hardy inequality for the pure fractional power L s α . Theorem 1.7. For α ≥ 0 and 0 < s < 1, we have In the same way, we denote by W s,2 (X, dµ α ) the Sobolev space consisting of all L 2 (X, dµ α ) functions f such that L s/2 α f ∈ L 2 (X, dµ α ), which is a Sobolev space naturally associated to L α . Again, we have that an L 2 (X, dµ α ) function f belongs to W s,2 (X, dµ α ) if and only if L α,s/2 f ∈ L 2 (X, dµ α ). Non-homogeneous version of fractional Hardy inequality of L α,s is listed in the following.
Moreover, the inequality is optimal and the equality is Our proof employs the Cowling-Haagerup formula and Schur test, whose approach is different from the homogeneous case in Theorem 1.6. The Schur test lemma plays an important role in the proof and also in Ronal and Thangavelu [10].
Again, it can be shown that the operator U α,s = L α,s L −s α is bounded. From Theorem 1.8, we can immediately obtain a non-homogeneous Hardy inequality for the pure fractional power L α,s . Theorem 1.9. For α ≥ 0, 0 < s < α+2 2 and δ > 0, we have We can also deduce Heisenberg type uncertainty inequalities for L α,s and Λ α,s from our Hardy inequalities. Theorem 1.10. For α ≥ 0 and all functions f ∈ W s,2 (X, dµ α ), we have Lα,sf, f α provided 0 < s < α+2 2 and δ > 0. In the smaller range 0 < s < 1, we have and The paper is organized as follows. In Section 2 the fractional power of the Grushin operator G s and modified fractional power of the Grushin operator are naturally introduced by spectral decomposition on the orthogonal projection responding to the Hermite operator on R n . The related fractional power of the generalized sublaplacian is presented by the Laguerre functions of type α. It also includes the relation of the Laguerre translation operator and the modified heat kernel of generalized sublaplacian. In Section 3 we demonstrate Theorem 1.6 and 1.7 by combing the fundamental solution, the integral representation formula and the ground state representation. We also deduce Theorem 1.8 and 1.9 by employing the Cowling-Haagerup formula and Schur test. Finally, we obtain Theorem 1.10 from Theorem 1.6 and 1.8. In Section 4 by applying Spherical harmonics and Hecker-Bochner formula, we prove Theorem 1.1 and 1.3, then Theorem 1.2, 1.4 and 1.5 follow similarly.

Preliminaries.
2.1. Fractional powers of the Grushin operator on R n+1 (n ≥ 2). The Grushin operator G on R n+1 (n ≥ 2) is defined by where (x,w) ∈ R n+1 equipped with Lebesgue measure and |x| is the Euclidean norm of x. It is well known that G is a self-adjoint positive operator on L 2 (R n+1 ), which is shown to be closely related to the scaled Hermite operators on R n defined as In fact, for any f ∈ L 2 (R n+1 ), let f λ stand for the Fourier transform of f in the variable Applying the operator G to the inverse Fourier transform of f λ (x) in the variable λ On the other hand, we can write the spectral decomposition of H(λ) as where P k (λ) stands for the orthogonal projection of L 2 (R n ) onto the k-th eigenspace corresponding to the eigenvalue (2k + n)|λ| of H(λ). More precisely, for any φ ∈ L 2 (R n ), where for λ ∈ R * and each β ∈ N n , Φ λ β (x) = |λ| n 4 Φ β ( |λ|x), x ∈ R n . Here, Φ β is the normalized Hermite function on R n (see [16,Chapter 1.4]).
Hence, the spectral decomposition of the Grushin operator G is given by Therefore, for any s > 0, a natural way to define fractional powers of the Grushin operator G is by spectral decomposition However, it is convenient to investigate the following modified fractional powers G s defined by i.e., G s corresponds to the spectral multiplier Moreover, the inverse of the operator G s is given by Note that G −1 s = G −s . As in [10], in order to prove a version of Hardy inequality for fractional powers of the Grushin operator with a homogeneous weight function, we do not deal directly with G s but the related operator where dx and dw are the standard Lebesgue measures. We denote by L p (X, dµ α ) (1 ≤ p ≤ ∞) the classical Lebesgue spaces with respect to the measure dµ α endowed with the norm · p . Let ·, · α stand for the inner product derived by L 2 (X, dµ α ). For convenience, we also denote the elements of X by Greek letters ξ = (x, w) and η = (y, v).

2.3.
Fractional powers of generalized sublaplacian. For α ≥ 0, generalized sublaplacian L α on X is defined by which is self-adjoint and positive on L 2 (X, dµ α ). By the following identity for Laguerre polynomials k,λ is the eigenfunction for L α with the eigenvalue (2k + α + 1)|λ|. Thus, for any s > 0, we can naturally define fractional powers of generalized sublaplacian L s α via spectral decomposition by Note that L s α f (α, λ, k) = (2k + α + 1)|λ| sf (α, λ, k). However, it is convenient to work with the following modified fractional powers L α,s defined by which implies that the spectral multiplier of L α,s is Also, the inverse of the operator L α,s is given by Note that L −1 α,s = L α,−s . We shall prove that L α,s has an explicit fundamental solution, which makes it more suitable than L s α , since the fundamental solution of L s α can not be written down explicitly. Nevertheless, L α,s is not very different from L s α . We will see that L α,s = U α,s L s α , where U α,s is a bounded operator on L 2 (X, dµ α ). As [1], in order to prove a version of Hardy inequality with a nonhomogeneous weight function, they do not deal directly with L s α but L α,s . We also need to study the related operator Modified heat kernel for generalized sublaplacian. Generalized sublaplacian L α is a self-adjoint positive operator on L 2 (X, dµ α ), which generates a heat semigroup e −tLα defined by If we define h α,t by the relation . The function h α,t is called the heat kernel associated with L α , which is positive and In addition, we have the explicit expression for h λ For 0 < s < 1 and t > 0, we define the modified heat kernel K α,t,s (x, w) by is an even function in the second variable. We shall state the behavior of this kernel in the following lemma.
Proof. Using (2.3) and (2.4), letting λ go to 0, we have From this, it follows that Then by [12, Lemma 3.1], we have Using the identity (see [9, p. 411 we obtain T η * α 1(ξ) = 1. Consequently, the left side of (2.6) immediately comes out. In addition, since K α,t,s is an even function in the second variable, it can be easily checked that Therefore, which concludes the right side of (2.6).
We need to introduce an auxiliary funtion: for a, b ∈ R + and c ∈ R, According to [4,Proposition 3.6], the function L satisfies the following identity for all a, b ∈ C and λ > 0.
Proof. Though it has been already proved in [1, Proposition 3.1]), we repeat it for the sake of completeness. We start with the generating function identity for the Laguerre functions of type α, valid for |r| < 1 (see [16, 1.4 Therefore, taking r = y y+|λ| (y > 0) and For two appropriate functions f, g on R + , recall the definition of their Laplace transforms by In addition, by the following formula (see [4,Lemma 3.4 and taking a = x 2 2 , it reduces that Since u α,s,δ (x, w) is even in the second variable, the last left integral denotes u λ α,s,δ (x) and the last equality becomes We can rewrite the expansion (3.2) by Plugging this to (3.3), we get where the coefficients are given by In view of (3.1), it can be easily checked that On the other hand, by (2.2) and (3.4), we obtain u α,s,δ (α, λ, k) = Γ(α + 1)Γ(k + 1) So, combining with the spectral multiplier of L α,s , it follows that u α,s,δ (α, λ, k).
Our result immediately comes out.
Proof. From the proof of Lemma 3.1, we have Let δ tend to 0 in the above formula and we get One can easily check that Together with the spectral multiplier of L α,s , it gives that which implies that g α,s is a fundamental solution for the operator L α,s .

3.2.
Integral representations. In this subsection we find an integral representation for the operator Λ α,s . For convenience, we work with Λ α,1−s = L −1 α,s L α and state the results for this operator. In terms of the kernel K α,t,s , we define another kernel K α,s by which is a positive function and will be explicitly calculated. In order to compute the kernel K α,s , we shall exploit several formulas and identities. We state them here together.

2
. Combined with the representation for the associated Legendre function (3.8), it becomes On the other hand, the integral J 2 can be evaluated using (3.9) and taking ν = α−s+2 2 and β = 1. Together with the representation for the Gegenbauer polynomial (3.10), we have Besides, we obtain sin γ = 4w 1+4w 2 because cos γ = 1−4w 2 1+4w 2 . Thus, we get which gives Finally, plugging (3.17) into (3.15), it gives and by (3.12), where the constant c α,s is given by By applying Legendre's duplication formula , and after simplification, we obtain It completes the proof of the Proposition.
Theorem 3.5. Let α ≥ 0 and 0 < s < 1. Then for all f ∈ W 1−s,2 (X, dµ α ), we have Moreover, the following pointwise representation is valid for all f ∈ C ∞ 0 (X), Proof. Use the identity (3.18) and take ν = s − 1, β = 1, which gives that It follows from the above identity that So we have Indeed, for any > 0, consider the integral which converges to 0 as → 0. In view of (3.19) and (3.20), we obtain Hence, by taking µ = 2k + α + 1 and the change of variable t → |λ|t, we get

Multiplying both sides by |λ| α−s+2
πΓ(α+1)f (α, λ, k)φ α k,λ (x), and summing over k, we have We multiply the above identity both sides by e −iλw and integral over λ variable, we obtain Because of (1−s) Γ(s) = 1 |Γ(s−1)| , we have proved the representation Therefore we get Under the assumption that f ∈ C ∞ 0 (X), by using the stratified mean value theorem (see [5]), we can exchange the order of integration, which gives our desired representation.
The next proposition follows from the above Theorem.
Proposition 3.6. Let α ≥ 0 and 0 < s < 1. Then for all f, g ∈ W 1−s,2 (X, dµ α ), we have Proof. Let f, g ∈ C ∞ 0 (X). The integral representation obtained in last proposition gives that Noting that (2.7), by Fubini's theorem, we obtain Therefore, we have By using a density argument as [10, Theorem 5.4], we complete the proposition.
It follows that f, ν α,s,δ converges to On the other hand, since F is supported away from 0, the right hand side of (3.21) converges to Using Proposition 3.6, we have which completes the proof of the theorem.

The Hardy inequalities and uncertainty principle.
Proof of Theorem we can pass to the limite as δ → 0. Since ν α,s,δ converges in the sense of distribution to a constant multiple of g α,s , we obtain our desired inequality.
In order to prove Theorem 1.8, we need to introduce a version of Schur test.
Moreover, we define another operator on L 2 (X, dµ α ) by It can be easily checked that T α,s,δ = N α,s,δ N * α,s,δ , which leads to Noting that L α,s f , we immediately get the inequality in Theorem 1.8 on a dense subspace. On the other hand, if we take f = u α,−s,δ in the theorem, both sides of the inequality equal C α,s,δ u α,s,δ , u α,−s,δ , which proves the optimality of the constant C α,s,δ .
Proof of Theorem 1.7 and 1.9. By estimating the norms of certain bounded operators V α,s and U α,s , we can immediately deduce the Hardy inequalities for L s α . We will deal with the norm of V α,s in detail and in a similar way we can also estimate U α,s . We have V α,s = L −1 α,1−s L α L −s α , which corresponds to the mutiplier 2k + α + 1 2 We shall show that V α,s is bounded on L 2 (X, dµ α ) and more precisely, Indeed, from the formula (see [18,Section 7]) Therefore, if γ ≥ 0 and x > 0, we obtain that , β = 2−s 2 and γ = s 2 , we have 2k + α + 1 2 Similarly, it can be shown that U α,s is bounded on L 2 (X, dµ α ) and where [·] and (·) are the part and the fractional part of a real number respectively. Indeed, since Γ(x + 1) = xΓ(x), x > 0, we obtain which yields the estimate of U α,s .
Proof of Theorem 1.10. Let W(x, w) denote either the homogeneous weight (x 4 + 4w 2 ) s 2 or the non-homogeneous weight ((δ + x 2 2 ) 2 + w 2 ) s . By Cauchy-Schwarz inequality, we obtain By the Hardy inequality in Theorem 1.6 and 1.8, the last integral is bounded by Λ α,s or L α,s times the corresponding constant. Finally, we obtain the uncertainty principle.

4.1.
Spherical harmonics and Kecke-Bochner formula. Above all, we recall some facts about spherical harmonic analysis and solid harmonics. For details, we refer to [11,Chapter 4]. For m ∈ N, let Y m denote the space of spherical harmonics of degree m. Let a m = dim Y m and {Y m,j } am j=1 be the orthonormal basis of Y m . We know that L 2 (S n−1 ) = ⊕ ∞ m=0 Y m and the collection {Y m,j : m ∈ N, j = 1, 2, · · · , a m } forms an orthonormal basis for L 2 (S n−1 ). Corresponding to each spherical harmonic, we define solid harmonics on R n by P m,j (x) = |x| m Y m,j (x/|x|). Let h m denote the space consisting of linear combination of functions of the form g(|x|)P m,j (x), where g is radial such that g(|x|)P m,j (x) ∈ L 2 (R n ). With these definitions, we have L 2 (R n ) = ⊕ ∞ m=0 h m . Moreover, for any f ∈ L 2 (R n+1 ), we have where f m,j (|x|, w) = S n−1 f (|x|ω, w)P m,j (|x| −1 ω)dσ(ω). By an easy calculation, we obtain Moreover, we recall the Hecke-Bochner formula for the Hermite projection operators (see [15]).

4.2.
Proof of the main theorem.
Proof of Theorem 1.1. Suppose that f ∈ C ∞ c (R n+1 ) has a form of f (x, w) = g(|x|, w) P (x), where P is a solid harmonics of degree m. Applying the spectral decomposition of Λ s and Lemma 4.1, we obtain Thus, by the orthogonality of solid harmonics with respect to inner product inherited from L 2 (S n−1 ), we obtain Λ s f, f = R +∞ k=0 d m,k (s)|A λ m,k (g)| 2 |λ| n/2+m+s dλ.
Since u n/2,−s,δ is the optimizer of the inequality in Theorem 1.8 for α = n/2 − 1, we can see that Therefore, the constant involved in the inequality is sharp and the equality is achieved by f (x, w) = (δ + |x| 2 /2) 2 + w 2 − n/2−s+1 2 .
In a similar way to generalized sublaplacian, we can also prove Theorem 1.2 and 1.4 and 1.5.