Weighted Hardy-Sobolev Type Inequality for Generalized Baouendi-Grushin Vector Fields and Its Application

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1 44Æ 3 «Vol.44 No ADVANCES IN MATHEMATICS(CHINA) May 05 doi: 0.845/sxjz.03075b Weighted Hady-Sobolev Type Ieuality fo Geealized Baouedi-Gushi Vecto Fields ad Its Applicatio ZHANG Shutao HAN Yazhou DOU Jigbo (. Depatmet of Mathematics College of Sciece Chia Jiliag Uivesity Hagzhou Zhejiag 3008 P. R. Chia;. School of Statistics Xi a Uivesity of Fiace ad Ecoomics Xi a Shaaxi 7000 P. R. Chia) Abstact: This wok is devoted to geealizig a class of Caffaelli-Koh-Niebeg type ieualities fo the geealized Baouedi-Gushi vecto fields. Ispied by the idea of Che ad Li a fuctio tasfomatio is itoduced. Combiig some elemetay ieualities ad accuate estimates we establish a class of weighted Hady-Sobolev type ieualities ad the pove a class of Caffaelli-Koh-Niebeg type ieualities fo the geealized Baouedi-Gushi vecto fields. Keywods: weighted Hady-Sobolev type ieuality; Caffaelli-Koh-Niebeg type ieuality; Baouedi-Gushi vecto field MR(00) Subject Classificatio: 6D0; 35H0 / CLC umbe: O75.9 Documet code: A Aticle ID: (05) Itoductio It is well kow that Hady ieuality Sobolev ieuality ad Hady-Sobolev ieuality o the Euclidea space play impotat oles i may mathematical ad applied fields. I paticula these ieualities wee extesively studied ad applied to vaious iteestig poblems i patial diffeetial euatios such as eigevalue poblem existece poblem of euatio with sigula weight egulaity poblem etc. (see [ 4 8 ]). Caffaelli et al. obtaied a class of fist ode itepolatio ieualities with weights (kow as the Caffaelli-Koh-Niebeg (CKN) ieuality) as follows (see []): Theoem 0. Let pλβγσa satisfy p > 0 0 a p + λ + β + γ > 0 (0.) whee γ = aσ + ( a)β. The thee exists a positive costat C such that the followig ieuality holds fo all u C 0 (R ) x γ u L C x λ u a L p x β u a L (0.) Received date: Revised date: Foudatio item: Suppoted by NSFC (No No. 039) ad Natual Sciece Foudatio of Zhejiag Povice (No. Y608). taoe558@63.com; Coespodig Autho: yazhou.ha@gmail.com; djbm@6.com

2 4 Å 44Æ if ad oly if the followig elatios hold: + γ ( = a (this is dimesioal balace) p + λ ) ( +( a) + β ) (0.3) 0 λ σ if a > 0 λ σ if a > 0 ad p + λ = + γ. Futhemoe o ay compact set i a paamete space i which (0.) (0.3) ad 0 σ hold the costat C is bouded. Simila esults fo highe ode case wee give by Li [0] i 986. Fo the special case a = (0.) is Hady-Sobolev ieuality. I [] Badiale ad Taatello had established a class of moe geeal Hady-Sobolev ieualities whee the sigula weight is depedet oly o patial vaiables. Though a caeful aalysis of the elatio of all paametes Zhao ad the secod autho (see [8]) foud that Hady-Sobolev ieuality is the key ole ad othe cases ca be tasfomed ito Hady-Sobolev ieuality. These facts solve a techical difficulty. I fact usig the classical symmety techiue is a impotat step i []. But the techiue does ot hold o ocommutative vecto fields. So based o these obsevatio a class of CKN type ieualities fo the geealized Baouedi-Gushi vecto fields has bee established i [8]. To state ou esults we itoduce thee otatios fist (moe details ca be foud i Sectio ). Let = +m(+) ( > 0) be the homogeeous dimesio be the distace fuctio ad L = d = d(xy) = (+) +(+) y ) (+) ( x be the geealized gadiet. x x x x ) x y y y m I the followig we fist state some kow esults ad the give ou mai esults o the geealized Baouedi-Gushi vecto fields. I 007 Niu ad Dou [] obtaied Hady-Sobolev ieualities fo the geealized Baouedi-Gushi vecto fields as follows: Theoem 0. Let (xy) = (x y x y ) = (z z ) whee z = (x y ) R k+l z = (x y ) R k+m l k l m. Deote d = d(x y ) = ( x (+) + ( + ) y ) (+). Assume that s satisfies 0 s < k +(+)l ad put (s) = ( s). The thee exists a positive costat C = C(skl) such that fo evey u D ( ) R+m x s d s u (s) d s ( ) s dxdy C L u dxdy (0.4)

3 3 ² ³ : Weighted HS-type Ieuality ad Its Applicatio 43 whee D ( ) is the completio of C 0 (R+m ) ude the om ( u = L u dxdy Usig foegoig idea ad combiig (0.4) we have give a class of CKN type ieuality [8] as follows: Theoem 0.3 Set pβγσa such that whee γ = aσ +( a)β. ). < p < 0 a > γ > β γ+ > 0 β + > 0 ) Necessity. If fo ay u C 0 (R+m ) thee exists a positive costat C such that (R+m x γ ) d γ dγ u dxdy ( )a ( C L u p p x β dxdy R+m d β dβ u dxdy ) a (0.5) holds the + γ ( = a p ) ( +( a) + β ) (0.6) 0 σ whe a > 0 (0.7) σ whe a > 0 p = + γ. (0.8) ) Sufficiecy. Fo the case p = if (0.6) (0.8) hold the CKN type ieuality (0.5) is tue. Note that the weight i the gadiet tem is abset i (0.5) ad the the above esult is ot completely aalogous to CKN ieuality (0.). This pape is maily devoted to geealizig the above esult. Ou mai esult is the followig weighted CKN ieuality fo the geealized Baouedi-Gushi vecto fields fo the case p = : Theoem 0.4 Set λβγσa such that > 0 0 a +(λ γ) > 0 +(λ β) > 0 γ+ > 0 β + > 0 λ+ > 0 whee γ = aσ+( a)β. (0.9)

4 44 Å 44Æ The thee exists a positive costat C such that the followig ieuality holds fo all u C 0 ( ): ) (λ γ) d γ u dxdy) ( ) a ( C d λ L u dxdy ) (λ β) d β u dxdy) a (0.0) if ad oly if the followig elatios hold: + γ = a ( + λ ) ( +( a) + β ) (0.) λ σ 0 whe a > 0 (0.) λ σ whe a > 0 + λ = + γ. (0.3) To pove Theoem 0.4 we employ the idea of [8]. So we fist eed to establish a class of weighted Hady-Sobolev type ieuality fo the geealized Baouedi-Gushi vecto fields as follows: Theoem 0.5 If 0 s < = +m( +) λ > the thee exists a positive costat C = C(sλ) such that fo evey u D λ (R+m ) R+m x s d λ u (s) ( ) s d s d s dxdy C d λ L u dxdy (0.4) whee D λ (R+m ) is the closue of C0 ( ) with espect to the om u λ = d λ L u dxdy. Remak 0. I paticula whe λ = 0 (0.0) ad (0.4) become (0.5) ad (0.4) espectively. So the above two esults geealize the esult of [8 ]. Remak 0. Whe a = the coditios of Theoem 0.4 imply 0 λ σ = λ γ + σ = + λ ad the =. So thee exists t [0] such that = t+( t) = ( t) ad λ σ = t. Substitutig ad λ σ (0.0) is tasfomed ito (0.4) amely weighted Hady-Sobolev ieuality. Remak 0.3 I additio due to the eeds of academic a class of CKN type ieualities hadbee established othe H-typegoup [3]. Sicewe employthe sameidea the poofissimila. Howeve fo the coveiece of eades the detailed poof is give i Sectio 3.

5 3 ² ³ : Weighted HS-type Ieuality ad Its Applicatio 45 The pape is ogaized as follows. I Sectio we itoduce some basic facts fo the geealized Baouedi-Gushi vecto fields. Sectio is devoted to the poof of Theoem 0.5. By usig the idea of [8] CKN type ieuality fo the geealized Baouedi-Gushi vecto fields amely Theoem 0.4 is obtaied i Sectio 3. Pelimiay We itoduce some defiitios ad basic esults elated to the geealized Baouedi-Gushi (BG) vecto fields (see [4 8 ] ad efeece theei). The geealized BG vecto fields ae defied as Z i = Z +j = x i = j = m x i y j whee > 0. Deote the coespodig geealized gadiet ad BG opeato with ad L = (Z Z Z Z + Z +m ) L = x + x y = +m i= espectively. A atual family of dilatios is defied by Z i = L L δ λ (xy) = (λxλ + y) λ > 0 ξ = (xy) ad = + m(+ ) is the coespodig homogeeous dimesio. Defie the distace fuctio ad the d = d(xy) = ( x (+) +(+) y ) (+) L d = x d = ψ. By the pola coodiate tasfomatio itoduced by D Ambosio [4 5] ad Dou et al. [6 ] we have dxdy = ρ dρdσ whee dσ = ( + )m siθ + cosθ m dθdω dω m ω ad ω m deote the usual suface aea measues o S ad S m espectively. Moeove they gave the followig citeia fo the itegability of x p d. ) If p+ + > 0 ad p+ > 0 the B x p d dxdy < + ; ) If p+ + < 0 ad p+ > 0 the \B x p d dxdy < +. Deote ( ) A = d λ L u dxdy B = ) (λ β) d β u dxdy)

6 46 Å 44Æ ad thus (0.0) ca be ewitte as ( ) x (λ γ) d γ u dxdy) CA a B a. Rescalig u such that A = B = ou goal becomes to ivestigate the followig ieuality ( ) x (λ γ) d γ u dxdy C. I the seuel we take Φ(xy) (0 Φ ) which is a fixed C0 fuctio o satisfyig { if d < Φ(xy) = (.) 0 if d >. Weighted Hady-Sobolev Type Ieualities Fist we itoduce the followig Hady type ieuality established by D Ambosio [4]. Theoem. Let p > m ad βγ R such that +m(+) > γ β p ad > p β. The fo ay ope subset Ω ad evey u D p (Ω x β p d (+)p γ ) it follows c p pβγ u p x β Ω d γ dxdy L u p x β p d (+)p γ dxdy (.) Ω whee c pβγ = +β γ p D p (Ω x β p d (+)p γ ) is the completio of C0 (Ω) ude the om ( ) u = L u p x β p d (+)p γ p dxdy. Ω If (00) Ω the the costat c p pβγ of (.) is shap. Poof of Theoem 0.5 Because of λ > we kow (s)µ s+ > 0 ad µ+ > 0 which esue that the ight ad left itegal of (0.4) ae well defied. Fo ay u C0 (R+m ) take w = d λ u ad the R+m L w dxdy = d λ L u dxdy λ(+λ ) L d u R d dλ dxdy. +m Fom (.) we see that L w dxdy < + i.e. w D ( ). A staightfowad computatio deduces that ad R+m x s d λ u (s) R+m x s w (s) d s d s dxdy = d s d s dxdy (.) d λ L u dxdy = d λ d λ L w λd λ w L d dxdy R +m = ( L w +λ x w ) d d λd L d L w dxdy = L w dxdy +λ( +λ) R+m x d w d dxdy. (.3)

7 3 ² ³ : Weighted HS-type Ieuality ad Its Applicatio 47 Notig +λ > + > 0 it is easy to check that (0.4) holds usig (0.4) if λ 0. If λ < 0 we obtai fom (.) that d λ L u dxdy C L w dxdy whee C = +µ( +µ)( ) = ( ) ( +µ) > 0. Theefoe (0.4) holds though (0.4). 3 CKN Type Ieualities With p = Sice the poof of the ecessity of Theoem 0.4 is simila to the pocess i [ 8] we omit hee. I the seuel we oly show the sufficiecy. I additio if a = 0 the (0.0) obviously holds; if a = we complete the poofi Remak0.. So we oly eed to teat the case0 < a <. 3. Sufficiecy Whe 0 < a < 0 λ σ Notig ( + λ σ ) a agumet simila to the oe i Remak 0. shows that thee exists t (0 t ) such that ( + λ σ ) = ( t) Hece By (0.9) ad (0.6) = a( + λ σ ) (λ σ)( +λ σ ( C d λ L u dxdy )+ a ( ) x (λ γ) d γ u dxdy ( C ) (λ σ)( +λ σ ( ) x (λ β) d β u dxdy ( ) x (λ γ) d γ u dxdy d λ L u dxdy ) d λ u ( +λ σ ) d (λ σ)( +λ σ ) t. ad (λ σ)( + λ σ ) = t. ) dxdy. So fo > we have )a ) d λ u ( +λ σ ) dxdy d (λ σ)( +λ σ ) ) a ) ) (λ γ) d γ u dxdy) ) a( +λ σ ) ) (λ β) d β u dxdy) a ad hece (0.0) holds. Usig the same lie we ca also get the case =.

8 48 Å 44Æ 3. Sufficiecy Whe 0 < a < λ σ > (0.3) tells us a < + λ + γ. Usig the discussio i the ed of Sectio ad assumig A a B a = we eed to pove that ( ) x (λ γ) d γ u dxdy (3.) is bouded. I Subsectio 3. (0.0) has bee checked fo λ σ = ad λ σ = 0. Hece we coclude that ( ) x (λ δ)s ( ) x d δs u s (λ ε)t dxdy C d εt u t dxdy C (3.) whee δsε ad t satisfy δ = bλ+( b)β s = b + b b ε = d(λ )+( d)β t = d + d fo some choices of b ad d 0 bd ad the { +(λ δ)s > 0 δs+ > 0 +(λ ε)t > 0 εt+ > 0. (3.3) (3.4) Obviously t + ε ( = d + λ ) ( +( d) + β ) + γ ( = a + λ ) ( +( a) + β ) s + δ ( = b + λ ) ( +( b) + β ). Now to fiish the poof we discuss two cases. ) Case + λ < + β. Take b < a < d ad the t + ε < + γ < s + δ. (3.5) It is easy to veify that ( s = (a b) p ) ( t = (a d) p ) + + a(λ σ) a(λ σ )

9 3 ² ³ : Weighted HS-type Ieuality ad Its Applicatio 49 ( + (λ γ) ( + (λ γ) ) ( t + (λ ε) ( = (a d) p + ) (β + λ) ) ( s + (λ δ) ) = (a b) Sice a > 0 λ σ > we see that ) ( p + (β λ) ) ( +a(λ σ ) + ) ( +a(λ σ) + ). a(λ σ ) 0 < < a(λ σ) ( 0 < a(λ σ ) + ) ( < a(λ σ) + ) ad fo sufficietly small d a ad b a > s > t + (λ γ) + (λ γ) Combiig (3.5) ad (3.6) oe gets Hölde ieuality implies that ( ) x (λ γ) d γ Φ u dxdy) > t + (λ ε) > s + (λ δ). +(ε γ) t t > 0 (γ ε) + > 0 t t (3.6) +(δ γ) s s (3.7) > 0 (γ δ) s s + < 0. ad C C ) (λ ε)t d εt u t dxdy) t ) (λ ε)t d εt u t dxdy) t ) (λ γ) d γ ( Φ) u dxdy) ) (λ δ)s d δs u s dxdy) s ) (λ δ)s d δs u s dxdy) s d< d> ( d + x ( d + x ) (γ ε)t t dxdy ) t ) (γ δ)s ) s s dxdy (3.8) (3.9)

10 40 Å 44Æ whee Φ is defied i (.). Combiig (3.) (3.8) ad (3.9) we deduce that (3.) holds. ) Case + λ > + β. Take d < a < b such that d a ad b a ae sufficietly small. The (3.3) (3.7) hold. This implies that (3.) (3.8) ad (3.9) ae tue. So (3.) holds. Ackowledgemets The authos expess the gatitude to the efeees fo thei valuable metios ad suggestios. Refeeces [] Badiale M. ad Taatello G. A Sobolev-Hady ieuality with applicatios to a oliea elliptic euatio aisig i astophysics Ach. Ratio. Mech. Aal (4): [] Caffaelli L. Koh R. ad Niebeg L. Fist ode itepolatio ieualities with weights Compos. Math (3): [3] Che J.-L. ad Li C.-S. Miimizes of Caffaelli-Koh-Niebeg ieualities with the sigulaity o the bouday Ach. Ratio. Mech. Aal (): [4] D Ambosio L. Hady ieualities elated to Gushi type opeatos Poc. Ame. Math. Soc (3): [5] D Ambosio L. ad Lucete S. Noliea Liouville theoems fo Gushi ad Ticomi opeatos J. Diffeetial Euatios (): [6] Dou J.B. Niu P.C. ad Ha J.. Pola coodiates fo the Geealized Baouedi-Gushi opeato ad applicatios J. Patial Diffe. Eu (4): [7] Gaofalo N. Uiue cotiuatio fo a class of elliptic opeatos which degeeate o a maifold of abitay codimesio J. Diffeetial Euatios (): [8] Ha Y.Z. ad Zhao. A class of Caffaelli-Koh-Niebeg type ieualities fo geealized Baouedi- Gushi vecto fields Acta Math. Sci. Se. A Chi. Ed. 0 3(5): 8-89 (i Chiese). [9] Ji Y.Y. ad Ha Y.Z. Impoved Hady ieuality o the Heisebeg goup Acta Math. Sci. Se. A Chi. Ed. 0 3(6): (i Chiese). [0] Li C.-S. Itepolatio ieualities with weights Comm. Patial Diffeetial Euatios 986 (4): [] Niu P.C. ad Dou J.B. Hady-Sobolev type ieualities fo geealized Baouedi-Gushi opeatos Miskolc Math. Notes 007 8(): [] Niu P.C. Dou J.B. ad Zhag H.. Noexistece of weak solutios fo the p-degeeate subelliptic ieualities costucted by geealized Baouedi-Gushi vecto fields Geogia Math. J. 005 (4): [3] Zhag S.T. Ha Y.Z. ad Dou J.B. A class of Caffaelli-Koh-Niebeg type ieualities o the H-type goup Red. Semi. Mat. Uiv. Padova 04 3: º Baouedi-Gushi µ ¾ ¼ Hady-Sobolev» À (. ± 3008;. 7000) ÐÉ ÚÉØ Baouedi-Gushi Õ ÄÓ ÑÅ Caffaelli-Koh-Niebeg Ã Æ. Ò Ý Che Ë Li Å Ô Ù ÇÊ ÂÎ; Ì ÖÏÁÅÃÆ Ë È Å Hady-Sobolev ÃÆ ; ÍÜ Û ÚÉØ Baouedi-Gushi Õ ÄÓ ÑÅ Caffaelli-Koh-Niebeg ÃÆ. ¹½ Hady-Sobolev ÃÆ ; Caffaelli-Koh-Niebeg ÃÆ ; Baouedi-Gushi Õ Ä

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