OPTIMAL ESTIMATES IN LORENTZ SPACES OF SEQUENCES WITH AN INCREASING WEIGHT

Transcription

1 THE PUBLISHIG HOUSE PROCEEDIGS OF THE ROMAIA ACADEMY Series A OF THE ROMAIA ACADEMY Volue 4 uber / OPTIMAL ESTIMATES I LORETZ SPACES OF SEQUECES WITH A ICREASIG WEIGHT Soria BARZA Aca-icoleta MARCOCI 2 Liviu-Gabriel MARCOCI 2 Lars-Eri PERSSO 3 Karlstad Uiversity Deartet of Matheatics SE-6588 Karlstad Sede 2 Techical Uiversity of Civil Egieerig Bucharest Deartet of Matheatics ad Couter Sciece RO-2396 Bucharest Roaia 3 Luleå Uiversity of Techology Deartet of Matheatics SE Luleå Sede arvi Uiversity College PO BOX ARVIK oray E-ail: acaarcoci@gailco We study the eighted Loretz saces of sequeces to fid otial estiates betee differet equivalet quasi-ors We also obtai the best costat i the triagle iequality Key ords: eighted Loretz sequece saces orability equivalet ors decoositio or aial or dual or shar costats ITRODUCTIO Let < < For a sequece =( ) c (the sace of ull sequeces) the decreasig rearrageet of is obtaied by rearragig ( ) i decreasig order A oegative sequece of real ubers = ( ) ill be called a eight sequece Without loss of geerality e ay suose that l ( ) see eg [6] We recall the defiitio of the eighted Loretz saces of sequeces d ( )={ : := ( ) < } = It is roved i [6] that is a or if ad oly if is a decreasig sequece Also d ( ) is equivaletly orable if ad oly if / + C / = 2 = W W () here W = = This is also the coditio hich characterizes the boudedess of the discrete Hardy oerator fro d ( ) to l ( ) Here l ( ) is the classical eighted Lebesgue sace of sequeces I the recet aers [3] ad [4] the authors cosidered estiates betee a dual or (defied i ters of Köthe duality) decoositio or ad the usual or i the cotiuous case The ai reaso for these cosideratio as that the equivalet or defied i ters of aial fuctio (or Hardy oerator) does ot give the best costat i the triagle iequality I [3] it as cosidered the case of the classical Lorez s saces L ith < s hile i [4] siilar results ere roved for the eighted Loretz saces Γ ( ) here is a icreasig eight fuctio Usig siilar techiques the sae relatios betee ors o s Loretz saces of sequeces l ere roved i [5] I this aer e eted the results roved i [5] to the case of ore geeral Loretz saces of sequeces ith a icreasig eight Sice e eed the sace

2 2 Otial estiates i Loretz saces of sequeces ith a icreasig eight 2 d( ) to be orable e ill ecessarily have to assue that the eight satisfies the coditio () Here ad i the sequel e deote by = Also deotes the cojugate ide of aely is such that + = Observe that if is a icreasig sequece the is a decreasig sequece As a cosequece of the fact that is equivalet to a or it is easy to see that it is a quasi-or satisfyig the triagle iequality uiforly i the ubers of ters eressed as follos: there eists a ( ) costat C > such that for every fiite collectio { } d( ) it yields that C (2) ( ) ( ) = = It ca be roved also that the coverse result holds Moreover a alterative equivalet or is give by eas of the folloig decoositio or: { } ( ) ( ) ( ) = = := if : = for all I hat follos e use the otatios y if = i y i i= i ad y if y for all here =( ) ad y =( y ) The aer is orgaized as follos: I Sectio 2 e state soe leas hich ill be used i the subsequet Sectios I Sectio 3 e rove our ai results A iortat result i Sectio 3 is Theore 3 hich gives us otial costats betee ad Aother result of this Sectio is that the dual or of = ( ) d( ) coicides ith (see Theore 33) I Theore 34 e obtai that C ' here C is the shar costat We rove also that the dual or coicides ith the decoositio or (see Theore 35) I the last Sectio e reset soe alicatios eg the triagle iequality for the quasi-or 2 PRELIMIARY RESULTS We eed the folloig stateet related to the dual or: LEMMA 2 Let =( ) d( ) here is a arbitrary ositive sequece hich satisfies coditio () The ' = ' A roof ca be foud eg i [2] LEMMA 22 Let =( ) d( ) ad a eight hich satisfies coditio () The the folloig stateets hold: (a) The equality =if (3) ( ) ( ) ( ) holds here the ifiu is tae over all fiite o-egative sequeces = (b) If y the y ( ) ( ) ( ) ( ) ( ) (c) If y ad y he the y ( ) ( ) Proof The roof is siilar ith the roof of Lea 27 i [3] so e oit the details

3 22 S Barza A Marcoci LG Marcoci LE Persso 3 LEMMA 23 For each d( ) e have that (4) ( ) ( ) Proof The roof is siilar ith the roof of Lea 28 i [3] The cocet of level sequece ith resect to aother sequece as used i aalogy ith the level fuctio for the study of siilar robles i the fraeor of classical Loretz saces of sequeces i [5] or [] The uique sequece =( ) i Theore 33 i [5] is called the level sequece of =( ) ith resect to ϕ= ( ϕ ) This defiitio is aalogous to the oe give i the aer [3] for the cotiuous case For ore geeral cases of level fuctios see eg [] Let > For a eight sequece =( ) hich satisfies for soe costat C ad ay e deote by + + C (5) = = + + su = = C = the otial costat i the above iequality The folloig Lea ill be iortat i the roof of Theore 3 LEMMA 24 Let > ad =( ) be a icreasig eight sequece hich satisfies (5) such that is decreasig for ay oegative iteger The + C = su = = (6) Proof It is clear that iequality Let us fi Observe that for = C su = = We rove o the coverse e have equality so e ay suose that > If e deote z = = by = z = + ad r = < e have to rove that r r r r z = = Observe that both ( ) ad ( z ) are decreasig sequeces by hyothesis Deote by X X = = Z = z = ad C = Z X X X Sice is decreasig (see eg [8]) e have that = C for Alyig the Z Z Z discrete versio of the ajorizatio ricile also o as Karaata's iequality (see eg [8]) for the decreasig sequeces ( ) ad y = Cz ad for the cove fuctio Φ ()= t t r r < e get that ( Cz ) or C z r r r r r = = = =

4 4 Otial estiates i Loretz saces of sequeces ith a icreasig eight 23 r r = r r The last iequality ilies that ( = ) ( = ) sice < z r Sice as arbitrary z = e get the desired iequality ad the roof is colete Rear 25 The above Lea reais true also i the case of decreasig sequeces if e cosider = I the et Sectios e assue that all of our eights are icreasig sequeces hich satisfies () ad the hyotheses of Lea 24 3 MAI RESULTS THEOREM 3 Let < < ad be a icreasig eight Let =( ) d( ) be a oegative ad decreasig sequece ad =( ) be the level sequece ith resect to the sequece ϕ=( ϕ ) here ϕ = The e have that C (7) here C is defied by (6) The costats i (7) are otial Proof First e cosider the left had side i (7) By Theore 33 i [5] e have that j = λ all j I Alyig Hölder's iequality e obtai that j for ( ) = = I I I I I = I I I λ = This estiate ad Theore 33 i [5] yields the first iequality i (7) Let us o cosider = ( ) here = for all ad let = ( ) be the level sequece of = ( ) ith resect to ϕ=( ϕ ) ϕ = for all By Theore 33 i [5] Lea 36 i [] 9 ad Hölder iequality e obtai that = = = = = = To obtai the secod iequality i (7) it is sufficiet to rove that ~ (8) C (9) here the costat C is as i Lea 24 Let E:= { : = } The e have that Z E = + I here { I } are disjoit ad such that = for all I () I + I

5 24 S Barza A Marcoci LG Marcoci LE Persso 5 By Hölder's iequality e get I I I () We also have that ( ) = ( ) = Hece by () ad () it yields that E E E ( ) C I ( I ) + I I I I Fro Lea 24 it follos that (9) holds hich eas that the right had side iequality i (7) is roved It oly reais to rove the sharess of the obtaied iequalities We ote that the left had side iequality i (7) becoes equality if for a fied e tae = ( ) if here = i i= otherise For the right had side iequality i (7) e obtai equality for = ( ) ad fied here the e have that ( ) if otherise if = otherise = ad i ( = ) i= = = It is easy to verify that = ( ) here = Sice is arbitrary e get that also the costat o the right had-side iequality (7) is otial The roof is colete Recall that for a sequece = ( ) ( ) < < its dual or is defied by ' :=su{ y : y =( y ) y = } here the sureu is tae over all sequeces y =( y ) ( ) ith y = Fro Lea 2 ad the Hardy-Littleood iequality (see eg [2] 44) for ay sequece =( ) d( ) < < e have that ' = su y : y = = (2) here the sureu is tae over all oegative ad o-icreasig sequeces y =( y ) d( ) ith y = ad =( ) deotes the o-icreasig rearrageet of the sequece = ( ) I the et Proositio e suarize soe ell o results for < < ad =( ) d( ) For the roof see eg [2] or [6] PROPOSITIO 32 Let < < (a) ' (b) If is a decreasig sequece the The the folloig stateets hold:

6 6 Otial estiates i Loretz saces of sequeces ith a icreasig eight 25 ' = (3) (c) If the sequece ( ) is o-icreasig the (3) holds (d) If =( ) is a arbitrary oegative sequece i d( ) the e have that if z z s I the case of Loretz saces L ( R µ ) it as roved by I Haleri i [7] (see also [] Theore 365) that equality i (4) holds i the case of real fuctios defied o the iterval () ad that the s ifiu is attaied For < s a colete roof i the case of L ( R µ ) here ( R µ ) deotes a σ -fiite oatoic easure sace as give i the recet aer [3] I the case he ( R µ ) is a totally -fiite easure sace coletely atoic ith all atos havig the sae easure the equality as roved i [5] Our et result eteds the result fro [5] THEOREM 33 Let < < Suose that =( ) d( ) is a oegative ad o-icreasig sequece Let ϕ=( ϕ ) here ϕ = The e have that ' = if z = z here =( ) is the level sequece of =( ) ith resect to the sequece ϕ=( ϕ ) Proof I vie of (4) i Proositio 32 ad Theore 33 (b) i [5] it is sufficiet to rove that (4) ' (5) We deote by E ={ : = } Accordig to Theore i [5] it yields that E = I here I are such that = (6) I I We first cosider =( ) here =( ) for all As before e have that = = = We choose y= ( y ) here y = hich ilies that y = Fro Theore i [5] for each e have that for I = λ here λ is a costat Thus = λ ad y = λ = λ = ( ) Moreover e have that I I I I = ( ) y E E Thus e obtai that = = y This ilies (5) ad the roof is colete The fial result i this Sectio gives the shar estiate of the stadard or via the dual or THEOREM 34 Let < < The for ay sequece =( ) d( ) ad for ay icreasig eight hich satisfies coditio () it yields that C (7) ' here C is defied by (6) The costat is otial The roof follos iediately fro Theore 33 ad Theore 3

7 26 S Barza A Marcoci LG Marcoci LE Persso 7 The et Theore shos the coicidece of the dual ad the decoositio ors THEOREM 35 Let < < ad ( ) be a icreasig eight The for ay sequece =( ) d( ) e have ' = (8) ( ) Proof The roof is siilar ith Theore 52 fro [5] COROLLARY 36 Let =( ) d( ) < The = (9) ( ) ( ) Proof The equality (9) follos iediately fro (8) ad Lea 2 4 APPLICATIOS Fro Theore 34 ad Theore 35 e get the folloig "triagle iequality": THEOREM 4 Let < < a icreasig eight hich satisfies () ad suose that ( ) ( ) =( ) d( ) = The the folloig iequality holds here C give by (6) is the otial costat Proof Let < < C (2) ( ) ( ) = = Let us ote that the iequality (2) is equivalet to the iequality C (2) ( ) here = ( ) is i d( ) Iequality (2) follos directly fro Theore 34 ad Theore 35 if Moreover for =( ) ith = for a fied e obtai that otherise ( ) = = Fro Theore 33 ad Theore 35 e have that = ( ) / = ad therefore e get equality i (2) Thus also the sharess stateet is roved ad the roof is colete I the et Theore e rove a discrete versio of the Miosi iequality: THEOREM 42 Let < < ad =( ) be a oegative icreasig sequece of real ubers Assue that =( ) is a sequece ith the roerty that for ay the sequece =( ) belogs to d( ) ad deote by X =( X ) the sequece give by X = = v = 2 here v is a oegative sequece such that v = = < The X C v = here C is defied i (24) ad it is the otial costat i the iequality Proof By Theore 34 e have

8 8 Otial estiates i Loretz saces of sequeces ith a icreasig eight 27 C (22) ' here C is defied i (6) Let o y d( ) such that y = By Hölder's iequality e get = = X = y v y y v y v v By (22) e get our iequality The iequality is shar ad this coletes the roof of our theore Rear 43 Iequality (2) as roved i [9] i a ore geeral for but ith differet techiques ACKOWLEDGMETS The first aed author ould lie to tha the grou of Matheatical Modellig of Karlstad Uiversity for suort The secod ad the third aed authors ere artially suorted by CCSIS- UEFISCSU roject uber 538/29 PII-IDEI code 95/28 REFERECES G BEETT Factorizig the Classical Iequalities Me Aer Math Soc C BEETT R SHARPLEY Iterolatio of Oerators Acadeic Press Bosto S BARZA V KOLYADA J SORIA Shar costats related to the triagle iequality i Loretz saces Tras Aer Math Soc (29) 4 S BARZA J SORIA Shar costats betee equivalet ors i eighted Loretz saces J Austral Math Soc (2) 5 S BARZA A MARCOCI L-E PERSSO Best costats betee equivalet ors i Loretz sequece saces J Fuct Saces Al Vol 22 Article ID M J CARRO J A RAPOSO J SORIA Recet Develoets i the Theory of Loretz Saces ad Weighted Iequalities Me Aer Math Soc 87 Providece RI 27 7 I HALPERI Fuctio saces Caad J Math (953) 8 G J O JAMESO The q -cocavity costats of Loretz sequece saces ad related iequalities Math Z (998) 9 A KAMISKA A M PARRISH Coveity ad cocavity costats i Loretz ad Marcieieicz saces J Math Aal Al (28) G G LORETZ Berstei olyoials Uiv of Toroto Press Toroto 953 G SIAMO Saces defied by the level fuctio ad their duals Studia Math 9 52 (994) Received Deceber 28 2