Boundedness of Calderón-Zygmund Operator and Their Commutator on Herz Spaces with Variable Exponent

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1 Alied Mathematics htt://wwwscirorg/oural/am ISSN Olie: 5-79 ISSN Prit: Boudedess of Calderó-Zygmud Oerator ad Their Commutator o Herz Saces with Variable Exoet Omer Abdalrhma * Afif Abdalmoem Shuagig Tao College of Mathematics ad Statistics Northwest Normal Uiversity azhou Chia College of Educatio Shedi Uiversity Shedi River Nile State Suda Faculty of Sciece Uiversity of Dala Dala South ordofa Suda How to cite this aer: Abdalrhma O Abdalmoem A ad Tao SP (07 Boudedess of Calderó-Zygmud Oerator ad Their Commutator o Herz Saces with Variable Exoet Alied Mathematics htts://doiorg/046/am Received: March 07 Acceted: Aril 7 07 Published: Aril 0 07 Coyright 07 by authors ad Scietific Research Publishig Ic This wor is licesed uder the Creative Commos Attributio Iteratioal icese (CC BY 40 htt://creativecommosorg/liceses/by/40/ Oe Access Abstract The aim of this aer is to study the boudedess of Calderó-Zygmud oerator ad their commutator o Herz Saces with two variable exoets ( ( By alyig the roerties of the ebesgue saces with variable exoet the boudedess of the Calderó-Zygmud oerator ad the commutator geerated by BMO fuctio ad Calderó-Zygmud oerator is obtaied o Herz sace Keywords Calderó-Zygmud Oerator Commutator Herz Saces with Variable Exoet BMO Saces Itroductio Defiitio et T be a bouded liear oerator from S ( to S ( (see [] [] T is called a stadard oerator if T satisfies the followig coditios: T exteds to a bouded liear oerator o ( There exists a fuctio ( satisfies K xy defied by {( ( xy ( ; x y} ( K xy C x y ( C > 0 Tf g K x y f y g x dd x y = ( ( ( ( ( su( f su( g = for f g S( with DOI: 046/am Aril 0 07

2 O Abdalrhma et al A stadard oerator T is called a γ -Calderó - Zygmud oerator if K is a stadard erel satisfies: ( ( γ γ K x y K z y C x z x y ; ( ( ( γ γ K yx K yz Cx z x y ( if x z < x y for some 0< γ The bouded mea oscillatio BMO sace ad BMO orm are defied resectively by { } ( loc ( BMO( BMO = b : b < (4 ( ( B b = su B b x b B d x (5 BMO B:ball The commutator of the Calderó-Zygmud oerator is defied by [ b T] f ( x b( x Tf ( x T( bf ( x = (6 I 98 J- Joué roved γ -Calderó - Zygmud oerator is bouded o ( i [] Coifma Rochberg ad Weiss roved that commutator [bt] is bouded o ( ( < < (see [4] Kováci ad Ráosí itroduced ebesgue saces ad Sobolev saces with variable exoets (see [5] The fuctio saces with variable exoet has bee recetly obtaied a icreasig iterest by a umber of authors sice may alicatios are foud i may differet fields for examle i fluid dyamics (see [6] image restoratio (see [7] [8] [9] ad differetial euatios Herz saces lay a imortat role i harmoic aalysis After they were itroduced i [0] the boudedess of some oerators ad some characterizatios of Herz saces with variable exoets were studied extesively (see []-[6] I 05 Wag ad Tao itroduced the Herz saces with two variable exoets ( ( ad studied the arameterized ittlewood-paley oerators ad their commutators o Herz saces with variable exoets i [7] I this aer we will discuss the boudedess of the Calderó-Zygmud oerator T ad their commutator [ bt ] are bouded o Herz saces with two variable exoets ( ( Defiitios of Fuctio Saces with Variable Exoet I this sectio we recall some defiitios et Ω be a measurable set i with Ω> 0 We firstly recall the defiitio of the ebesgue saces with variable exoet Defiitio [5] et ( : Ω [ be a measurable fuctio The ebesgue sace with variable exoet ( ( Ω is defied by x ( ( ( f x ( Ω = f is measurable : d x< for some costat > 0 ( Ω For all comact K Ω the sace ( ( loc Ω is defied by 49

3 O Abdalrhma et al ( ( ( { ( } Ω = f is measurable : f K ( loc The ebesgue saces f ( ( Ω ( ( Ω is a Baach saces with the orm defied by x ( f ( x = if > 0 : dx ( Ω We deote essif { ( x : x } ess su { ( x : x } ( Ω cosists of all ( satisfyig Hardy-ittlewood maximal oerator We deote ( Ω ( fuctio ( ( Ω satisfyig the M is bouded o ( Ω Defiitio [8] et ( ( ( Ω = Ω = Ω The ( > ad < et M be the to be the set of all The mixed ebesgue seuece sace with variable exoet ( ( is the collectio of all seueces { f } = 0 of the measurable fuctios o { f } = 0 ( ( ( such that f = if > 0 : Q ( ( < ( ζ = 0 x ( f ( x Q ( ( ( ({ f } 0 if 0; d = ζ x > = R = 0 x ( ζ et { } have that (4 B = x : x C = B \ B = for < we C ( ( ( ( = 0 = ( = 0 Q f f ( { } et B = { x : x } C = B \ B = C Defiitio [7] et α ( ( ( sace with variable exoet K α ( ( ( (5 The homogeeous Herz α ( Euied the orm is defied by { } ( ( ( loc ( { } ( ( K ( K = f \ 0 : f α < f K α ( = α f ( ( { } = 0 ( ( ( α ( f = if > 0 : ( = ( Remar [7] et ( ( ( satisfyig ( ( ad satisfy the followig results: K α α K ( If ( ( ( ( ( ( ( ( ad ( ( l For ay f K α ( ( ( by usig 40

4 O Abdalrhma et al emma 7 ad Remar we have = ( α f f ( α ( ( = ( ( ( ( ( ( = v = α f α f α f > miv av v v= 0 * = maxv av > v v= 0 ( ( This imlies that K α α ( ( K ( Remar et v a 0 < The we have ( ( v v av ah v= 0 v= 0 * v h ( ( miv av v v= 0 * = maxv av > v v= 0 Proerties ad emmas of Variable Exoet * I this sectio we recall some roerties ad some lemmas of variable exoet belogig to the class ( Proositio [9] If ( ( ( ( satisfies C x y x y ( ( og ( x y ; C x y y x Hece we have ( ( emma [5] Give ( : [ g C = og ( e x ( ( have that for all fuctios f ad ( ( ( ( ( ( f x g x d x C f g ( 4

5 O Abdalrhma et al emma [5] Suose that ( ( ( ( ( ( ( ( f g whe ( ( ( ( = ( ( ( ( f x g x C g x f x ( for ay we get ( ( ( (4 C = Proositio [0] et ( ( oerator The we have Tf ad T be a Calderó - Zygmud ( ( ( ( f emma [0] et ( ( b BMO fuctio ad T be a Calderó - Zygmud oeratorthe emma 4 [] et BMO ( (5 [ bt ] f ( ( ( C b BMO( f ( b If i with i C b su ( ( b b BMO B B ( ( ( Cb BMO B ( B ( ( ( ( ( ( ( ( b b C i b B i B BMO B emma 5 [] et ( ( ( u = 0 < ι ι < ad C > 0 such that for all balls u u subset R B R u B u u ( ( ( ( emma 6 [] If ( ( for ay balls B i we have < the we have (6 the there exist costats B ad all measurable u ι ( u R ι R u ( R C C B B u B ( ( (7 there exist a costat C > 0 such that B ( ( ( B B ( emma 7 [7] Suose that ( ( ( If f ( ( the ( ( f ( ( f ( ( f f ( ( f ( ( ( ( (8 mi max (9 4 The Mai Theorems ad Their Proofs Theorem 4 Suose that ( ( ( ( ( ( ( If ι < α < ι with oerator T is bouded from ( K α ( ( to ( K α ( ( Proof et ( h x K α ( ( ( We write with ι ι as defied i emma 5 the the ( = ( = ( h x h x h x = = 4

6 O Abdalrhma et al By Defiitio we have Sice ad ( K α ( ( ( α ( T( h = if > 0 : ( = ( ( α ( α T( h T( h = ( ( i= i α = ( ( ( ( α = α = α = T( h = = = ( ( ( α = T( h = = ( ( ( α = T( h = = i i= ( ( ( ( ( ( ( ( ( ( ( ( ( (4 (4 (4 (44 Thus We easily see that = ( α ( ( ( ( = C α ( ( i ( K i= (46 This imlies that we oly eed to rove 0 = h ( ( K α ( Deote C h α K ( ( ( 4

7 O Abdalrhma et al First we cosider By virtue of emma 7 we get = = = α α α = 0 = 0 = ( 0 ( ( ( ( ( ( ( ( ( (47 ( ( α T( h ( = 0 ( ( = ( α T ( h = ( > 0 ( ( I the above we use the Proositio ad Remar Sice ( ( ( ( h x K α we have = α h 0 ( ( ( = α h 0 we get α = = = ( 0 = 0 α 0 h α ad ( ( ( α h = 0 α h = 0 h ( ( ( ( ( ( ( ( ( ( ( * ( ( 44

8 O Abdalrhma et al Here ( ( ( ad * N ( = mi That is ( C 0 C h α ( ( K ( (48 et us ow tur to estimate Notig that x A ad by the geeralized Hölder's ieuality ad the Miowsi s ieuality we get By emmas 5-7 ad the fact that = = = = α = 0 ( A ( ( A ( ( d Th x K x y h y dy ( α h = 0 α = α h = 0 h ( ( α ι 0 C h y x y dy C h y y A h ( ( ( ( ( ( ( α 0 ( h ( B ( ( ( ( ( ( ( ( B h ( ( we easily see that ( ( α B B B = = 0 α h B ( ( α α = = 0 ( B ( = = α h 0 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (49 (40 45

9 O Abdalrhma et al ( ( α T( h = ( 0 ( ( = ( α T ( h = ( 0 ( ( Therefore if ( = mi * If ( > < ad ( ( ( = we ca get 0 ( ( ( ( * ( α h ( ( α ι = 0 = ( ( ( ( α = ad ( ( ( By Remar ad alyig the geeralized Hölder s ieuality we obtai = α = ( ( C ( = = ( ( = = mi * 0 ( ( α ι ( ( ( α ι ( (( α h ( ( ( = 0 = ( ( ( ( Hece we see that α h 0 ( ( α ι ( ( ( ( ( ( * have C C h α ( ( K ( 0 Fially we estimate Notig that for each (4 x A ad we 46

10 O Abdalrhma et al ( ( ( ( A A ( Th x K x y h y dy C h y x y dy C h (4 = C = α = 0 By emma 7 ad ( ( ( ( α h 0 ( ( we get ( α α h h = = ( C α = = 0 0 = 0 ( ( h ( ( ( ( ( ( = = 0 ( ( B ( ( ( ( B h α B B B = = 0 α B = = 0 α h α α h ( B ( ( ( ( ( B ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( α h ( ( α ι = = 0 ( ( ( ( α T( h ( = 0 ( ( = ( α T( h ( = > 0 ( ( ( (4 The we have C0 C h α ( ( K ( by usig the same argumet i Thus we rove Theorem 4 Theorem 4 et b BMO ( Suose that ( ( ( ( ( with ( ( ι ι as defied i lemma 5 the the commutator [ ] ( K α ( ( K α ( with from ( to ( If ι < α < ι bt is bouded 47

11 O Abdalrhma et al We write α Proof et ( h x K ( ( ( b BMO ( ( = ( = ( h x h x h x = By virtue of the defiitio of K α ( ( ( Sice [ ]( = we have ( α [ bt ]( h bt h α ( ( = if > 0 : K ( ( = ( ( ( α α [ bt ]( h [ bt ]( h = ( ( i= i ( ( ( α [ bt]( h [ bt]( h = = α ( ( ( ( ( [ bt]( h α = ( ( ( (44 (45 ad et α = [ bt ]( h = = = ( ( ( α = [ bt ]( h = = ( ( ( α = [ bt ]( h = = i i= ( ( ( (46 (47 (48 Therefore we ca obtai Thus it follows that = [ bt]( h α [ ]( ( ( ( α ( ( i ( bt h C = C K i= (40 Hece b h α BMO( ( ( K ( Deotig 0 = C h K α ( ( ( firstly we estimate as i Theorem 4 Alyig emma we imme- diately arrive at 48

12 O Abdalrhma et al = ( α [ bt ]( h = b 0 BMO ( ( ( C So we ca get that C b Cb h α ( ( ( ( K ( 0 BMO BMO Next we estimate et x A [ ] ( ( ( ( ( bt h K x y b x b y h y dy A ( ( ( ( C b x b y h y x y dy A ( B ( ( ( B C b x b h y dy b b y h y dy ( A ( ( ( ( C b x b h b b h A B B (4 (4 = Thus from emmas 4-7 We obtai that α = b = C = = [ bt]( h 0 BMO α = ( α α = = ( 0 BMO h ( ( ( ( b x b h b ( B ( b( b h ( b 0 BMO ( 0 ( ( B BMO h α C = = 0 h α C = = 0 = α ( = ( ( ( ( ( ( ( ( ( ( ( b b x b ( ( B ( ( ( B ( ( ( ( ( B ( ( ( B ( ( ( ( α B h ( ( ( ( B B B α 0 ( B ( ( ( ( 49

13 O Abdalrhma et al Therefore we get = α = b [ bt]( h 0 BMO ( ( ( ( ( ( α ( ( ( ( h α ι C = = 0 ( ( ( (4 ( This for ( that * If ( = ( α [ bt ]( h ( = 0 ( ( = ( α [ bt ]( h = ( 0 > ( ( < ( ( ( α = 0 b alog with Remar tells us [ bt]( h BMO ( ( ( ( ( α h ( ( ( α ι C C = 0 = ( ( N ( = mi = ( it is follows from Remar ad Hölder s ieuality that α = b ( [ bt]( h 0 BMO C = = ( ( ( ( α ι ( ( ( ( ( ( ( α (( ι ( = α h = 0 ( = ( ( α h 0 ( ( * ( ( ( ( ( ( ( α ι ( * 440

14 = C C * N ( = mi ( This imlies that C b Cb h α 0 ( ( ( ( BMO BMO K ( O Abdalrhma et al (44 Fially we estimate for ay x A by the same way to argumet i we obtai that bt h K x y b x b y h y dy ad [ bt]( h α = b ( = = [ ] ( ( ( ( ( A C ( b( x b( y h ( y x y dy A ( B ( A B A ( ( C b( x bb h b b B h ( C b x b h y dy b b y h y dy ( α = BMO ( = 0 ( = α h ( b( b h ( b 0 BMO ( B ( ( ( α ( B ( = 0 ( ( B ( ( ( ( α h B ( ( α α ( = 0 ( B ( ( ( ( ( α ( ( ( ( h α ι = 0 ( ( ( C = C = ( BMO ( ( ( b x b h b ( ( ( B ( ( ( (45 (46 ( ( α [ bt ]( h = ( 0 ( ( = ( α [ bt ]( h = ( 0 ( ( > 44

15 O Abdalrhma et al Hece we arrive at that similar argumet i the roof Theorem 4 This comletes the roof of Theorem 4 C b Cb h α 0 BMO( BMO( ( ( K ( by the Acowledgemets This aer is suorted by Natioal Natural Foudatio of Chia (Grat No 5606 Refereces [] Calderó A ad Zygmud A (956 O Sigular Itegrals America Joural of Mathematics htts://doiorg/007/757 [] Calderó A ad Zygmud A (978 O Sigular Itegral with Variable Kerels Alied Aalysis 7-8 htts://doiorg/0080/ [] Joué J- (98 Calderó-Zygmud Oerators Pseudo-Differetial Oerators ad the Cauchy Itegral of Calderó I: ecture Notes i Math Vol 994 Sriger-Verlag Berli Heidelberg [4] Coifma R Rochberg R ad Weiss G (976 Factorizatio Theorems for Hardy Saces i Several Variables Aals of Mathematics htts://doiorg/007/ [5] Kováči O ad Ráosí J (99 O Saces (x ad W (x Czechoslova Mathematical Joural [6] Dieig ad Ruica M (00 Calderó-Zygmud Oerators o Geeralized ebesgue Saces ( ad Problems Related to Fluid Dyamics Joural für die Reie ud Agewadte Mathemati htts://doiorg/055/crll0008 [7] Che Y evie S ad Rao M (006 Variable Exoet iear Growth Fuctioals i Image Restoratio SIAM Joural o Alied Mathematics htts://doiorg/07/ [8] i F i Z ad Pi (00 Variable Exoet Fuctioals i Image Restoratio Alied Mathematics ad Comutatio [9] Harulehto P Hästö P atvala V ad Toivae O (0 Critical Variable Exoet Fuctioals i Image Restoratio Alied Mathematics etters [0] Izui M (009 Herz ad Amalgam Saces with Variable Exoet the Haar Wavelets ad Greediess of the Wavelet System East Joural o Aroximatios [] Izui M (00 Boudedess of Commutators o Herz Saces with Variable Exoet Redicoti del Circolo Matematico di Palermo htts://doiorg/0007/s [] Izui M (00 Fractioal Itegrals o Herz-Morrey Saces with Variable Exoet Hiroshima Mathematical Joural [] Wag ad Tao S (04 Boudedess of ittlewood-paley Oerators ad Their Commutators o Herz-Morrey Saces with Variable Exoet Joural of Ieualities ad Alicatios 04 7 htts://doiorg/086/09-4x-04-7 [4] Wag ad Tao S (05 Parameterized ittlewood-paley Oerators ad Their Commutators o ebegue Saces with Variable Exoet Aalysis i Theory ad Alicatios -4 [5] Ta J ad iu Z (05 Some Boudedess of Homogeeous Fractioal Itegrals 44

16 O Abdalrhma et al o Variable Exoet Fuctio Saces ACTA Mathematics Sciece (Chiese Series [6] Omer A Afif A ad Tao S (06 The Boudedess for Commutators of Calderó-Zygmud Oerator o Herz-Tybe Hardy Saces with Variable Exoet Joural of Alied Mathematics ad Physics htts://doiorg/046/am06460 [7] Wag ad Tao S (06 Parameterized ittlewood-paley Oerators ad Their Commutators o Herz Saces with Variable Exoets Turish Joural of Mathematics htts://doiorg/0906/mat-4-5 [8] Cuz-Uribe D Fioreza A Martell JM ad Perez C (006 The Boudedess of Classical Oerators o Variable Saces Aales Academiae Scietiarum Feicae-Mathe- matica 9-64 [9] Almeida A Hasaov J ad Samo S (008 Maximal ad Potetial Oerators i Variable Exoet Morrey Saces Georgia Mathematical Joural [0] Cruz-Uribe D ad Fioreza A (0 Variable ebesgue Saces: Foudatios ad Harmoic Aalysis I: Alied ad Numerical Harmoic Aalysis Sriger New Yor [] Dieig (005 Maximal Fuctio o Musiela-Orlicz Saces ad Geeralized ebesgue Saces Bulleti des Scieces Mathématiues htts://doiorg/006/bulsci00000 Submit or recommed ext mauscrit to SCIRP ad we will rovide best service for you: Accetig re-submissio iuiries through Faceboo iedi Twitter etc A wide selectio of ourals (iclusive of 9 subects more tha 00 ourals Providig 4-hour high-uality service User-friedly olie submissio system Fair ad swift eer-review system Efficiet tyesettig ad roofreadig rocedure Dislay of the result of dowloads ad visits as well as the umber of cited articles Maximum dissemiatio of your research wor Submit your mauscrit at: htt://aersubmissioscirorg/ Or cotact am@scirorg 44