He and Yin Journal of Inequalities and Applications (2016) 2016:129 DOI 10.1186/s13660-016-1070-8

RESEARCH

Open Access

Lp Hardy type inequality in the half-space on the H-type group Jianxun He and Mingkai Yin* *

Correspondence: [email protected] School of Mathematics and Information Sciences, Guangzhou University, Guangzhou, 510006, P.R. China

Abstract In the current work we studied Hardy type and Lp Hardy type inequalities in the half-space on the H-type group, where the Hardy inequality in the upper half-space Rn+ was proved by Tidblom in (J. Funct. Anal. 221:482-495, 2005). Keywords: H-type group; Hardy type inequality; Green’s function

1 Introduction In recent years a lot of authors studied the Hardy inequalities (see [–]). They are the extensions of the original inequality by Hardy []. The Heisenberg group, denoted by Hn , is also very popular in mathematics (see [–]). By Hn,+ = {(z, t) ∈ Hn |z ∈ C n , t > } is denoted the half-space on the Heisenberg group. A Hardy type inequality on Hn,+ in [] is stated as follows. For u ∈ C∞ (Hn,+ ), we have 

 |∇Hn u| dz dt ≥ Hn,+

Hn,+

(Q + )(Q – ) |z|  |u| dz dt +  t 

 ρ – |z| |u| dz dt, Hn,+



where ρ = (|z| + t  )  and Q = n +  is the homogeneous dimension of the Heisenberg group. We know that the H-type group, denoted by H = {(z, t) ∈ H|z ∈ C n , t ∈ Rm }, is the nilpotent Lie group introduced by Kaplan (see []). We also know that Hn is a nilpotent Lie group with homogeneous dimension n + . The homogeneous dimension of H is n + m. Kaplan introduced the H-type group as a direct generalization of the Heisenberg group, which motivates us to study the H-type group. In this paper we prove the Hardy type inequality in the half-space on the H-type group (see Theorem .). The half-space on the H-type group is given by H+ = {(z, t) ∈ H|tm > }. For u ∈ C∞ (H+ ), we have  H+

|∇H u| dz dt ≥



|z|  |u| dz dt  H + tm  (Q – )(Q + ) d(z, t)– |z| |u| dz dt +  H+  m–   (k)  tk |u| U z, U (m) z d(z, t)– dz dt, – (Q + ) tm H+

 

k=

© 2016 He and Yin. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

He and Yin Journal of Inequalities and Applications (2016) 2016:129

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

where d(z, t) = (|z| + |t| )  and Q = n + m is the homogeneous dimension of the Htype group. In [], the Lp Hardy inequalities in the upper half-space Rn+ were studied. So we are also interested in the Lp Hardy type inequalities in the half-space on the H-type group. In the remainder of this section we give a basic concept of H-type group and a useful theorem. Let (z, t), (z , t  ) ∈ H, U (j) is a n × n skew-symmetric orthogonal matrix and U (j) satisfy U (i) U (j) + U (j) U (i) = , i, j = , , . . . , m with i = j. The group law is given by       (z, t) z , t  = z + z , t + t  +  zz ,  where ((zz ))j = z, U (j) z , z, U (j) z is the inner product of z and U (j) z on Rn . The left invariant vector fields are given by

n  m ∂   ∂ (k) Xj = + si Ui,j , j = , , . . . , n, ∂xj  ∂tk i= k=

n  m ∂ ∂   (k) + si Ui,j+n , j = , , . . . , n, Yj = ∂yj  ∂t k i= k=

Tk =

∂ , ∂tk

k = , , . . . , m,

where si = xi for i = , , . . . , n and si = yi–n for i = n + , n + , . . . , n. The sub Laplacian L is defined by

L=–

n     Xj + Yj . j=

We write ∇H = (X , . . . , Xn , Y , . . . , Yn ) and divH (f , f , . . . , fn ) =

n  (Xj fj + Yj fj+n ). j=

We define the Kohn Laplacian H by H =

n  

 Xj + Yj .

()

j=

On the Heisenberg group, a fundamental solution for the sub Laplacian was studied in []. Similarly, we give a fundamental solution for H below. For  < r < ∞ and (z, t) ∈ H, we define δr (z, t) = (rz, r t). Theorem . A fundamental solution for H with source at  is given by cn,m d(z, t)–Q+ , where  c– n,m = (n + m + )( – n – m)

H

  –n–m– |z| d(z, t) +   dz dt.

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For u(z, t) ∈ C∞ (H), we have   H u(z, t), cn,m d(z, t)–Q+ L (H) = u(, ). 

Proof For ε > , let dε (z, t) = (d(z, t) + ε )  , similar to [], by equation () and a direct calculation, we have   H dε (z, t)–Q+ = ε–Q φ δ  (z, t) ,

()

ε

where   –n–m– φ(z, t) = (n + m + )( – n – m)|z| d(z, t) +   . From this, it follows that, for all u(z, t) ∈ C∞ (H),     H u(z, t), cn,m d(z, t)–Q+ L (H) = lim H u(z, t), cn,m dε (z, t)–Q+ L (H) ε→

  = lim u(z, t), cn,m H dε (z, t)–Q+ L (H) ε→



= u(, ). For ε > , the Green’s function on the half-space on the H-type group is given by G(z, t, ε) =

(|z|

+ 

m–



 j= tj

Q–

+ (tm – ε) ) 

–

(|z|

+ 

m–



 j= tj

+ (tm + ε) )

Q– 

.

2 Result We give the main results of this paper in this section. Theorem . For u ∈ C∞ (H+ ), we have 



|z|  |u| dz dt  H + tm  (Q – )(Q + ) d(z, t)– |z| |u| dz dt +  H+  m–   (k)  tk |u| U z, U (m) z d(z, t)– dz dt. – (Q + ) tm H+

 |∇H u| dz dt ≥ +  H 

k=

The theorems below show us the Lp Hardy type inequalities in the half-space on the H-type group. Theorem . Let u ∈ C∞ (H+ ) and  < p < ∞, then 

 p– pp |u|p |z| |∇H u| dz dt ≥ dz dt p p  H + tm H+   p p– pp–  p– |u|p dz dt. – p p |z| p  p– H+ tm



p

()

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Theorem . Let u ∈ C∞ (H+ ) and  < p < ∞, then  p– pp |u|p |z| dz dt p p  H + tm   p

p 

p– pp– p– p– |z| p– |u|p dz dt. – p p  – tm p  p– H+ tm



 H+

|∇H u|p dz dt ≥

()

We also study the Lp Hardy type inequalities in the H-type group. Theorem . Let u ∈ C∞ (H) and  < p < ∞, then  p– p |u|p |z| |∇H u| dz dt ≥ p(Q – ) dz dt (–Q+)p+Q p H H d p   p |u|p |z| p– p– p (p – )(Q – ) p– dz dt. – p (–Q+)p+ p– Q p H d 



p

()

Theorem . Let u ∈ C∞ (H) and  < p < ∞. Then  |∇H u|p dz dt H



 

p |u|p |z| p –  p– –

p– p

≥ cn,m u() + p(Q – ) dz dt (–Q+)p+Q p p H d p p   p p– p |u|p | – d(–Q+)(p–) | p– |z| p– – (p – )(Q – ) p– dz dt. p (–Q+)p+ p– Q p H d

()

3 Hardy type inequality This section is to show the Hardy type inequality in H+ . 

Proof of Theorem . Let v(z, t) = G(z, t, ε)–  u(z, t). Write t ε = (, . . . , , ε). We know that  G(, t ε , ε) = ∞, so we have v(, t ε ) =  and u(z, t) = G(z, t, ε)  v(z, t). Then we obtain  ∇H u =

 ∇H G ∇H v + u  G v

and  H+

|∇H u| dz dt

 =  = = =

     

 H+



H+



H+



H+

 ∇H G, ∇H v  |∇H v|  |u| dz dt + |u| dz dt Gv v H+ H+   |∇H G|  |u| dz dt + v ∇ G, ∇ v

dz dt + |∇H v| G dz dt H H G H+ H+      |∇H G|   ∇H G, ∇H v dz dt + |u| dz dt + |∇H v| G dz dt G  H+ H+  |∇H G|   –   ε  , t + |u| dz dt + v |∇H v| G dz dt c + G  n,m H |∇H G|  |u| dz dt + G



He and Yin Journal of Inequalities and Applications (2016) 2016:129

=

 



 ≥ 

H+

|∇H G|  |u| dz dt + G

H+

|∇H G|  |u| dz dt. G



Page 5 of 10

 H+

|∇H v| G dz dt

Using L’Hospital’s rule, we also have lim

ε→+

G(z, t, ε) = (Q – )tm d(z, t)–Q– ε

and

∇H G(z, t, ε)  

 

= (Q – )  t  ∇H d(z, t)–Q–  lim m

ε→+ ε   + d(z, t)–Q– tm ∇H d(z, t)–Q– , ∇H tm    + d(z, t)–Q– |∇H tm | . Because  n n n n     (m) (m)  (m) (m) si Ui, , . . . , si Ui,n , si Ui,+n , . . . , si Ui,n ,  i=  i=  i=  i=

∇H tm =

from this we can see that

|∇H tm | =

 (m) si Ui,  i=

+ =

n



 (m) si Ui,+n  i= n

+ ··· +



 (m) si Ui,n  i=

+ ··· +

n



 (m) si Ui,n  i= n

  

 |z|. 

By a direct calculation, we get

∇H d(z, t)  =

|z| . d(z, t)

Thus we have

∇H d(z, t)–Q–  = (Q + ) d(z, t)–Q– |z| , |∇H tm | =

  |z| , 

()

and

m      –Q– –Q– (k) (m) ∇H d(z, t) U z, U z tk . , ∇H tm = (–Q – )d(z, t) k=

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Consequently, we have



∇H G(z, t, ε)    

= (Q – ) lim (Q + ) d(z, t)–Q– |z| tm

ε→+ ε    + (–Q – )d(z, t)–Q– U (m) z, U (m) z tm + d(z, t)–Q– |z|   m–   –Q– (k) (m) U z, U z tm tk . + (–Q – )d(z, t) k=

This finishes the proof of the theorem.



4 Lp Hardy type inequality In this section, we are going to consider the Lp Hardy type inequalities in H+ and H, respectively. Let  be a domain in H. We write (z, t) = dist((z, t), ∂). Similar to [], we have the lemma below. Lemma . Let u ∈ C∞ (), l ∈ {, , , . . .},  < p < ∞, s ∈ (–∞, lp – ), Fj ∈ C  (), j = , , . . . , n, F = (F , F , . . . , Fn ) and w ∈ C  () be a nonnegative weight function. We write C(p, l, s) = ( lp–s– )p , then we have p 

|∇H u|p w dz dt ≥ C(p, l, s) (l–)p–s



p|u|p |∇H | w dz dt lp–s   p|u|p H w – C(p, l, s) dz dt lp–s–  (lp – s – )  p divH F|u|p w + C(p, l, s) dz dt lp – s –   

p p – 

lp–s– p– – C(p, l, s) ∇ – F |u|p w dz dt H lp–s     ∇H lp – s –  p– ∇H w F – lp–s– |u|p dz dt. + p 



Proof Applying Hölder’s inequality, we can deduce that

p  p–

∇H

p– p |∇H u|p w l–– ps

p dz dt F |u| w dz dt

l(p–)+ ps –s – (l–)p–s  

p

 

  ∇H w p–

. ∇ ≥ pp

– Fw sign(u)|u| u dz dt H

lp–s–  

p

On the other hand, by partial integration we get

 

p

  ∇H w p– p ∇H u dz dt

– Fw sign(u)|u| lp–s– 

 

H (lp – s – )|∇H |

= – lp–s– + divH F w lp–s 

p 

∇H |u|p dz dt

. + ∇H w F – lp–s– p

()

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Thus we obtain  |∇H u|p w p dz dt p (l–)p–s 

 

H (lp – s – )|∇H | ≥

– + div F w H lp–s lp–s– 

p 

∇H |u|p dz dt

+ ∇H w F – lp–s–

p  –p+

p– p

∇H l–– ps

× F |u| w dz dt .

l(p–)+ ps –s –  It is clear that

|a|p bp–

≥ pa – (p – )b for b > . Then we have equation ().



For F = , we have  

|∇H u|p w dz dt ≥ C(p, l, s) (l–)p–s



p|u|p |∇H | w dz dt lp–s   p|u|p H w – C(p, l, s) dz dt lp–s–  (lp – s – )  p p– p– |u|p w dz dt – C(p, l, s) |∇ | H lp–s  p–   ∇H w∇H p lp – s –  |u| dz dt. – lp–s– p 

Now, let us discuss the Lp Hardy type inequalities in H+ . Let l = , s = , and w = , we have by equation ()  p|u|p |∇H | p– p |∇H u| dz dt ≥ dz dt p p     p|u|p H p– p dz dt – p– p  (p – )   p divH F|u|p p– p dz dt + p p–   

p p– p p – 

∇H – p– F p– |u|p dz dt. – p p  



p

()

For  = H+ , we have = tm . So we get  p– p p|u|p |∇H tm | |∇H u| dz dt ≥ dz dt p p tm H+ H+   p– p p|u|p H tm dz dt – p– p H+ (p – )tm   p divH F|u|p p– p dz dt + p p– H+  

p p– p p – 

p– p– ∇H tm – tm F |u|p dz dt. – p p H + tm 



p

()

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Proof of Theorem . We know that  |∇H tm | = |z|  and H tm = . Set F = , using equation (), then we obtain equation ().



Proof of Theorem . Set F = ∇H tm . Since U (m) is a n × n skew-symmetric orthogonal matrix, we have

divH F =

n  j=

=

   (m) si Ui,j(m) + Yj si Ui,j+n  i=  j= i= n

Xj

n

n

n n   (m)   (m) Uj,j + U   j+n,j+n j= j=

= . Using equation (), we have equation ().



Now we are going to deal with the Lp Hardy type inequalities in H. Lemma . Let u ∈ C∞ (H), l ∈ {, , , . . .},  < p < ∞, s ∈ (–∞, lp – ), Fj ∈ C  (H), j = , , . . . , n, F = (F , F , . . . , Fn ) and w ∈ C  (H) be a nonnegative weight function. Then we have  H

|∇H u|p w –Q+ (d )(l–)p–s

dz dt

 p|u|p |∇H d–Q+ | w p divH F|u|p w dz dt dz dt + C(p, l, s) –Q+ lp–s (d ) lp – s –  H H  p lp–s– p–  p – 

– C(p, l, s) F |u|p w dz dt ∇H d–Q+ – d–Q+ –Q+ lp–s ) H (d  p–   lp – s –  ∇H d–Q+ + ∇H w F – –Q+ lp–s– |u|p dz dt, p (d ) H 

≥ C(p, l, s)

)p . where C(p, l, s) = ( lp–s– p Proof Similar to Lemma ., we have 

|∇H u|p w dz dt –Q+ )(l–)p–s H (d  p|u|p |∇H d–Q+ | w ≥ C(p, l, s) dz dt (d–Q+ )lp–s H  p|u|p H d–Q+ w dz dt – C(p, l, s) –Q+ )lp–s– H (lp – s – )(d

()

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p divH F|u|p w dz dt lp – s –  H  p lp–s– p–  p – 

∇H d–Q+ – d–Q+ F |u|p w dz dt – C(p, l, s) –Q+ lp–s ) H (d p–    ∇H d–Q+ lp – s –  ∇H w F – –Q+ lp–s– |u|p dz dt. + p (d ) H + C(p, l, s)

()

We know that cn,m d(z, t)–Q+ is a fundamental solution for H . So we have 

p |u|p H d–Q+ w (Q–)(lp–s–)

dz dt = c– = . n,m u() w()d() (d–Q+ )lp–s–

H



For l = , s = , and w = , we have  |∇H u|p dz dt H



≥

 p– p p|u|p |∇H d–Q+ | dz dt p (d–Q+ )p H   p divH F|u|p p– p dz dt + p p– H   p  –Q+ p– p– p– p p – 

–Q+

|u|p dz dt. ∇ d – d F – H –Q+ )p p H (d

()

Set F = , then we get  p|u|p |∇H d–Q+ | p– p dz dt p (d–Q+ )p H  

p p – 

p– p –Q+ p– d |u|p dz dt. ∇ – H –Q+ )p p H (d



 |∇H u|p dz dt ≥ H

()

Proof of Theorem . It is obvious that |∇H d| =

|z| . d

So we have 

∇H d–Q+  = (Q – ) d(–Q+) |z| . d

()

From this together with (), we get equation ().



Proof of Theorem . Let F = ∇H d–Q+ . Then we have divH F = divH ∇H d–Q+ = H d–Q+ . From equations () and (), it follows that  |∇H u|p dz dt H

≥



p– p

p



|u|p |z|

p(Q – ) H

d(–Q+)p+Q

dz dt

He and Yin Journal of Inequalities and Applications (2016) 2016:129



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p 

pH d–Q+ |u|p dz dt p– H p p   p |u|p | – d(–Q+)(p–) | p– |z| p– p– p (p – )(Q – ) p– dz dt, – p (–Q+)p+ p– Q p H d +

p– p

which implies that  |∇H u|p dz dt H

≥

 

p p– p p –  p– –

|u|p |z| cn,m u() + p(Q – ) dz dt (–Q+)p+Q p p H d p p   p |u|p | – d(–Q+)(p–) | p– |z| p– p– p p– (p – )(Q – ) dz dt. – p (–Q+)p+ p– Q p H d



So we have equation ().



Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Acknowledgements The work for this paper is supported by the National Natural Science Foundation of China (No. 11271091, 11471040). Received: 25 November 2015 Accepted: 19 April 2016 References 1. Tidblm, J: A Hardy inequality in the half-space. J. Funct. Anal. 221, 482-495 (2005) 2. Barbatis, G, Filippas, S, Tertikas, A: A unified approach to improved Lp Hardy inequalities with best constants. Trans. Am. Math. Soc. 356, 2169-2196 (2004) 3. Hoffmann-Ostenhof, M, Hoffmann-Ostenhof, T, Laptev, A: A geometrical version of Hardy’s inequality. J. Funct. Anal. 189, 539-548 (2002) 4. Luan, JW, Yang, QH: A Hardy type inequality in the half-space on Rn and Heisenberg group. J. Math. Anal. Appl. 347, 645-651 (2008) 5. Tidblm, J: Lp Hardy inequalities in general domains. http://www.math.su.se/reports/2003/4/ (2003) 6. Hardy, G: Note on a theorem of Hilbert. Math. Z. 6, 314-317 (1920) 7. Folland, GB: Harmonic Analysis in the Phase Space. Princeton University Press, Princeton (1989) 8. Geller, D: Fourier analysis on the Heisenberg group. I. Schwartz space. J. Funct. Anal. 26, 205-254 (1980) 9. Geller, D, Stein, EM: Singular convolution operators on the Heisenberg group. Bull. Am. Math. Soc. 6, 99-103 (1982) 10. Stein, EM: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993) 11. Thangavelu, S: Harmonic Analysis on the Heisenberg Group. Birkhäuser, Boston (1998) 12. Kaplan, A: Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms. Trans. Am. Math. Soc. 258, 147-153 (1980) 13. Folland, GB: A fundamental solution for a subelliptic operator. Bull. Am. Math. Soc. 79, 373-376 (1973)