Equivalence of K-andJ -methods for limiting real interpolation spaces

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1 Available online a wwwciencedireccom Journal of Funcional Analyi ) wwweleviercom/locae/jfa Equivalence of K-andJ -meho for limiing real inerpolaion pace Fernando Cobo a,,1, Thoma Kühn b,1 a Deparameno de Análii Maemáico, Faculad de Maemáica, Univeridad Compluene de Madrid, Plaza de Ciencia 3, Madrid, Spain b Mahemaiche Iniu, Fakulä für Mahemaik und Informaik, Univeriä Leipzig, Johannigae 26, D Leipzig, Germany Received 29 July 2010; acceped 31 Augu 2011 Available online 13 Sepember 2011 Communicaed by Alain Conne Abrac We conider limiing real inerpolaion pace defined by uing power of ieraed logarihm and how heir decripion by mean of he J -funcional Our reul allow o complemen ome eimae on approximaion of ochaic inegral 2011 Elevier Inc All righ reerved Keywor: Limiing inerpolaion pace; J -funcional; K-funcional; Lorenz Zygmund pace; Beov pace 1 Inroducion The real inerpolaion mehod A 0,A 1 ) θ,q play an ouanding role in applicaion of inerpolaion heory o funcional analyi, harmonic analyi, approximaion heory, parial differenial equaion and ome oher domain of mahemaic We refer, for example, o he book by Buzer and Beren [7], Bergh and Löfröm [5], Triebel [29], Benne and Sharpley [4], Brudnyĭ and Krugljak [6], Conne [12] and Amrein, Boue de Monvel and Georgecu [2] The mo familiar definiion of A 0,A 1 ) θ,q are hoe given by Peere K- and J -funcional The equivalence heorem ee [5, Secion 33]) how ha hee conrucion produce he ame * Correponding auhor addree: cobo@maucme F Cobo), kuehn@mahuni-leipzigde T Kühn) 1 Auhor have been uppored in par by he Spanih Minierio de Ciencia e Innovación MTM ) /$ ee fron maer 2011 Elevier Inc All righ reerved doi:101016/jjfa

2 F Cobo, T Kühn / Journal of Funcional Analyi ) pace Thi fac i an imporan iem in he developmen of he heory In fac, i yiel deniy of A 0 A 1 in A 0,A 1 ) θ,q if q<, and i i ueful o eablih he reieraion heorem or he dualiy heorem, among oher reul The definiion of A 0,A 1 ) θ,q require 0 <θ<1, bu cerain queion in funcion pace have moivaed he inveigaion of limiing real inerpolaion pace Then θ = 0orθ = 1, and he definiion may alo include logarihmic erm See, for example, he book by Milman [25] and he paper by Gomez and Milman [22], Dokorkii [13], Evan and Opic [15], Evan, Opic and Pick [16], Gogaihvili, Opic and Trebel [21], Cobo, Fernández-Cabrera, Kühn and Ullrich [9], Cobo, Fernández-Cabrera and Mayło [10], Fernández-Marínez and Signe [17] and Ahmed, Edmun, Evan and Karadzhov [1] Working wih limiing meho, i i naural o reric o ordered couple, where A 0 A 1 If θ = 0 hee meho produce pace which are very cloe o A 0, and when θ = 1 hey are very near o A 1 Among he limiing meho, K-meho have been more widely ued han J -meho Equivalence beween limiing K- and J -conrucion i only known for ome choice of parameer and, on he conrary o he cae of he real mehod, an adjumen i needed wih he effec ha a logarihmic facor hould be inered o obain he K-decripion of a limiing J - pace ee [9, Theorem 42] and [10, Corollary 510]) In Sepember 2009, Sefan Gei [18] aked one of he preen auhor abou he J -decripion of he K-pace defined by he condiion up e K,a) < 11) log ) α Here α>0 i arbirary The moivaion for hi queion i he poenial ue of a limiing equivalence heorem in ochaic finance, more pecifically, in approximaion of ochaic inegral ee [19,20]) K-pace defined by 11) have been conidered in [15,16,21,17] Even more general K- pace have been udied in hee reference, including in he definiion power of ieraed logarihm and no only he L -norm bu alo he L q -norm However, heir J -decripion ha no been inveigaed ye, excep for he cae α = 1 in 11) ee [9]) Accordingly, in hi paper we eablih J -decripion for all hee K-pace Our reul reveal a new phenomenon relaed o he facor of correcion: i change wih logarihm which define he K-pace The paper i organized a follow In Secion 2 we fix noaion and inroduce he K- and J -pace ha we are dealing wih Secion 3 conain ome baic properie of limiing pace a well a ome example The main reul of he paper are in Secion 4 where we prove he equivalence heorem Our argumen are baed on a modified verion of he o-called fundamenal lemma of inerpolaion heory and do no ue diviibiliy reul or rong form of he fundamenal lemma Finally, in Secion 5, we eablih limiing formulae for couple of vecorvalued equence pace and we how ha our reul can be ued o complemen ome eimae of C Gei, S Gei and M Hujo [19,20] on approximaion of ochaic inegral uing deerminiic equidian ne 2 Preliminarie For ieraed logarihm we ue he noaion L 0 ) =, L 1 ) = log and L j ) = log L j 1 ) ) if j>1

3 3698 F Cobo, T Kühn / Journal of Funcional Analyi ) Similarly, we wrie LL 1 ) = 1 + log and LL j ) = 1 + log LL j 1 ) ) if j>1 For ieraed exponenial funcion we pu E 0 ) =, E 1 ) = e and E j ) = e E j 1) if j>1 If ᾱ = α 0,α 1,,α m ) R m+1,wee ϕᾱ) = m L j ) α j j=0 A uual, given any 1 q, he conjugae index q i defined by 1/q + 1/q = 1, where we pu 1/ =0 If X, Y are quaniie depending on cerain parameer, by X Y we mean ha here i a conan c>0 independen of all parameer uch ha X cy We pu X Y if X Y and Y X Similarly, given wo non-negaive funcion f),g) defined on G R, we ay ha hey are equivalen and we wrie f g if here are conan c 1,c 2 > 0 uch ha c 1 f) g) c 2 f) for all G Le A 0,A 1 be Banach pace wih A 0 A 1 Here he ymbol mean coninuou embedding For >0, Peere K- and J -funcional are defined by and K,a) = K,a; A 0,A 1 ) = inf { } a 0 A0 + a 1 A1 : a = a 0 + a 1,a j A j, a A1, J,a)= J,a; A 0,A 1 ) = max { a A0, a A1 }, a A0 Definiion 21 Le A 0,A 1 be Banach pace wih A 0 A 1 Suppoe ha 1 q, m N {0} and le ᾱ = α 0,α 1,,α m ) R m+1 uch ha he funcion ϕᾱ aifie 1 d ϕᾱ) q < if 1 q<, 21) 1 up ϕᾱ) < if q = a ᾱ,q;k = The K-pace A 0,A 1 )ᾱ,q;k coni of all a A 1 which have a finie norm up K,a) ϕᾱ) K,a) ϕᾱ) ) ) q 1/q d if 1 q<, if q =

4 F Cobo, T Kühn / Journal of Funcional Analyi ) The pace decribed in Definiion 21 are pecial cae of he general K-mehod udied in [6] and [27] For a A 1, i follow from ha a A1 K1,a) K,a), ) ) 1 q 1/q d a A1 ϕᾱ) a ᾱ,q;k 22) So A 0,A 1 )ᾱ,q;k ={0} if 21) doe no hold By 22), we have ha A 0,A 1 )ᾱ,q;k A 1 Moreover, uing ha i i eaily checked ha A 0 A 0,A 1 )ᾱ,q;k K,a) a A0 if a A 0, Example 22 If α 0 > 0, or α 0 = 0 and α 1 > 1/q hen ϕᾱ aifie 21) Condiion 21) alo hol if α 0 = 0, α 1 = =α r 1 = 1/q and α r > 1/q for ome 2 r m Example 23 If α 0 > 1 hen we have A 0,A 1 )ᾱ,q;k = A 1 becaue a ᾱ,q;k a A1 ϕᾱ) ) ) q 1/q d c a A1 Example 24 When m = 0 and ᾱ = θ) wih 0 <θ<1, we recover he claical real inerpolaion pace A 0,A 1 ) θ,q realized a a K-pace ee [5,29,4]) The cae m N and ᾱ R m+1 wih 0 <α 0 < 1 i a pecial cae of he well-known real mehod wih funcion parameer ee [23,24,28]) For hi reaon, we are only inereed here in he cae α 0 = 0 See [16,21,17] for oher reul on pace A 0,A 1 )ᾱ,q;k Nex we inroduce J -pace Definiion 25 Le A 0,A 1 be Banach pace wih A 0 A 1 Suppoe ha 1 q, m N {0} and le β = β 0,β 1,,β m ) R m+1 uch ha he funcion ϕ β aifie up ϕ β ) ) q d < if 1 <q, ϕ β ) < if q = 1 23)

5 3700 F Cobo, T Kühn / Journal of Funcional Analyi ) The pace A 0,A 1 ) β,q;j i formed by all hoe a A 1 for which here i a rongly meaurable funcion u) wih value in A 0 uch ha a = u) d convergence in A 1 ) 24) and J,u)) ϕ β ) ) ) q 1/q d < 25) he inegral hould be replaced by he upremum if q = ) The norm in A 0,A 1 ) β,q;j i a β,q;j = inf { J,u)) ϕ β ) ) ) q 1/q } d where he infimum i aken over all repreenaion u aifying 24) and 25) The pace A 0,A 1 ) β,q;j are paricular cae of he general J -mehod ee [6,27]) They are alo iing beween A 0 and A 1 Indeed, ince any a A 0 can be repreened a a = eem 1) ad/, embedding A 0 A 0,A 1 ) β,q;j follow by uing ha ϕ β i a coninuou funcion On he oher hand, given any a A 0,A 1 ) β,q;j and any J -repreenaion a = u) d/, we derive a A1 u) A1 d ϕ β ) ) q d J,u)) ϕ β ) ) ) q 1/q d By 23), he fir inegral in he la inequaliy i finie, o A 0,A 1 ) β,q;j A 1 Noe alo ha if 23) doe no hold hen a β,q;j = 0 for any a A 0 Thi fac can be hown by following he ame idea a in [7, Propoiion 327] Indeed, ake any a A 0 and le ψ) be a non-negaive funcion aifying ha ψ)/ϕ d β ))q )1/q = 1 Pu u) = ψ)/ a ψ) Then a = d u) Uing ha J,u)) u) A 0 for, we obain

6 F Cobo, T Kühn / Journal of Funcional Analyi ) ψ) ϕ β ) ϕ β ) ) a β,q;j = ψ) ψ) ) a β,q;j ) ) ψ) q d ϕ β ) Taking he upremum over all poible funcion ψ we derive ha ϕ β ) Thi yiel ha a β,q;j = 0 if 23) i no aified ) q a β,q;j a A 0 J,u)) ϕ β ) a A0 = a A0 ) ) q 1/q d Example 26 Condiion 23) i aified if β 0 < 1, or β 0 = 1 and β 1 < 1/q,orβ 0 = 1, β 1 = =β r 1 = 1/q and β r < 1/q for ome 2 r m Example 27 When m = 0 and β = θ) wih 0 <θ<1, we ge again he real inerpolaion pace A 0,A 1 ) θ,q bu hi ime in he form of a J -pace ee [5,29]) When m N and β R m+1 wih 0 <β 0 < 1 we recover pace decribed by he real mehod wih funcion parameer ee [23,24]) If m = 0 = β 0, we obain pace udied in [9] and [10] 3 Some properie and example of limiing K-pace Le B = B 0,B 1 ) be anoher couple of Banach pace wih B 0 B 1 By T LĀ, B) we mean ha T i a bounded linear operaor from A 1 ino B 1, whoe rericion o A 0 give a bounded linear operaor from A 0 ino B 0 We pu T Ai,B i for he norm of T acing from A i ino B i i = 0, 1) I i eay o check ha he rericion T : A 0,A 1 )ᾱ,q;k B 0,B 1 )ᾱ,q;k i alo bounded Nex we eimae i norm in he limi cae α 0 = 0 We hall need he funcion LL j Given any real number α, we pu α + = max{α, 0} Theorem 31 Le Ā = A 0,A 1 ), B = B 0,B 1 ) be couple of Banach pace wih A 0 A 1, B 0 B 1 Suppoe ha 1 q,m N {0} and le ᾱ = α 0,α 1,,α m ) R m+1 uch ha α 0 = 0 and α 1 > 1/q, orα 0 = 0,α 1 = =α r 1 = 1/q and α r > 1/q for ome 2 r m If T LĀ, B) and M i = T Ai,B i for i = 0, 1, we have T A0,A 1 )ᾱ,q;k,b 0,B 1 )ᾱ,q;k { M0 if M 1 M 0, M 0 mj=1 LL α+ j j M 1 /M 0 ) if M 0 M 1

7 3702 F Cobo, T Kühn / Journal of Funcional Analyi ) Proof Since K,Ta; B 0,B 1 ) M 0 KM 1 /M 0,a; A 0,A 1 ), > 0, a A 1, and K,a) i an increaing funcion of,form 1 M 0 we obain Ta ᾱ,q;k M 0 KM1 /M 0,a) ϕᾱ) Suppoe now M 0 M 1 I follow by inducion ha ) ) q 1/q d M 0 a 0 ᾱ,q;k L k M 1 /M 0 ) L k )LL k M 1 /M 0 ), E k 1), k N 31) We obain Ta ᾱ,q;k M 0 M 0 M 0 up m j=1 KM1 /M 0,a) ϕᾱm 1 /M 0 ) { } ϕᾱm1 /M 0 ) ϕᾱ) ϕᾱm 1 /M 0 ) ϕᾱ) a ᾱ,q;k LL α+ j j M 1 /M 0 ) a ᾱ,q;k, ) ) q 1/q d where we have ued 31) in he la inequaliy The cae q = can be reaed analogouly In he pecial cae m = 1,α 0 = 0 and α 1 = 1, we recover a norm eimae eablihed in [9, Theorem 49] Nex we give example of limiing inerpolaion pace Le Ω, μ) be a σ -finie meaure pace and le wx) be a weigh on Ω, ha i, a poiive meaurable funcion on Ω A uual, we pu L q w) = { f : f Lq w) = wf Lq < } Le w 0,w 1 be weigh on Ω uch ha w 0 x) w 1 x) μ-ae Then L q w 0 ) L q w 1 ) Replacing w 1 x) by w 1 x)/, which only produce an equivalen norm in L q w 1 ), we may aume ha w 1 x) w 0 x) μ-ae in Ω Theorem 32 Le Ω, μ) be a σ -finie meaure pace, le 1 q, m N {0} and le ᾱ = α 0,α 1,,α m ) R m+1 uch ha α 0 = 0 and α 1 > 1/q in hi cae we le r = 1), or α 1 = =α r 1 = 1/q and α r > 1/q for ome 2 r m

8 F Cobo, T Kühn / Journal of Funcional Analyi ) Le w 0 and w 1 be weigh on Ω uch ha w 1 x) w 0 x) μ-ae Pu wx) = w 0 x)l 1/q r ) w0 x) / m L α j j w 1 x) j=r ) w0 x) w 1 x) Then we have wih equivalen norm Lq w 0 ), L q w 1 ) ) ᾱ,q;k = L qw) Proof Since K,f; L q w 0 ), L q w 1 ) ) Ω [ min { w0 x), w 1 x) } fx) ] q dμx), we obain f ᾱ,q;k Ω fx) q Le I be he inerior inegral We have [ min{w0 x), w 1 x)} ϕᾱ) ] q d dμx) I = w 1 x) q w 0 x)/w 1 x) ) q d ϕᾱ) + w 0 x) q w 0 x)/w 1 x) ) 1 q d = I 1 + I 2 ϕᾱ) To eimae I 2, noe ha ) [ 1 q m ) q r 1 1 = L α j j ϕᾱ) ) L j )] j=r j=1 We le r 1 j=1 L j ) = 1ifr = 1 Take any ε>0 uch ha α r ε) > 1/q Uing ha and L ε r ) m j=r+1 L α j j ) i equivalen o a non-decreaing funcion L ε r ) m j=r+1 L α j j ) i equivalen o a non-increaing funcion

9 3704 F Cobo, T Kühn / Journal of Funcional Analyi ) we ge w 0 x)/w 1 x) 1 m j=r L α j j ))q [ m d L r 1 ) L 1 ) L α j j j=r ) ] q ) w0 x) w0 x) L r w 1 x) w 1 x) Hence ) [ I 2 w 0 x) q w0 x) m ) ] q L r L α j w0 x) j = wx) q w 1 x) w j=r 1 x) Nex we conider I 1 Clearly, I 1 0 Uing ha for any 0 <ε<1 he funcion ε /ϕ α ) i equivalen o a non-decreaing funcion, we obain ) I 1 w 1 x) q w0 x) q )) w0 x) q ϕᾱ w 1 x) w 1 x) [ r 1 = w 0 x) q j=1 ) ] 1[ w0 x) m L j L α j j w 1 x) j=r ) ] q w0 x) w 1 x) ) [ w 0 x) q w0 x) m ) ] q L r L α j w0 x) j I 2 w 1 x) w j=r 1 x) Conequenly, I wx) q Thi yiel ha f ᾱ,q;k Ω [ wx) fx) ] q dμx) Nex we wrie down a paricular cae of Theorem 32 which refer o weighed equence pace Corollary 33 Le 1 q, m N {0} and le ᾱ = α 0,α 1,,α m ) R m+1 uch ha α 0 = 0 and α 1 > 1/q in hi cae we le r = 1), or α 1 = =α r 1 = 1/q and α r > 1/q for ome 2 r m Then we have wih equivalen norm lq n 1/2 ) ),l q ᾱ,q;k = l qw n )

10 F Cobo, T Kühn / Journal of Funcional Analyi ) where w n = n 1/2 L 1/q r In paricular, for α>0, we derive l n 1/2 ) ),l n 1/2 ) m j=r L α j j n 1/2 ) 0,α), ;K = l n 1/2 1 + log n) α) In Secion 5 we will derive a vecor-valued verion of Corollary 33 which i ueful o approximae ochaic inegral When α 0 = 0 and α 1 = 1, Theorem 32 give back a reul eablihed in [9, Theorem 48] The following reieraion formulae beween he real mehod and he limiing K-mehod will allow u o how more example Theorem 34 Le A 0,A 1 be Banach pace wih A 0 A 1,le0 <θ<1, 1 q, m N {0} and le ᾱ = α 0,α 1,,α m ) R m+1 uch ha α 0 = 0 and α 1 > 1/q in hi cae we le r = 1), or α 1 = =α r 1 = 1/q and α r > 1/q for ome 2 r m Le ρ = ρ 0,ρ 1,,ρ m ) R m+1 he m + 1)-uple defined by ρ 0 = θ and { ρ1 = α 1 1/q and ρ j = α j for 2 j m if r = 1, ρ 1 = =ρ r 1 = 0, ρ r = α r 1/q, ρ j = α j, r + 1 j m if 2 r m Then we have wih equivalen norm ) A0,A 1 ) θ,q,a 1 ᾱ,q;k = A 0,A 1 ) ρ,q;k Proof By Holmed formula ee [5, Corollary 362]) we have Hence K ) 1/1 θ) [,a; A 0,A 1 ) θ,q,a 1 θ K,a; A 0,A 1 ) ] q a ᾱ,q;k = = 1/1 θ) 0 [ 0 0 [ θ K,a) max{, 1 θ } ϕᾱ) 1 ϕᾱ) ] q ) q d d ] θ K,a) ) q

11 3706 F Cobo, T Kühn / Journal of Funcional Analyi ) Proceeding a for I 2 in Theorem 32, we derive ha 1 ) q d ϕᾱ) = 1 m j=r L α j j ))q m d r 1 j=1 L j ) j=r L α j j )) ql r ) where = max{, 1 θ } I follow ha a ᾱ,q;k 0 θ ) ) K,a) q 1/q mj=r L α j j )Lr 1/q ) For 0 <<Em 1/1 θ) 1), wehavek,a) a A1 Whence E 1/1 θ) m 1) 0 θ K,a) ) q mj=r L α j j E m1))l 1 r ) E 1/1 θ) m 1) 0 ) 1 θ)q a q A 1 Conequenly, = c a q A 1 a ᾱ,q;k E 1/1 θ) m 1) E 1/1 θ) m 1) E 1/1 θ) m 1) 1 ϕ ρ) ) ) q a q A 1 θ K,a) mj=r L α j j 1 θ )Lr 1/q 1 θ ) K,a) ϕ ρ ) ) q θ K,a) ) ) q 1/q mj=r L α j j 1 θ )Lr 1/q 1 θ ) ) ) q 1/q Nex we wrie down a concree cae of hi reul for funcion pace Le Ω, μ) be a finie meaure pace For 1 <p, 1 q and ν j R for 1 j m, he generalized Lorenz Zygmund pace L p,q;ν1,,ν m coni of all equivalen clae of) meaurable funcion f on Ω

12 F Cobo, T Kühn / Journal of Funcional Analyi ) which have a finie norm μω) [ f = Lp,q;ν1,,νm 0 1/p m j=1 LL ν j j )f ) ] q d Here = max{,1/},f ) = 1/) 0 f ) and f i he non-increaing rearrangemen of f defined by f ) = inf { >0: μ { x Ω: fx) > }) } Thee pace are udied in [26] and [14] They are a pecial cae of o-called Lorenz Karamaa pace in [14] If m = 1, he pace L p,q;ν1 coincide wih he Lorenz Zygmund pace L p,q log L) ν1 inroduced in [3] Corollary 35 Le Ω, μ) be a finie meaure pace, le 1 <p<, 1 q,m N {0} and le ᾱ = α 0,α 1,,α m ) R m+1 uch ha α 0 = 0 and α 1 > 1/q in hi cae we le r = 1), or α 1 = =α r 1 = 1/q and α r > 1/q for ome 2 r m Then we have wih equivalence of norm L p,q,l 1 )ᾱ,q;k = L p,q;ν1,,ν m, where ν 1 = =ν r 1 = 0, ν r = 1/q) α r and ν j = α j for r + 1 j m Proof A i i well known, L p,q = L,L 1 ) 1/p,q Le ρ = 1/p, r 1 {}}{ 0,,0,α r 1/q, α r+1,,α m ) I follow from Theorem 33 ha L p,q,l 1 )ᾱ,q;k = L,L 1 ) ρ,q;k Uing he ideniy ee [5,29]), we derive 1/ K,f; L,L 1 ) = f ) = f 1/) 0 f Lp,q,L 1 )ᾱ,q;k f L,L 1 ) ρ,q;k [ / m = 1/p f 1/) L α j j )L 1/q r j=r ) ] q d

13 3708 F Cobo, T Kühn / Journal of Funcional Analyi ) The equivalence heorem E 1 m 1) 0 μω) 0 = f Lp,q;ν1,,νm [ / m ] q 1/p f 1/) LL α j d j )LL 1/q r ) j=r [ / m ] q 1/p f ) LL α j d j )LL 1/q r ) j=r [ 1/p LL 1/q) α r r ) m j=r+1 LL α j j )f ) ] q d Thi ecion i he cenral par of he paper I conain he J -decripion of limiing K-pace For hi aim, we fir inroduce pace baed on a modified K-funcional and he Lebegue meaure on 0, ) inead of d/ For n N, we pu K n, a) = K E n ), a ) = inf { a 0 A0 + E n ) a 1 A1 : a = a 0 + a 1,a j A j } Definiion 41 Le A 0,A 1 be Banach pace wih A 0 A 1 Suppoe ha 1 q, n N, m N {0} and le γ = γ 0,γ 1,,γ m ) R m+1 uch ha he funcion ϕ γ aifie We pu A 0,A 1 ) n) γ,q;k = { a A 1 : a n) γ,q;k = ) ) 1 q 1/q < 41) ϕ γ ) ) ) Kn, a) q 1/q < } ϕ γ ) A imilar argumen o he one given afer Definiion 21 how ha if 41) doe no hold hen A 0,A 1 ) n) γ,q;k ={0} Hence, if γ 0 < 1/q he pace i {0} On he oher hand, 41) i aified if γ 0 > 1/q, orγ 0 = γ 1 = =γ r 1 = 1/q and γ r > 1/q for ome 1 r m The nex lemma decribe he relaionhip beween Definiion 21 and 41 Lemma 42 Le A 0,A 1 be Banach pace wih A 0 A 1 Suppoe ha 1 q, n N, m N {0} and le γ = γ 0,γ 1,,γ m ) R m+1 uch ha he funcion ϕ γ aifie 41) Then A 0,A 1 ) n) γ,q;k = A 0,A 1 )ᾱ,q;k

14 F Cobo, T Kühn / Journal of Funcional Analyi ) where ᾱ i he m + n + 1)-uple given by n 1 ᾱ = {}}{ ) 0, 1/q,,1/q, γ 0,γ 1,,γ m Proof Aume fir 1 q< We make he change of variable = E n ), o = L n ) and We obain ) Kn, a) q = ϕ γ ) = = E m+n 1) E m+n 1) d L n 1 ) L 1 ) K,a) ϕ γ L n ))L n 1 ) L 1 ) ) K,a) q d ϕᾱ) To deal wih he cae q =, we wrie again = E n ) We ge K n, a) up = up ϕ γ ) E m+n 1) K,a) ϕ γ L n )) = up E m+n 1) K,a) ϕᾱ) ) q d n {}}{ where now ᾱ = 0,,0,γ 0,γ 1,,γ m ) Remark 43 Le ᾱ = α 0,α 1,,α m ) R m+1 uch ha α 0 = 0 and α 1 > 1/q in hi cae we le r = 1), or α 1 = =α r 1 = 1/q and α r > 1/q for ome 2 r m Pu γ = α r,α r+1,,α m ) Then, according o Lemma 42, we have A 0,A 1 )ᾱ,q;k = A 0,A 1 ) r) γ,q;k Conequenly, he K-pace we are inereed in can alway be repreened in he new cale A 0,A 1 ) r) γ,q;k wih γ 0 > 1/q Subequenly, we conider only K r -pace of hi kind Remark 44 Noe alo ha he pace defined by condiion 11) in he Inroducion coincide wih A 0,A 1 ) 1) γ, ;K, where m = 0 and γ = γ ) wih γ>0 Now we conider pace defined by a modified J -funcional For n N,wee J n, a) = J E n ), a ) = max { a A0, E n ) a A1 }

15 3710 F Cobo, T Kühn / Journal of Funcional Analyi ) Definiion 45 Le A 0,A 1 be Banach pace wih A 0 A 1,le1 q, n N, m N {0} and le δ = δ 0,δ 1,,δ m ) R m+1 The pace A 0,A 1 ) n) δ,q;j i formed by all hoe a A 1 which can be repreened by a = v) convergence in A 1 ) 42) where v) i a rongly meaurable funcion wih value in A 0 and ) ) Jn, v)) q 1/q < 43) ϕ δ ) We pu { a n) δ,q;j = inf ) Jn, v)) q 1/q } ) ϕ δ ) where he infimum i aken over all v aifying 42) and 43) In order o compare he modified K- and J -pace, we fir prove a limi verion of he o-called fundamenal lemma of inerpolaion heory Lemma 46 Le A 0,A 1 be Banach pace wih A 0 A 1,len N and m N {0} Aume ha a A 1 wih lim K,a)/ = 0 Then here i a rongly meaurable funcion v :[, ) A 0 wih and a = v) convergence in A 1 ) ) K n 4,a) J n,v) c,, where c>0 i a conan independen of a and v Proof Chooe r 0 N uch ha 2 r 0 1 <2 r 0, e 0 = E n 1) and for k N pu k = E n 2 k ) and I k =[ k 1, k ) Find vecor a 0,k 1 A 0 and a 1,k 1 A 1 wih a = a 0,k 1 + a 1,k 1 and a 0,k 1 A0 + r0 +k a 1,k 1 A1 2K r0 +k,a)

16 F Cobo, T Kühn / Journal of Funcional Analyi ) Le v 0 = a 0,0 and v k = a 0,k a 0,k 1 for k N Then v k 1 ) A 0 and N a v j = a a 0,N A1 = a 1,N A1 2 A1 K r 0 +N,a) 0 r0 +N j=0 an Hence a = k=1 v k 1, where he erie converge in A 1 Nex define I i clear ha v) = { v 0 2 r 0 if [, 2 r 0), v k 2 r 0 +k 1 if [2 r0+k 1, 2 r0+k ), k = 1, 2, Moreover, for [, 2 r 0) we have v) = v k 1 = a k=1 ) 1 J n,v) 2 r 0 J ) n 2 r 0,v 0 1 [ ] a0,0 2 r A0 + r0 a a 1,0 A1 0 1 [ 2Kn 2 r r,a ) ] + r0 a A1 0 c 1 K n 2 r 0+1,a) 2 r 0 For k N and [2 r 0+k 1, 2 r 0+k ), we derive K n 4,a) c 1 ) 1 J n,v) 2 r 0+k 1 J n 2 r 0 +k ),v k 1 [ )] a0,k 2 r A0 + a 0,k 1 A0 + r0 +k a1,k A1 + a 1,k 1 A1 0+k 1 4K r 0 +k+1,a) 2 r 0+k 1 8K n4,a) = 4K n2 r 0+k+1,a) 2 r 0+k 1 The proof i complee

17 3712 F Cobo, T Kühn / Journal of Funcional Analyi ) Now we can eablih he equivalence heorem beween K n - and J n -pace Theorem 47 Le A 0,A 1 be Banach pace wih A 0 A 1 Suppoe ha 1 q, n N, m N {0} and le γ = γ 0,γ 1,,γ m ) R m+1 uch ha γ 0 > 1/q Le δ = γ 0 1,γ 1,,γ m ) Then we have wih equivalen norm A 0,A 1 ) n) γ,q;k = A 0,A 1 ) n) δ,q;j Proof For laer ue we noe ha he aumpion on γ 0 implie ha ϕ γ ) = j=0 L j ) γ j equivalen o an increaing funcion and ha here i a conan c 1 > 0 wih i Take any a A 0,A 1 ) n) value in A 0 uch ha γ,q;k ϕ γ 2) c 1 ϕ γ ), 44) By Lemma 46, here i a rongly meaurable funcion v) wih a = v) ) K n 4,a) wih J n,v) c, Thi inequaliy and 44) yield ) Jn, v)) q c ) Kn 4,a) q 1/q ) ϕ δ ) = c c 2 ϕ δ ) ) Kn 4,a) q 1/q ) ϕ γ ) ) Kn 4,a) q ϕ γ 4) a n) γ,q;k Therefore, a belong o A 0,A 1 ) n) wih a n) a n) δ,q;j δ,q;j γ,q;k Take now a A 0,A 1 ) n) δ,q;j and le a = v) be a J n-repreenaion of a wih We have ) Jn, v)) q ϕ δ ) 2 a n) δ,q;j

18 F Cobo, T Kühn / Journal of Funcional Analyi ) K n, a) = K n,v) ) min { 1, E n )/E n ) } J n,v) ) J n,v) ) + En ) J n, v)) E n ) Thu a n) γ,q;k = + ) Kn, a) q 1/q d) ϕ γ ) 1 ϕ γ ) = I 1 + I 2 E n ) ϕ γ ) J n,v) ) ) q d J n, v)) E n ) ) q d We fir eimae I 1 Chooe ε>0 uch ha ρ = γ 0 ε>1/q Uing Hölder inequaliy, we ge J n,v) ) = Jn, v)) ϕ δ ) Jn, v)) ϕ δ ) ) q 1/q ) ε ) q 1/q ) ε ϕ δ ) ε ϕ γ ) 1+ε ) q ) q Since ρ<γ 0, he funcion ϕ γ )/ ρ i equivalen o an increaing funcion I follow ha Conequenly, ϕ γ ) ρ ) 1 q 1+ε ρ ϕ γ ) ρ 1 q 1 ε+ρ = ϕ γ ) ε 1 q

19 3714 F Cobo, T Kühn / Journal of Funcional Analyi ) I 1 = εq 1 [ ϕ δ ) Jn, v)) ϕ δ ) ) ) q 1/q ] ε d ) Jn, v)) q ) ε εq 1 d ) Jn, v)) q ϕ δ ) 2 a n) δ,q;j Nex we proceed wih I 2 Fir we conider he cae q = 1 We have I 2 = J n, v)) E n ) ) E n ) ϕ γ ) d The econd inegral behave like E n )/ϕ γ) becaue E n )/e /2 ϕ γ) i equivalen o an increaing funcion and o I follow ha E n ) e /2 ϕ γ ) e/2 d E n) e /2 ϕ γ ) e /2 d E n) ϕ γ ) I 2 = J n, v)) ϕ γ ) J n, v)) ϕ δ ) 2 a n) δ,1;j J n, v)) ϕ γ ) Suppoe now ha 1 <q By Hölder inequaliy we derive I 2 2 [ E n ) ϕ γ ) [ E n ) ϕ γ ) ) Jn, v)) q 1/q ) ϕ δ ) ϕ δ ) E n ) ϕ δ ) E n ) ) q ] q d ) q a n) δ,q;j ] q d Since e /2 ϕ δ )/E n) i equivalen o a decreaing funcion, he inner inegral behave like

20 F Cobo, T Kühn / Journal of Funcional Analyi ) ϕ δ )/E n) and o I 2 2 q d a n) δ,q;j The la inegral i finie becaue 1 <q Conequenly, a n) γ,q;k I 1 + I 2 a n) δ,q;j Thi complee he proof Nex we how he relaion of J n -pace o he uual J -pace Lemma 48 Le A 0,A 1 be Banach pace wih A 0 A 1 Suppoe ha 1 q, n N, m N {0} and le β = β 0,β 1,,β m ) R m+1 Then A 0,A 1 ) n) β,q;j = A 0,A 1 ) ρ,q;j {}}{ where ρ i he m + n + 1)-uple given by ρ = 0, 1/q,, 1/q,β 0,β 1,,β m ) Proof Le a = v) Making he change of variable = L n) wih we obain where a = Moreover, if q =,wehave E m+n 1) n 1 = d/l n 1 ) L 1 ) vl n )) d = L n 1 ) L 1 ) E m+n 1) u) = v L n ) ) /L n 1 ) L 1 ) u) d J m, v)) J,vL n ))) up ϕ β ) = up E m+n 1) ϕ β L n)) J,u)) = up E m+n 1) ϕ β L n))l n 1 ) L 1 )) 1 = up E m+n 1) J,u)) ϕ ρ )

21 3716 F Cobo, T Kühn / Journal of Funcional Analyi ) Similarly, for 1 q<, we derive ) Jn, v)) q ϕ β ) = = E m+n 1) E m+n 1) J,u)) ) q d ϕ β L n))[l n 1 ) L 1 )] 1 L n 1 ) L 1 ) J,u)) ϕ ρ ) ) q d Thi yiel ha A 0,A 1 ) n) β,q;j A 0,A 1 ) ρ,q;j To check he convere incluion, ake any a A 0,A 1 ) ρ,q;j wih J -repreenaion a = d E m+n1) u) We make he change of variable = E n) wih = d/l n 1 ) L 1 ) Pu v) = ue n ))E n 1 ) E 1 ) Then Moreover, a = u E n ) ) E n 1 ) E 1 ) = v) E m+n 1) J,u)) ϕ ρ ) ) q d = = J n, v)) ϕ ρ E n )) n 1 k=1 E k) ) Jn, v)) q ϕ β ) ) q n 1 E k ) k=1 Thi complee he proof Now we eablih he equivalence heorem for limiing K-pace Corollary 49 Le A 0,A 1 be Banach pace wih A 0 A 1,le1 q, m N {0} and uppoe ha ᾱ = α 0,α 1,,α m ) R m+1 wih α 0 = 0 and α 1 > 1/q in hi cae we le r = 1), or α 1 = =α r 1 = 1/q and α r > 1/q for ome 2 r m Then we have wih equivalen norm r 1 A 0,A 1 )ᾱ,q;k = A 0,A 1 ) β,q;j {}}{ where β = 0, 1/q,, 1/q,α r 1,α r+1,,α m )

22 F Cobo, T Kühn / Journal of Funcional Analyi ) Proof Le γ = α r,α r+1,,α m ) R m r+1 and ω = α r 1,α r+1,,α m ) R m r+1 Uing Lemma 42, Theorem 47 and Lemma 48, we derive A 0,A 1 )ᾱ,q;k = A 0,A 1 ) r) γ,q;k = A 0,A 1 ) r) ω,q;j = A 0,A 1 ) β,q;j Corollary 49 how ha in he equivalence heorem for he limiing cae, i i needed he correcion facor Ψ)= r L j ) j=1 which depen on he logarihm ha define he K-pace bu i doe no depend on he L q -norm: Condiion K,a) mj=1 L j ) α j L q ) d i equivalen o he fac ha a can be J -repreened by ome u) uch ha Ψ)J,u)) mj=1 L j ) α j ) d L q A a direc conequence we ge he following J -decripion for he K-pace menioned in he Inroducion Corollary 410 Le A 0,A 1 be Banach pace wih A 0 A 1,le1 q, α>0 and ρ> 1/q Then we have wih equivalen norm and A 0,A 1 ) 0,α), ;K = A 0,A 1 ) 0,α 1), ;J, A 0,A 1 ) 0,1/q,ρ),q;K = A 0,A 1 ) 0, 1/q,ρ 1),q;J For q< we have alo he following deniy reul Corollary 411 Le A 0,A 1 be Banach pace wih A 0 A 1,le1 q<, m N {0} and uppoe ha ᾱ = α 0,α 1,,α m ) R m+1 wih α 0 = 0 and α 1 > 1/q, orα 0 = 0,α 1 = = α r 1 = 1/q and α r > 1/q for ome 2 r m Then A 0 i dene in A 0,A 1 )ᾱ,q;k Proof By Corollary 49, we have wih equivalence of norm A 0,A 1 )ᾱ,q;k = A 0,A 1 ) β,q;j r 1 {}}{ where β = 0, 1/q,, 1/q,α r 1,α r+1,,α m ) Take any a A 0,A 1 ) β,q;j and le a = u) d/ be a J -repreenaion of a Given any ε>0, ince q<, i follow from 25)

23 3718 F Cobo, T Kühn / Journal of Funcional Analyi ) ha here i > uch ha Moreover u) d A0 J,u)) ϕ β ) J,u)) ϕ β ) ϕ β )d J,u)) ϕ β ) ) ) q 1/q d <ε ) ) q 1/q d d ϕ β )q < becaue ϕ β )q / i a coninuou funcion Le w = u) d/ Then we have ha w A 0 wih a w = u) d/ and ) ) J,u)) q 1/q a w β,q;j d ϕ β ) <ε 5 Remark on approximaion of ochaic inegral In hi la ecion we will how ome complemen o he reul of C Gei, S Gei and M Hujo [19,20] Le F be a Banach pace, le λ n ) be a equence of poiive number and 1 q We define l q λ n F) a he collecion of all equence x = x n ) F having a finie norm ) q x lq λ n F) = λn x n F We are inereed in inerpolaion properie of he couple l q n 1/2 F),l q F )) Since for any x = x n ) l F ) we have n=1 K,x; l q n 1/2 F ),l q F ) ) { = min n 1/2, } ) q x n F, 51) n=1 we can proceed a in he calar cae ee Theorem 32 and Corollary 33) and obain, for 1 q and α>1/q, he formula lq n 1/2 F ),l q F ) ) 0,α),q;K = l q n 1/2 1 + log n) 1/q) α F ) 52) The following reul refer o he non-diagonal cae

24 F Cobo, T Kühn / Journal of Funcional Analyi ) Theorem 51 Le F be a Banach pace, le 1 q< and α>1/q Then l n 1/2 F ),l F ) ) 0,α),q;K l q n 1/2 1/q 1 + log n) α F ) l n 1/2 1 + log n α F ) Proof Le x = x n ) l F ) Fixn N and le n 1/2 By 51), we have Whence x 0,α),q;K K,x) n 1/2 x n F Taking he upremum over n N we obain 1 K,x) LL α 1 ) n 1/2 x n F n ) ) q 1/q d d LL αq 1 ) n 1/2 1 + log n) 1/q) α x n F l n 1/2 F ),l F ) ) 0,α),q;K l n 1/2 1 + log n α F ) On he oher hand, ince K,x) n 1/2 x n F for n 1/2 <n+ 1) 1/2, we derive Thi how ha ) x q K,x) q 0,α),q;K d LL α 1 ) 1 n+1) 1/2 n 1/2 ) q d x n F LL αq n=1 n 1/2 1 ) n 1/2 ) q 1 1 x n F n 1/2 n n=1 1/2 LL αq 1 n) = n 1/2 1/q 1 + log n) α ) q x n F n=1 l n 1/2 F ),l F ) ) 0,α),q;K l q n 1/2 1/q 1 + log n) α F ) and complee he proof

25 3720 F Cobo, T Kühn / Journal of Funcional Analyi ) Remark 52 Noe ha he pace A = l q n 1/2 1/q 1 + log n) α), B = l n 1/2 1 + log n α) are no comparable Indeed, ake any N N and conider he equence x = x n ) defined by { n x n = 1/q 1/2 1 + log n) α for n N, 0 ele Then x A = N n=1 1 = N 1/q bu x B = max n1 + log n) = N 1/q 1 + log N 1 n N So A i no coninuouly embedded in B On he oher hand, ake he equence y = y n ) defined by y n = n 1/2 1 + log n) α 1/q Clearly y belong o B wih y B = 1, bu i doe no belong o A becaue n 1/2 1/q 1 + log n) α ) q 1 y n = n1 + log n) = n=1 In order o relae he previou reul on limiing inerpolaion pace o approximaion of ochaic inegral, le γ be he andard Gauian meaure on he real line, le h n ) n=1 be he orhonormal bai of L 2 γ ) given by Hermie polynomial and conider he Sobolev pace { 1/2 } D 1,2 γ ) = f = λ n h n L 2 γ ): f D1,2 γ ) = n + 1)λn) 2 < n=1 By real inerpolaion in he ordered couple D 1,2 γ ), L 2 γ )) we obain he Beov pace B 1 θ 2,q γ ) = D 1,2γ ), L 2 γ )) θ,q ee [29,5,30]) Beov pace wih a more general moohne have been alo conidered in he lieraure o deal wih a number of problem in funcion pace ee, for example, [8,31,32,11]) We are inereed here in limiing Beov pace generaed by he 0,α),q; K)-mehod wih α>1/q n=1 n=1 B 1,α 2,q γ ) = D 1,2 γ ), L 2 γ ) ) 0,α),q;K 53) The following reul i a conequence of 53), Theorem 51 and he inerpolaion propery Theorem 31) Corollary 53 Le T : L 2 γ ) l L 2 γ )) be a bounded linear operaor uch ha he rericion o D 1,2 γ ) define a bounded operaor T : D 1,2 γ ) l nl 2 γ )) If1 q and α>1/q, hen he rericion are bounded T : B 1,α 2,q γ ) l q n 1/2 1/q 1 + log n) α L 2 γ ) ), T : B 1,α 2,q γ ) l n 1/2 1 + log n α L 2 γ ) )

26 F Cobo, T Kühn / Journal of Funcional Analyi ) Corollary 53 allow u o complemen he reul by C Gei, S Gei and M Hujo [19,20] on approximaion of ochaic inegral uing deerminiic equidian ne of cardinaliy n + 1 They howed, in paricular, ha for f D 1,2 γ ) he opimal approximaion rae n 1/2 i aained, while for f B 1 θ 2, γ ) wih 0 <θ<1 he rae i only n 1/2+θ/2 In he limiing cae θ = 0, Corollary 53 provide addiional informaion Indeed, by our reul and he argumen given in [20, pp ], i follow ha for f B 1,α 2, γ ) wih α>0he approximaion rae i n 1/2 1 + log n) α, o he opimal rae n 1/2 i achieved up o a logarihmic facor Reference [1] I Ahmed, DE Edmun, WD Evan, GE Karadzhov, Reieraion heorem for he K-inerpolaion mehod in limiing cae, Mah Nachr ) [2] WO Amrein, A Boue de Monvel, V Georgecu, C 0 -Group, Commuaor Meho and Specral Theory of N- Body Hamilonian, Progr Mah, vol 135, Birkhäuer, Bael, 1996 [3] C Benne, K Rudnick, On Lorenz Zygmund pace, Dieraione Mah ) 1 67 [4] C Benne, R Sharpley, Inerpolaion of Operaor, Academic Pre, Boon, 1988 [5] J Bergh, J Löfröm, Inerpolaion Space An Inroducion, Springer, Berlin, 1976 [6] YuA Brudnyĭ, NYa Krugljak, Inerpolaion Funcor and Inerpolaion Space, vol 1, Norh-Holland, Amerdam, 1991 [7] PL Buzer, H Beren, Semi-Group of Operaor and Approximaion, Springer, New York, 1967 [8] F Cobo, DL Fernandez, Hardy Sobolev pace and Beov pace wih a funcion parameer, in: Funcion Space and Applicaion, in: Lecure Noe in Mah, vol 1302, Springer, Berlin, 1988, pp [9] F Cobo, LM Fernández-Cabrera, T Kühn, T Ullrich, On an exreme cla of real inerpolaion pace, J Func Anal ) [10] F Cobo, LM Fernández-Cabrera, M Mayło, Abrac limi J -pace, J London Mah Soc 2) ) [11] F Cobo, T Kühn, Approximaion and enropy number in Beov pace of generalized moohne, J Approx Theory ) [12] A Conne, Noncommuaive Geomery, Academic Pre, San Diego, 1994 [13] RYa Dokorkii, Reieraion relaion of he real inerpolaion mehod, Sovie Mah Dokl ) [14] DE Edmun, WD Evan, Hardy Operaor, Funcion Space and Embedding, Springer, Berlin, 2004 [15] WD Evan, B Opic, Real inerpolaion wih logarihmic funcor and reieraion, Canad J Mah ) [16] WD Evan, B Opic, L Pick, Real inerpolaion wih logarihmic funcor, J Inequal Appl ) [17] P Fernández-Marínez, T Signe, Real inerpolaion wih ymmeric pace and lowly varying funcion, Q J Mah 2010), doi:101093/qmah/haq009, in pre [18] S Gei, Privae communicaion, Sepember 2009 [19] C Gei, S Gei, On approximaion of a cla of ochaic inegral and inerpolaion, Soch Soch Rep ) [20] S Gei, M Hujo, Inerpolaion and approximaion in L 2 γ ), J Approx Theory ) [21] A Gogaihvili, B Opic, W Trebel, Limiing reieraion for real inerpolaion wih lowly varying funcion, Mah Nachr ) [22] ME Gomez, M Milman, Exrapolaion pace and almo-everywhere convergence of ingular inegral, J London Mah Soc ) [23] J Guavon, A funcion parameer in connecion wih inerpolaion of Banach pace, Mah Scand ) [24] S Janon, Minimal and maximal meho of inerpolaion, J Func Anal ) [25] M Milman, Exrapolaion and Opimal Decompoiion, Lecure Noe in Mah, vol 1580, Springer, Berlin, 1994 [26] B Opic, L Pick, On generalized Lorenz Zygmund pace, Mah Inequal Appl ) [27] J Peere, A Theory of Inerpolaion of Normed Space, Noa Ma, vol 39, In Ma Pura Apl, Rio de Janeiro, 1968

27 3722 F Cobo, T Kühn / Journal of Funcional Analyi ) [28] L-E Peron, Inerpolaion wih a parameer funcion, Mah Scand ) [29] H Triebel, Inerpolaion Theory, Funcion Space, Differenial Operaor, Norh-Holland, Amerdam, 1978 [30] H Triebel, Theory of Funcion Space, Birkhäuer, Bael, 1983 [31] H Triebel, The Srucure of Funcion, Birkhäuer, Bael, 2001 [32] H Triebel, Theory of Funcion Space III, Birkhäuer, Bael, 2006