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Advances in Nonlinear Analysis

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Weighted Caffarelli–Kohn–Nirenberg type inequalities related to Grushin type operators

Manli Song
  • School of Natural and Applied Sciences, Northwestern Polytechnical University, Xi’an, Shaanxi 710129, P. R. China
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/ Wenjuan Li
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  • School of Natural and Applied Sciences, Northwestern Polytechnical University, Xi’an, Shaanxi 710129, P. R. China
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Published Online: 2016-12-20 | DOI: https://doi.org/10.1515/anona-2016-0146


We consider the Grushin type operator on xd×yk of the form


and derive weighted Hardy–Sobolev type inequalities and weighted Caffarelli–Kohn–Nirenberg type inequalities related to Gμ.

Keywords: Grushin type operator; weighted Hardy–Sobolev inequality; weighted Caffarelli–Kohn–Nirenberg type inequality

MSC 2010: 26D10; 35H20

1 Introduction

Hardy–Sobolev inequalities and Caffarelli–Kohn–Nirenberg inequalities on the Euclidean space play an important role in mathematics and applied fields. They are very useful tools to study various interesting problems in partial differential equations, such as eigenvalue problems, existence problems of equations with singular weights, regularity problems, etc.

The initial work of first order interpolation inequalities with weights (Caffarelli–Kohn–Nirenberg inequalities) was given by Caffarelli, Kohn and Nirenberg [2]. The result is stated as follows.

Theorem 1.

Let p, q, r, α, β, γ, σ and a satisfy


where γ=aσ+(1-a)β. Then there exists a positive constant C such that for all uC0(Rn) the inequality


holds if and only if the following relations hold:

1r+γn=a(1p+α-1n)+(1-a)(1q+βn)(this is dimensional balance),(1.3)


0α-σif a>0α-σ1if a>0 and 1p+α-1n=1r+γn.

Furthermore, on any compact set in the parameter space in which (1.1), (1.3) and 0α-σ1 hold, the constant C is bounded.

When a=1, we see that (1.2) are reduced to Hardy–Sobolev inequalities, i.e., Hardy–Sobolev inequalities are special cases of Caffarelli–Kohn–Nirenberg inequalities. Later, Lin [13] generalized (1.2) to cases including derivatives of any order. Badiale and Tarantello [1] derived a class of more general Hardy–Sobolev inequalities with singular weights depending only on partial variables. Recently, Hardy–Sobolev type inequalities have been extended to noncommutative field vectors. In the Heisenberg group setting, we refer the readers to [3, 7, 8, 9, 10], etc.

In this paper, we shall prove weighted Hardy–Sobolev type inequalities and weighted Caffarelli–Kohn–Nirenberg type inequalities related to the Grushin type operator


where xd and yk. Let Q=d+(1+μ)k be the homogeneous dimension, let


be the distance function from the origin to (x,y) on xd×yk and let μ=(x,|x|μy) be the gradient operator. We have Gμ=μμ.

For the Grushin type operator, D’Ambrosio [4] proved some Hardy type inequalities and gave sharp estimates in some cases. Here, we recall a result from [4].

Theorem 2.

Let p>1 and αR satisfy 1p+αQ>1Q. Then there exists a positive constant C=(pQ-p+αp)p such that for any uDα1,p(Rd+k) we have


where Dα1,p(Rd+k) denotes the closure of C0(Rd+k) with respect to the norm


Niu and Dou [14] established Hardy–Sobolev inequalities related to Gμ. Zhang et al. [15] obtained a class of weighted Hardy–Sobolev inequalities and a class of weighted Caffarelli–Kohn–Nirenberg inequalities in the special case p=2. In the sequel, let p*(s,p,Q)=p(Q-s)Q-p for any 1<p<Q. The weighted Hardy–Sobolev inequalities are listed as follows.

Theorem 3 (see [15]).

If 0s2<Q, α>2-Q2, there exists a positive constant C=C(s,α,μ,Q) such that for any uDα1,2(Rd+k) we have


The weighted Caffarelli–Kohn–Nirenberg inequalities related to Gμ for the case p=2 are given by the following theorem.

Theorem 4 (see [15]).

Let q, r, α, β, γ, σ and a satisfy


where γ=aσ+(1-a)β. Then there exists a positive constant C such that for all uC0(Rd+k) the inequality


holds if and only if the following relations hold:

1r+γQ=a(12+α-1Q)+(1-a)(1q+βQ)(this is dimensional balance),


0α-σif a>0α-σ1if a>0 and 12+α-1Q=1r+γQ.

In this paper, we are going to establish the following weighted Caffarelli–Kohn–Nirenberg inequalities related to Gμ for 1<p<Q.

Theorem 5.

Let p, q, r, α, β, γ, σ and a satisfy




Then there exists a positive constant C such that for all uC0(Rd+k) the inequality


holds if and only if the following relations hold:

1r+γQ=a(1p+α-1Q)+(1-a)(1q+βQ)(this is dimensional balance),(1.7)


0α-σif a>0(1.8)α-σ1if a>0 and 1p+α-1Q=1r+γQ.(1.9)

To prove Theorem 5 by employing the idea in [7, 15], we first need to obtain a class of weighted Hardy–Sobolev type inequalities for 1<p<Q.

Theorem 6.

If 1<p<Q, 0sp and α>p-Qp, there exists a positive constant C=C(s,p,α,μ,Q) such that for any uDα1,p(Rd+k) we have


Remark 7.

When a=1, the conditions of Theorem 5 imply


and then prp*=QpQ-p. Therefore, there exists t[0,1] satisfying


Replacing r and α-σ into (1.6), we easily see that (1.6) is reduced to (1.10), which is exactly a weighted Hardy–Sobolev type inequality.

Remark 8.

Since the methods based on radial symmetry in [2] are no longer suitable for Gμ, Zhang, Han and Dou [15] adopted a different idea. They first proved weighted Hardy–Sobolev type inequalities related to Gμ for the case p=2 and then derived the associated weighted Caffarelli–Kohn–Nirenberg inequalities for the special case. Inspired by their work, we extended their results to all cases 1<p<Q. However, it is still open for the cases pQ.

This paper is organized as follows: Section 2 introduces some definitions and basic facts related to Gμ. In Section 3, we establish a Sobolev–Stein embedding theorem and Hardy–Sobolev type inequalities related to Gμ. Furthermore, we prove Theorem 6. Section 4 is devoted to the proof of Theorem 5.

2 Preliminary

We shall introduce some notions and basic facts about the Grushin type operators. Let μ be a positive real number and (x,y)xd×yk=d+k with d,k1. We denote by |x| (resp. |y|) the Euclidean norm in d (resp. k), i.e., |x|2=i=1dxi2 (resp. |y|2=j=1kyj2). The symbol x (resp. y) and Δx (resp. Δy) stand respectively for the usual gradient operator and the Laplace operator on d (resp. k).

The Grushin type vector fields are defined by


and the corresponding gradient operator and divergent operator are denoted respectively by


Denote the Grushin type operator by


A family of dilations {δλ:λ>0} on d+k is defined by


and Q=d+(1+μ)k is the corresponding homogeneous dimension. It is easy to see that the vector fields Xi and Yj are homogeneous of degree one with respect to the dilation, i.e., Xi(δλ)=λδλ(Xi), Yj(δλ)=λδλ(Yj), and hence μ(δλ)=λδλ(μ) and Gμ(δλ)=λ2δλ(Gμ).

The distance function from the origin to (x,y) on d+k is defined by


It is not difficult to check that ρ is homogeneous of degree one with respect to δλ and


Furthermore, Γ=Cμρ2-Q is the fundamental solution at the origin of Gμ (see [5]).

Denote the open ball of radius R centered at the origin by


Recalling the explicit polar transform defined by D’Ambrosio [3], we have


where dσ=(11+μ)k|sinθ|d2-1|cosθ|k-1dθdωddωk, and ωd and ωk denote the usual surface measures on d and k, respectively. In addition, the criteria for the integrability of |x|pρq was given as follows:

  • (i)

    If p+d>0 and p+q+Q>0, then


  • (ii)

    If p+d>0 and p+q+Q<0, then


3 Proof of Theorem 6

Firstly, we need to prove the Sobolev–Stein embedding result related to Grushin type operators.

Theorem 1.

If 1<p<Q, there exists a positive constant C=C(p,μ,Q) such that for any uD01,p(Rd+k) we have


Next, we shall prove the associated Hardy–Sobolev type inequalities.

Theorem 2.

If 1<p<Q, 0sp, there exists a positive constant C=C(s,p,μ,Q) such that for any uD01,p(Rd+k) we have


In order to prove Theorem 1, we consider the fractional integral operator


The Hardy–Littlewood–Sobolev theorem for Iν holds (see [6]).

Theorem 3.

Let 0<ν<Q and 1p<Qν. Then we have the following:

  • (i)

    If 1<p<Q , then the condition 1p-1q=νQ is necessary and sufficient for the boundedness of Iν from Lp(d+k) to Lq(d+k).

  • (ii)

    If p=1 , then the condition 1-1q=νQ is necessary and sufficient for the boundedness of Iν from L1(d+k) to Lq,(d+k) , where Lq, denotes the weak Lq space.

Proof of Theorem 1.

For any uC0(d+k), using the integral representation formula for the fundamental solution of Gμ, we have


Noting Gμ=μμ and μ*=-μ, and integrating by parts on the right-hand side of (3.1), we obtain




we obtain


Now, the application of Theorem 3 yields




and C is a suitable positive constant depending only on p, μ and Q. ∎

Proof of Theorem 2.

Recall p*(s,p,Q)=p(Q-s)Q-p, where 1<p<Q and 0sp. If s=0, then we have p*(p,p,Q)=p and Theorem 2 is reduced to Theorem 1. If s=p, then p*(0,p,Q)=p* and Theorem 2 is reduced to Theorem 2 in the case α=0. Therefore, it suffices to deal with the case 0<s<p.

Denoting p*(s,p,Q)=(1-sp)p*+s, by the Hölder inequality, (1.4) in the case α=0 and Theorem 1, we have




To prove Theorem 6, we shall introduce two results.

Lemma 4 (see [11]).

Let p1. For all ξ1,ξ2Rn the following inequalities hold:

  • (i)

    If p2 , then


  • (ii)

    If p>2 , then


    where , represents the common inner product on the Euclidean space n.

Lemma 5 (see [12]).

For any ξ,ηRn and λ>0, we have


and the equality holds if and only if ξ=η.

Proof of Theorem 6.

The condition α>p-Qp implies


which ensures that the left and right integral of (1.10) are well defined on C0(d+k). For any uDα1,p(d+k), taking w=ραu, by (2.1), we have


It follows from (1.4) that


which implies wD01,p(d+k). In addition, a straightforward computation gives


Therefore, (1.10) is equivalent to the following inequality:


By Theorem 2, it suffices to prove


for some suitable constant C>0.

According to Lemma 4, we will investigate (3.2) in the case 1<p2 and p>2, respectively.

Case 1: 1<p2. Taking ξ1=αρ-1wμρ and ξ2=μw-αρ-1wμρ in the first case of Lemma 4, we obtain


Integrating by parts, we have




by (2.1), a straightforward computation implies


Putting (3.4) and (3.5) into (3.3), we have


Note that the condition α>p-Qp implies that Q-p+(p-1)α>0.

If α0, it follows from (1.4) and (3.6) that




Taking ξ=p-Qp, η=α and λ=p-1>0 in Lemma 5, we obtain C1>0.

If α>0, then (3.7) holds naturally with C1=1.

In conclusion, we prove (3.2) with C=C(p)-1C1>0.

Case 2: p>2. A direct calculation gives


As in Case 1, applying the estimate in the second case of Lemma 4 to the above inequality, we deduce


Therefore, argued as in Case 1,


holds for a suitable C1>0.

In addition, exploiting the Hölder inequality and the Minkowski inequality, we deduce


We conclude from (1.4)


Hence, by (3.9) and (3.10),


Combining (3.8) and (3.11), we deduce




In conclusion, (3.2) is proved. ∎

4 Proof of Theorem 5

4.1 Necessity

Necessity of (1.7). Let 0uC0(d+k) satisfy (1.6). Then uλ=uδλ(λ>0) also satisfies (1.6). A direct computation shows


Applying (1.6) to uλ, we deduce


which is true for any λ>0, so the powers of λ on the two sides must be equal, i.e.,


which is exactly (1.7).

Necessity of (1.8). Let 0uC0(B1) satisfy (1.6). Take (x0,y0)d+k, x00 and for sufficiently large λ>0 define






and further


applying (1.6) to uλ, we have


This yields γ=aσ+(1-a)βaα+(1-a)β, namely (1.8).

Necessity of (1.9). We conclude from (1.7) that


Choose the function

uε={0for ρ1,ρ-γ-Qrlog1ρfor ερ1,ε-γ-Qrlog1εfor ρε.

In polar coordinates, this leads to




and further


Applying (1.6) to uε, we have


which implies that


This immediately leads to


The combination of (1.5), (1.7) and (4.1) yields (1.9).

4.2 Sufficiency

If a=0, inequality (1.6) obviously holds true. If a=1, the proof is complete by Remark 7. In the sequel, we deal only with the case 0<a<1.

Case (I): 0<a<1, 0α-σ1. In this case, we have p(1p+α-σ-1Q)-1p*. Analogous to the argument in Remark 7, there exists t[0,1] satisfying


Applying (1.10) with 0s=tpp, we obtain


From (1.5) and (1.7) we have


By the Hölder inequality and (4.2), there follows


Case (II): 0<a<1, α-σ>1. Putting


we see that (1.6) can be written as


Rescaling u such that AaB1-a=1, we aim to prove


Note that in Case (I), inequality (1.6) has been proved for α-σ=0 and α-σ=1. Therefore,


provided that δ, s, ε and t satisfy


for some choices of b and c, 0b,c1 and


One computes that


Since 0<a<1, α-σ>1, this ensures 1p+α-1Q1r+γQ from (1.9) and then 1p+α-1Q1q+βQ.

Case (II.1): 1p+α-1Q<1q+βQ. Taking b<a<c, we have


A direct computation shows




The conditions 0<a<1 and α-σ>1 imply


and for sufficiently small |b-a| and |a-c|,


Combining (4.4) and (4.5), we have


Choose a fixed C0(d+k) function Φ(x,y) (0Φ1) such that

Φ(x,y)={1if ρ(x,y)<1,0if ρ(x,y)>2.

We shall investigate the left-hand side of (4.3) by splitting it into two parts. One obtains by the Hölder inequality that




Moreover, (4.6) ensures that the integrals on the right-hand side in (4.7) and (4.8) are bounded, which easily leads to (4.3).

Case (II.2): 1p+α-1Q>1q+βQ. Take c<a<b such that |c-a| and |a-b| are sufficiently small. Now inequalities (4.4)–(4.8) still hold true, and then the desired result (4.3) is derived.


The authors deeply thank Professor Pengcheng Niu for his suggestions and encouragement on the paper.


  • [1]

    M. Badiale and G. Tarantello, A Sobolev inequality with applications to nonlinear elliptic equation arising in astrophysics, Arch. Ration. Mech. Anal. 163 (2002), 259–293.  CrossrefGoogle Scholar

  • [2]

    L. Cafferelli, R. Kohn and G. Nirenberg, First order interpolation inequalities with weights, Compos. Math. 53 (1984), 259–275.  Google Scholar

  • [3]

    L. D’Ambrosio, Some Hardy inequalities on the Heisenberg group, Manuscripta Math. 106 (2001), 519–536.  CrossrefGoogle Scholar

  • [4]

    L. D’Ambrosio, Hardy inequalities related to Grushin type operators, Proc. Amer. Math. Soc. 132 (2004), no. 3, 725–734.  CrossrefGoogle Scholar

  • [5]

    L. D’Ambrosio and S. Lucente, Nonlinear Liouville theorems for Grushin and Tricomi Operators, J. Differential Equations 193 (2003), no. 2, 511–541.  CrossrefGoogle Scholar

  • [6]

    G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups, Math. Notes 28, Princeton University Press, Princeton, 1982.  Google Scholar

  • [7]

    Y. Han, Weighted Caffarelli–Kohn–Nirenberg type inequality on the Heisenberg group, Indian J. Pure Appl. Math. 46 (2015), no. 2, 147–161.  CrossrefWeb of ScienceGoogle Scholar

  • [8]

    Y. Han and P. Niu, Hardy–Sobolev type inequalities on the H-type group, Manuscripta Math. 118 (2005), 235–252.  CrossrefGoogle Scholar

  • [9]

    Y. Han, P. Niu and S. Zhang, On first order interpolation inequalities with weights on the Heisenberg group, Acta Math. Sin. (Engl. Ser.) 27 (2011), no. 12, 2493–2506.  CrossrefGoogle Scholar

  • [10]

    Y. Han, S. Zhang and J. Dou, On first order interpolation inequalities with weights on the H-type group, Bull. Braz. Math. Soc. (N.S.) 42 (2011), no. 2, 185–202.  CrossrefWeb of ScienceGoogle Scholar

  • [11]

    Y. Jin and Y. Han, Improved Hardy inequality on the Heisenberg group, Acta Math. Sci. Ser. A Chin. Ed. 31 (2011), no. 6, 1591–1600.  Google Scholar

  • [12]

    T. Kusano, J. Jaroš and N. Yoshida, A Picone-type identity and Sturmian comparison and oscillation theorems for a class of half-linear partial differential equations of second order, Nonlinear Anal. 40 (2000), 381–395.  CrossrefGoogle Scholar

  • [13]

    S. Lin, Interpolation inequalities with weights, Comm. Partial Differential Equations 11 (1986), no. 14, 1515–1538.  CrossrefGoogle Scholar

  • [14]

    P. Niu and J. Dou, Hardy–Sobolev type inequalities for generalized Baouendi–Grushin operators, Miskolc Math. Notes 8 (2007), no. 1, 73–77.  CrossrefGoogle Scholar

  • [15]

    S. Zhang, Y. Han and J. Dou, Weighted Hardy–Sobolev type inequality for generalized Baouendi–Grushin vector fields and its application, Adv. Math. (China) 44 (2015), no. 3, 411–420.  Google Scholar

About the article

Received: 2016-06-27

Revised: 2016-09-14

Accepted: 2016-10-02

Published Online: 2016-12-20

Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11371036

Award identifier / Grant number: 11601427

The authors were supported by the National Natural Science Foundation of China (grant no. 11371036 and grant no. 11601427) and the Fundamental Research Funds for the Central Universities (grant no. 3102015ZY068).

Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 130–143, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2016-0146.

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