Research Article Weighted Hardy Operators in Complementary Morrey Spaces

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1 Hindawi Publishing Copoaion Jounal of Funcion Spaces and Applicaions Volume 212, Aicle ID , 19 pages doi:1.1155/212/ Reseach Aicle Weighed Hady Opeaos in Complemenay Moey Spaces Dag Lukkassen, 1 Las-Eik Pesson, 2 and Sefan Samko 3 1 Depamen of Technology, Navik Univesiy College, P.O. Box 385, 855 Navik, Noway 2 Depamen of Mahemaics, Luleå Univesiy of Technology, SE Luleå, Sweden 3 Depaameno de Maemaica, Univesidade do Algave, Fao, Pougal Coespondence should be addessed o Las-Eik Pesson, laseik@lu.se Received 23 July 212; Acceped 2 Sepembe 212 Academic Edio: Alois Kufne Copyigh q 212 Dag Lukkassen e al. This is an open access aicle disibued unde he Ceaive Commons Aibuion License, which pemis unesiced use, disibuion, and epoducion in any medium, povided he oiginal wok is popely cied. We sudy he weighed p q-boundedness of he mulidimensional weighed Hady-ype opeaos Hw α and H α w wih adial ype weigh w w x, in he genealized complemenay Moey spaces L p,ψ {} Rn defined by an almos inceasing funcion ψ ψ. Wepoveaheoem which povides condiions, in ems of some inegal inequaliies imposed on ψ and w, fo such a boundedness. These condiions ae sufficien in he geneal case, bu we pove ha hey ae also necessay when he funcion ψ and he weigh w ae powe funcions. We also pove ha he spaces L p,ψ {} Ω ove bounded domains Ω ae embedded beween weighed Lebesgue space Lp wih he weigh ψ and such a space wih he weigh ψ, peubed by a logaihmic faco. Boh he embeddings ae shap. 1. Inoducion Hady opeaos and elaed Hady inequaliies ae widely sudied in vaious funcion spaces, and we efe o he books 1 4 and efeences heein. They coninue o aac aenion of eseaches boh as an ineesing mahemaical objec and a useful ool fo many puposes: see fo insance he ecen papes 5, 6. Resuls on weighed esimaions of Hady opeaos in Lebesgue spaces may be found in he abovemenioned books. In he papes 7 9 he weighed boundedness of he Hady ype opeaos was sudied in Moey spaces. In his pape we sudy muli-dimensional weighed Hady opeaos Hwf x α : x α n f ( y ) dy w x y < x w ( y ), Hα wf x : x α w x y > x f ( y ) dy y n ( ), w y 1.1

2 2 Jounal of Funcion Spaces and Applicaions whee α, in he so called complemenay Moey spaces. The one-dimensional case will include he vesions x Hwf x α x α 1 f d w x w, Hα wf x x α f d w x x w, x > 1.2 adjused fo he half-axis R 1, so ha in he sequel R n wih n 1 may be ead eihe as R 1 o R 1. The classical Moey spaces L p,λ Ω, Ω R n, defined by he nom ( f p,λ 1 : sup x Ω,> λ B x, f ( y ) p dy ) 1/p, 1 p<, λ n, 1.3 whee B x, Ω B x,, ae well known, in paicula, because of hei usage in he sudy of egulaiy popeies of soluions o PDE; see fo insance he books 1 12 and efeences heein. Thee ae also known vaious genealizaions of he classical Moey spaces L p,λ, and we efe fo insance o he suveying pape 13. One of he diec genealizaions is obained by eplacing λ in 1.3 by a funcion ϕ, usually saisfying some monooniciy ype condiions. We also denoe i as L p,ϕ Ω wihou dange of confusion. Such spaces appeaed in 14, 15 and wee widely sudied in 16, 17. The spaces L p,ϕ Ω, defined by loc he nom f ( 1 p,ϕ;loc : sup > ϕ B x, f ( y ) p dy) 1/p, 1.4 whee x Ω, ae known as genealized local Moey spaces Hady ype opeaos 1.1 in he spaces L p,ϕ Ω, L p,ϕ Ω have been sudied in 7 9. loc The nom in Moey spaces conols he smallness of he inegal B x, f y p dy ove small balls B x, and also a possible gowh of his inegal fo in he case Ω is unbounded. Thee ae also known spaces L p,ψ {x } Ω, called complemenay Moey spaces, wih he nom conolling possible gowh, as, of he inegal Ω\B x, f ( y ) p dy 1.5 ove exeio of balls. Such spaces have sense in he local seing only. I is inoduced in 16, 17 ha he space L p,ψ {x } Ω is defined as he space of all funcions f Lp loc Ω\{x } wih he finie nom f L p,ψ sup ψ 1/p {x } Ω f L p Ω\B x,, > 1.6 whee admission of he muliplie ψ vanishing a conols he gowh of he nom f L p Ω\B x,. Such spaces wee also sudied in some lae papes, and we efe fo insance

3 Jounal of Funcion Spaces and Applicaions 3 o 18. We efe also o he pape 19 whee vaiable exponen complemenay spaces of such ype wee inoduced. Duing he las decades vaious classical opeaos, such as maximal, singula, and poenial opeaos wee widely invesigaed boh in classical and genealized Moey spaces, including he complemenay Moey spaces. The mapping popeies of Hady ype opeaos in complemenay Moey spaces wee no known. In his pape we obain condiions fo he p q-boundedness of Hady ype opeaos in complemenay Moey spaces. Thus we make ceain conibuions o he known heoy of Hady ype inequaliies; see, fo example, he books 2 4 and efeences given hee. We also pove a new popey fo he genealized complemenay Moey spaces; see Ω ove bounded domains Ω ae embedded beween he weighed Lebesgue space L p wih he weigh ψ, and such a space wih he weigh ψ, peubed by a logaihmic faco. In he case whee ψ was a powe funcion, his was poved in 19. The pape is oganized as follows. In Secion 2 we give definiions and necessay peliminaies, including condiions fo adial funcions o belong o complemenay Moey spaces. In Secion 3 we pove he abovemenioned embedding of he genealized Moey space beween Lebesgue weighed spaces. In Secion 4, which plays a cucial ole in he pepaaion of he poofs of he main esuls, we pove poinwise esimae fo he Hadyype consucions via he nom defining he complemenay Moey space. In Secion 5 Theoem 3.1, by showing ha he spaces L p,ψ {} we give he final heoems on he weighed p q-boundedness of Hady opeaos in complemenay Moey spaces. In he appendix we collec vaious popeies of weighs fom he Bay-Sechkin class which we need when we fomulae some sufficien condiions fo he boundedness in ems of he Mauszewska-Olicz indices of he funcions ψ and w. 2. Complemenay Moey Spaces and Thei Popeies 2.1. Definiions Le Ω be an open se in R n, Ω R n and l diam Ω,<l, B x, {y R n : x y <} and B x, B x, Ω. Lealsoψ be a coninuous funcion nonnegaive on,l wih ψ : lim ψ which may abiaily gow as.lealso1 p<. Definiion 2.1. Le 1 p<. The complemenay Moey space L p,ψ {x } Ω, whee x Ω, is he space of funcions f L p loc Ω \{x } such ha ( ) 1/p ( ) f L p, ψ : sup {x } Ω ψ f y p dy <. > Ω\ B x, 2.1 In he case of powe faco ψ λ, λ>, we also denoe L p, λ {x } Ω : L p, ψ {x } Ω ψ λ 2.2 wihou dange of confusion.

4 4 Jounal of Funcion Spaces and Applicaions The space L p, ψ {x } Ω is nonivial in he case of any locally bounded funcion ψ wih an abiay behaviou a infiniy, because bounded funcions wih compac suppo belong hen o his space. We will also use he following noaion fo he modula: ( M p, ψ f; x, ) ( ) : ψ f y p dy. Ω\ B x, 2.3 The funcion ψ, defining he complemenay Moey space, will be called definiive funcion. Remak 2.2. The funcion ϕ in 1.4, defining he Moey spaces, in some papes is called weigh funcion. We pefe o call boh ϕ and ψ as definiive funcions, keeping he wod weigh fo is naual use in he heoy of funcion spaces, ha is, fo he cases whee he funcion f iself is conolled by a weigh On Belongness of Radial Funcions o he Space L p, ψ {x } Ω Lemma 2.3. Fo a nonnegaive adial ype funcions u u x x, x Ω, he condiion l sup ψ u p n 1 d < <<l 2.4 is sufficien o belong o L p, ψ {x } Ω. (I is also necessay, when Ω Rn o Ω is a ball ceneed a x.) The poof of Lemma 2.3 is obvious. By means of his lemma we easily obain he following coollay. Coollay 2.4. Le Ω be bounded. A powe funcion x x γ wih γ / n/p belongs o he space Ω if and only if L p, ψ {x } sup min{n γp,} ψ <. > 2.5 In he case n γp, he same holds wih min{n γp,} eplaced by ln.whenω R n, he necessay and sufficien condiions ae n γp < and sup > n γp ψ <. As is known, hee may be given sufficien numeical inequaliies fo he validiy of he inegal condiion in 2.4 in ems of Mauszewska-Olicz indices m u and M u of he funcion u. We give below such sufficien inequaliies, bu in ode no o ineup he main body of he pape by noions conneced wih such indices and elaed Bay-Sechkin funcion classes Z α,β, we pu hese noions in he appendix.

5 Jounal of Funcion Spaces and Applicaions 5 Le now u be a nonnegaive funcion in Bay-Sechkin class, namely, ) u Z n/p,l if l<, u Z n/p, n/p (R 1 if l. 2.6 Then l up n 1 d n u p, <<l, so ha he condiion sup <<l n ψ u p < is sufficien fo u u x x o be in L p, ψ {x } Ω. Noealsoha 2.6 is equivalen o he inequaliies pm u n <, pm u n < in ems of he indices, whee he second inequaliy is o be used only in he case l ;see A Complemenay Moey Spaces Ae Closely Embedded beween Weighed Lebesgue Spaces The complemenay Moey spaces ae close o a ceain weighed Lebesgue space as saed in he following heoem, which povides shap embeddings of independen inees. The noaion fo he weighed space is aken in he fom L p Ω,ϱ : {f : Ω ϱ x f x p dx < }. The class W,l, used in his heoem, is defined in he appendix; see Definiions A.1-A.2 heein. In he case whee ψ is a powe funcion, Theoem 3.1 was poved in 19. Theoem 3.1. Le Ω be a bounded open se, 1 p<, ψ W,l, l diam Ω, and le ψ be absoluely coninuous and ψ P : sup <. 3.1 <<l ψ Then L p( Ω,ψ ( y x )) L p, ψ {x } Ω ε>l p( Ω,ψ ε ( y x )), 3.2 whee ψ ε ψ / ln A/ 1 ε, A>l. The lef-hand side embedding in 3.2 is sic, when ψ saisfies he condiion l d ψ d C ψ 3.3 (in paicula, if ψ λ, λ>); ha is, hee exiss a funcion f f x such ha f L p, ψ {x } Ω, bu f / L p( Ω,ψ ( y x )). 3.4 The igh-hand side embedding in 3.2 is sic, when ψ saisfies he doubling condiion ψ 2 Cψ, (in paicula, if ψ λ, λ ), hee exiss a funcion g g x such ha g L p( Ω,ψ ε x x ), ε> bu g / L p, ψ {x } Ω. 3.5

6 6 Jounal of Funcion Spaces and Applicaions Poof. Wihou loss of genealiy, we may ake x fo simpliciy, supposing ha Ω. 1. The lef-hand side embedding: since he funcion ψ is almos inceasing, we have ( Ω ψ ( ) ( ) 1/p ( y f y dy) p ψ ( ) ( ) y f y p dy Ω\ B, ( C ψ Ω\ B, ) 1/p ) 1/p ( ) f y p dy 3.6 so ha f L p Ω,ψ y C f L p,ψ {} Ω The igh-hand side embedding: we have B, ( ) f y p ( ) ψε y dy B, ( ( ) y ) f y p d d ψ ε d dy, 3.8 whee ψ ε ψ [ ψ ψ 1 ε ] ln A/ 1 ln A/ 1 ε, 3.9 so ha ψ ε C ε ψ ln A/, C 1 ε ε P 1 ε ln A/l. 3.1 Theefoe, B x, f ( y ) p ψ ε ( y ) dy l ψ ε ( ψ ε f p f p L p,ψ {x } Ω {y Ω:< x y <} l L p Ω\ B x, ds ψ ε ψ d, f ( y ) ) p dy d 3.11 whee he las inegal conveges when ε>since ψ ε /ψ C ε / ln A/ 1 ε. 3. The sicness of he embeddings: he coesponding couneexamples ae f x 1 ψ 1/p x x n/p, g x ln ln B/ x x n/p ψ 1/p x, B > lee. 3.12

7 Jounal of Funcion Spaces and Applicaions 7 Calculaions fo he funcion f, which is obviously no in L p Ω,ψ y, aeeasy. Indeed, l f p S n 1 d L ψ p,ψ {} Ω ψ, 3.13 whee he igh-hand side is bounded by 3.3. In he case of he funcion g we have g p L p Ω,ψ ε Ω ln p ln B/ x dx S n 1 l x n 1 ε ln A/ x ln p ln B/ d < 1 ε ln A/ 3.14 fo evey ε>. Howeve, fo small,δ/2, whee δ dis, Ω, weobain ψ g p x dx ψ ln p ln B/ x dx x Ω: x > x Ω: < x <δ x n ψ x S n 1 ψ δ S n 1 ψ 2 ln p ln B/ d ψ ln p ln B/ d. ψ 3.15 Taking ino accoun ha ψ is almos inceasing and saisfies he doubling condiion, we ge 2 ln p ln B/ d ψ C ψ ( ψ ψ 2 lnp ln B ) 2 2 d ( C ln p ln B ) as, 3.16 which complees he poof of he heoem. Remak 3.2. Unde he assumpion 3.3, he uppe embedding in 3.2 does no hold wih ε. The coesponding couneexample is f x 1/ x x n ψ x x which is in L p,ψ {x } Ω, bu f/ L p Ω,ψ ε y x ε. 4. Weighed Esimaes of Funcions in Complemenay Moey Spaces In he following lemmas we give a poinwise esimae of he Hady-ype consucions in ems of he modula M p,ψ f; x, wih x. These esimaes ae of independen inees and also cucial fo ou sudy of he Hady opeaos in he complemenay Moey space.

8 8 Jounal of Funcion Spaces and Applicaions Lemma 4.1. Le 1 p<, <s p,lev, ψ s/p v W,, and v 2 cv.then z < f z s v z dz C V s d M s/p p,ψ ( f;, ), >, 4.1 fo f L p Ω \{}, wheec>does no depend on f and, and loc V s n 1 s/p v ψ s/p. 4.2 Poof. We use he dyadic decomposiion as follows: z < f z s v z dz B k f z s v z dz, 4.3 whee B k {z : 2 k 1 < z < 2 k }. Since hee exiss a β such ha β v is almos inceasing, we obseve ha 1 v z C v ( 2 k 1 ) 4.4 on B k. Applying his in 4.3 and making use of he Hölde inequaliy wih he exponen p/s 1, we obain z < f z s ( 2 k 1 v z dz C ) n 1 s/p ( ) s/p v ( 2 k 1 ) f z p dz. B k 4.5 Hence z < f z s ( v z dz C 2 k 1 ) n 1 s/p M s/p ( p,ψ f;, 2 k 1 ) v ( 2 k 1 ) ψ s/p( 2 k 1 ). 4.6 On he ohe hand we have V s M s/p p,ψ ( f;, ) d 2 k 2 k 1 ( ) Ms/p n 1 s/p 1 p,ψ f;, v ψ S/p d. 4.7

9 Jounal of Funcion Spaces and Applicaions 9 The funcion 1/ ψ M p,ψ f;, is deceasing, and he funcion 1/ v is almos deceasing afe muliplicaion by some powe funcions. Theefoe, V s M s/p p,ψ ( ) ( f;, d C 2 k k 1 ) n 1 s/p M s/p ( p,ψ f;, 2 k ) v ( 2 k ) ψ S/p( 2 k ) ( ) n 1 s/p M s/p ( C 2 k p,ψ f;, 2 k ) v ( 2 k ) ψ s/p( 2 k ) ( ) n 1 s/p M s/p ( C 1 2 k 1 p,ψ f;, 2 k 1 ) v ( 2 k 1 ) ψ s/p( 2 k 1 ). 4.8 Then 4.1 follows fom 4.6 and 4.8. Lemma 4.2. Le 1 p<, s p, v W R 1 and v 2 Cv, ψ 2 Cψ. Le also V s : n 1 s/p v. 4.9 ψ s/p Then z > v z f z s dz C V s M s/p ( ) p,ψ f;, d, 4.1 whee C> does no depend on > and f. Poof. We use he coesponding dyadic decomposiion: v f z s dz v z f z s dz, z > B k 4.11 whee B k {z :2 k < z < 2 k 1 }. Since hee exiss a β R 1 such ha β v is almos inceasing, we obain B k v z ( ) f z s dz C v 2 k 1 f z s dz, Bk 4.12 whee C may depend on β bu does no depend on and f. Applying he Hölde inequaliy wih he exponen p/s,wege z > v z ( )( ) ( ) s/p f z s n 1 s/p dz C v 2 k 1 2 k f z p dz B k ( C v 2 k 1 )( n 1 s/p ψ 2 ) ( ) k s/p 2 k M s/p p,ψ ( ) f;, 2 k. 4.13

10 1 Jounal of Funcion Spaces and Applicaions On he ohe hand, he inegal on he igh-hand side of 4.1 can be esimaed as follows: V s M s/p p,ψ ( f;, ) d 2 k 1 2 k [ ( )] s/p Mp,ψ f;, n 1 ns/p v d ψ ( ) [ ( C v 2 k M p,ψ f;, 2 k 1 ) ] s/p ( ) n 1 s/p ψ ( 2 k 1 ) 2 k ( C v 2 k 1 ) ψ 1/p( )( 2 k 2 k ) n 1 s/p M ( ) s/p p,ψ f;, 2 k 1, 4.14 which complees he poof. Coollay 4.3. Le 1 p<, lew, wψ 1/p W,l,w 2 cw, <l and Ω. Le also V : 1 w p ψ Z Then M p,ψ ( f w ;, ) C w p M ( ) p,ψ f;,, <<l Weighed Hady Opeaos in Complemenay Moey Spaces 5.1. Poinwise Esimaions The poof of ou main esul of his Secion given in Theoems 5.3 and 5.6 is pepaed by he following Theoems 5.1 and 5.2 on he poinwise esimaes of he Hady-ype opeaos. Theoem 5.1. Le 1 p,lew, wψ 1/p W, and w 2 Cw. The condiion ε V d < wih ε>, whee V n/p w ψ 1/p 5.1 is sufficien fo he Hady opeao H α w o be defined on he space L p,ψ {} Rn. Unde his condiion, he poinwise esimae H wf x x α C x α n w x V d f L p,ψ {} 5.2 holds. Poof. Apply Lemma 4.1 wih s 1.

11 Jounal of Funcion Spaces and Applicaions 11 Theoem 5.2. Le 1 p and 1/w W,ow W and w 2 Cw. The condiion ε V 1 d <, wih ε>, whee V w ψ 1/p n/p 5.3 is sufficien fo he Hady opeao H α w o be defined on he space L p,ψ {x } Rn, and in his case he following poinwise esimae: H α wf x C x α w x x V d f L p,ψ {x } 5.4 holds. Poof. Apply Lemma 4.2 wih s Weighed p q Boundedness of Hady Opeaos in Complemenay Moey Spaces The Case of he Opeao H α w The L p L q -boundedness of he mulidimensional Hady opeaos wihin he famewoks of Lebesgue spaces he case ϕ 1in 1.4 wih 1 <p< and <q< is well known: see fo insance, 4, page 54. Fo Moey spaces, boh local and global, he boundedness of Hady opeaos was sudied in 7, 8. We call aenion of he eade o he fac ha, in conas o he case of Lebesgue spaces, Hady-ype inequaliies in boh usual and complemenay Moey spaces diffeen fom Lebesgue spaces i.e., in he case ϕ oψ admi he value p 1. Theoem 5.3. Le 1 p, 1 q and le ( ) w, ψ 1/p w W R 1, w 2 cw. 5.5 The opeao H α w is bounded fom L p,ψ {} Rn o L q,ψ {} Rn if sup > W <,whee W : ψ ϱ n 1 ( ϱ α n/p ψ 1/p ϱ V ϱ ϱ q V d) dϱ, 5.6 and V ishesameasin 5.1. Unde his condiion, H α w f L q,ψ {} C sup W 1/q f L p,ψ > {}. 5.7

12 12 Jounal of Funcion Spaces and Applicaions Poof. Fom he esimae 5.2 of Theoem 5.1 we have H wf x ( α C x α n/p 1 x V ψ q/p x V x d) f L q,ψ. 5.8 {} Hence, calculaing Hwf α L p,ψ, passing o pola coodinaes, we obain 5.6. {} Remak 5.4. Noe ha 1 ϱ V ( ϱ ) V d in 5.6 if we suppose ha V Z. The following coollay gives sufficien condiions fo he boundedness of he opeao H α w in ems of he Mauszewska-Olicz indices of he funcion ψ and he weigh w we efe o he appendix fo hese noions. Coollay 5.5. Le 1 p q< (wih q p admied in he case α ) and he condiions 5.5 be saisfied. Suppose also ha ψ C αpq/q p n 5.1 in he case α /. The opeao H α w is bounded fom L p,ψ {} Rn o L q,ψ {} Rn if min{m V,m V } >, min { m ( ψ ),m ( ψ )} > The condiion min{m V,m V } >, is guaaneed by he inequaliies M w < n p M ( ψ ), M w < n p M ( ψ ) In he powe case ψ λ, λ>, and w μ, he condiions educe o 1 p< n λ α, 1 q 1 p α n λ, μ < n p λ p ; 5.13 condiions 5.13 ae also necessay fo he opeao H α w o be bounded fom L p,ψ {} Rn o L q,ψ {} Rn. Poof. We have o check ha he condiion sup > W < of Theoem 5.3 holds unde he assumpions Fom he inequaliy min{m V,m V } > i follows ha W Cψ ϱ n 1 ϱq α n/p ψ q/p( ) dϱ ϱ

13 Jounal of Funcion Spaces and Applicaions 13 We epesen ψ q/p ϱ as ψ q/p ϱ ψ q/p 1 ϱ ψ ϱ and by 5.1 obain dϱ W Cψ ϱψ ( ϱ ), 5.15 which is bounded in view of he second assumpion in 5.11, by he popey A.4. The sufficiency of he condiions 5.13 in he case of powe funcions is hen obvious since m ψ m ψ λ and m w m w μ in his case. Le us show he necessiy of hese condiions. Fom he boundedness H α w L q,ψ {} Rn C f L p,ψ {} Rn wih ψ λ,λ >, and w μ, by sandad homogeneiy agumens wih he use of he dilaion opeao Π δ f x : f δx i is easily deived ha he condiions 1 p< n λ /α,1/q 1/p α/ n λ necessaily hold, via he elaion Πδ f L p,ψ {} Rn δ n λ /p f L p,ψ {} Rn, Hα wπ δ f δ α Π δ H α wf We ake ino accoun ha we excluded q in his heoem. The necessiy of he emaining condiion μ< n/p λ/p in 5.13 follows fom he fac ha y n λ /p L p,ψ {} Rn by Coollay 2.4, so ha his condiion is necessay fo he opeao H α w wih w μ o be defined on he space L p,ψ {} Rn The Case of he Opeao H α w Le W : ψ w q( ϱ ) ϱ qα n 1 ( ϱ q V d) dϱ, 5.17 whee V is he same as in 5.3. Theoem 5.6. Le 1 p<, 1 q< and ( ) wψ 1/p W R 1 o ( ) w W R 1, w 2 Cw The opeao H α w is bounded fom L p,ψ {} Rn o L q,ψ {} Rn if supw <, > 5.19 and hen H α w f L q,ψ {} C sup W 1/q f L p,ψ > {}. 5.2

14 14 Jounal of Funcion Spaces and Applicaions Poof. Fom he esimae 5.4 we obain M q,ψ ( H α w f;, ) Cψ ϱ n 1 q α n/p ψ q/p( ϱ )( 1 V ( ϱ ) ϱ ) q V d dϱ f q, 5.21 L p,ψ {} fom which 5.2 follows. As above, we povide also sufficien condiions fo he boundedness of he opeao H β w in ems of he Mauszewska-Olicz indices. Coollay 5.7. Le 1 p q< (wih q p admied in he case α and w and ψ saisfy he condiions The opeao H α w is bounded fom L p,ψ {} Rn o L q,ψ {} Rn if max{m V,M V } <, min { m ( ψ ),m ( ψ )} >, 5.22 and he condiion 5.1 holds; he assumpion max{m V,M V } < is guaaneed by he condiions m w > m( ψ ) n, m w > m ( ) ψ n p p In he case ψ λ,λ>, and w μ, he condiions 5.22 and 5.1 educe o 1 p< n λ α, 1 q 1 p α n λ, μ > n λ p ; 5.24 condiions 5.24 ae also necessay fo he opeao H α w o be bounded fom L p,ψ {} Rn o L q,ψ {} Rn. Poof. We have o find, in ems of he Mauszewska-Olicz indices, condiions sufficien fo he validiy of Fo he lae we have W Cψ Cψ ϱ n 1 q α n/p ψ q/p( ϱ )( 1 V ( ϱ ) V ϱ ϱ n 1 q α n/p ψ q/p( ϱ ) dρ d) q dρ 5.25 by he assumpion max{m V,M V } <. Wih ψ q/p ρ ψ q/p 1 ρ ψ ρ and he condiion 5.1 we aive a W Cψ dϱ/ϱψ ϱ, whee he boundedness of he ighhand side is guaaneed by he condiion min{m ψ,m ψ } >. The poof fo he case of powe funcions is simila o ha in Coollay 5.5.

15 Jounal of Funcion Spaces and Applicaions 15 Appendix A. Zygmund-Bay-Sechkin (ZBS) Classes and Mauszewska-Olicz (MO) Type Indices In he sequel, a nonnegaive funcion f on,l, < l, is called almos inceasing almos deceasing if hee exiss a consan C 1 such ha f x Cf y fo all x y x y, esp.. Equivalenly, a funcion f is almos inceasing almos deceasing if i is equivalen o an inceasing deceasing, esp. funcion g, hais,c 1 f x g x c 2 f x, c 1 >, c 2 >. Definiion A.1. Le <l< : 1 by W W,l we denoe he class of coninuous and posiive funcions ϕ on,l such ha hee exiss finie o infinie limi lim x ϕ x ; 2 by W W,l we denoe he class of almos inceasing funcions ϕ W on,l ; 3 by W W,l we denoe he class of funcions ϕ W such ha x a ϕ x W fo some a a ϕ R 1 ; 4 by W W,l we denoe he class of funcions ϕ W such ha ϕ / b is almos deceasing fo some b R 1. Definiion A.2. Le <l< : 1 by W W l, we denoe he class of funcions ϕ which ae coninuous and posiive and almos inceasing on l, and which have he finie o infinie limi lim x ϕ x, 2 by W W l, we denoe he class of funcions ϕ W such x a ϕ x W fo some a a ϕ R 1. By W R 1 we denoe he se of funcions on R 1 whose esicions ono, 1 ae in W, 1 and esicions ono 1, ae in W 1,. Similaly, he se W R 1 is defined. A.1. ZBS Classes and MO Indices of Weighs a he Oigin In his subsecion we assume ha l<. We say ha a funcion ϕ belongs o a Zygmund class Z β, β R 1,ifϕ W,l and x ϕ /1 β d c ϕ x /x β, x,l, and o a Zygmund class Z γ, γ R 1,ifϕ W,l and l x ϕ /1 γ d c ϕ x /x γ, x,l. We also denoe Φ β γ : Z β Z γ, A.1 he lae class being also known as Bay-Sechkin-Zygmund class 2.

16 16 Jounal of Funcion Spaces and Applicaions Fo a funcion ϕ W, he numbes m ( ϕ ) ln ( ( )) ( ( )) lim sup sup h ϕ hx /ϕ h ln lim suph lim ϕ hx /ϕ h, <x<1 ln x x ln x M ( ϕ ) ln ( ( )) ( ( )) lim sup sup h ϕ hx /ϕ h ln lim suph lim ϕ hx /ϕ h x>1 ln x x ln x A.2 ae known as he Mauszewska-Olicz ype lowe and uppe indices of he funcion ϕ. The popey of funcions o be almos inceasing o almos deceasing afe he muliplicaion division by a powe funcion is closely conneced wih hese indices. We efe o fo such a popey and hese indices. Noe ha in his definiion ϕ x need no o be an N-funcion: only is behaviou a he oigin is of impoance. Obseve ha m ϕ M ϕ fo ϕ W,and <m ϕ M ϕ fo ϕ W, and he following fomulas ae valid: m [ x a ϕ x ] a m ( ϕ ), M [ x a ϕ x ] a M ( ϕ ), a R 1, A.3 m ([ ϕ x ] a) ( ) am ϕ, ([ ] a ) ( ) M ϕ x am ϕ, a, A.4 ( ) 1 m M ( ϕ ) ( ) 1, M m ( ϕ ), ϕ ϕ A.5 m uv m u m v, M uv M u M v A.6 fo ϕ, u, v W. The poof of he following saemen may be found in 21, Theoems 3.1, 3.2, and 3.5. In he fomulaion of Theoems in 21 i was supposed ha β, γ>andϕ W.Iis evidenly ue also fo ϕ W and all β, γ R 1, in view of fomulas A.3. Theoem A.3. Le ϕ W and β, γ R 1.Thenϕ Z β m ϕ >βand ϕ Z γ M ϕ <γ. Besides his, m ϕ sup{μ > : ϕ x /x μ is almos inceasing}, and M ϕ inf{ν > : ϕ x /x ν is almos deceasing}, and fo ϕ Φ β γ he inequaliies c 1 x M ϕ ε ϕ x c 2 x m ϕ ε A.7 hold wih an abiaily small ε> and c 1 c 1 ε, c 2 c 2 ε. We define he following subclass in W : { W,b ϕ W : ϕ }, is almos inceasing, b R 1. A.8 b

17 Jounal of Funcion Spaces and Applicaions 17 A.2. ZBS Classes and MO Indices of Weighs a Infiniy Following Secion 2.2 in 29, we use he following noaion. Le <α<β<. WepuΨ β α : Ẑ β Ẑ α, whee Ẑ β is he class of funcions ϕ W saisfying he condiion x x/ β ϕ d/ cϕ x, x l,, andẑ α is he class of funcions ϕ W l, saisfying he condiion x l x/ α ϕ d/ cϕ x, x l,, whee c c ϕ > does no depend on x l,. The indices m ϕ and M ϕ esponsible fo he behavio of funcions ϕ Ψ β α l, a infiniy ae inoduced in he way simila o A.2 : m ( ϕ ) sup x>1 ln [ lim inf h ( ϕ xh /ϕ h )] ln x ( ) ln [ ( )] lim sup M ϕ inf h ϕ xh /ϕ h. x>1 ln x, A.9 Popeies of funcions in he class Ψ β α l, ae easily deived fom hose of funcions in Φ α,l because of he following equivalence β ϕ Ψ β α l, ϕ Φ β α,l, A.1 whee ϕ ϕ 1/ and l 1/l. Diec calculaion shows ha ( ) ( ) ( ) ( ) ( ) 1 m ϕ M ϕ, M ϕ m ϕ, ϕ : ϕ. A.11 By A.1 and A.11, one can easily efomulae popeies of funcions of he class Φ β γ nea he oigin, given in Theoem A.3 fo he case of he coesponding behavio a infiniy of funcions of he class Ψ β α and obain ha c 1 m ϕ ε ϕ c 2 M ϕ ε, l, ϕ W, ( ) } m ϕ sup {μ R 1 : μ ϕ is almos inceasing on l,, ( ) } M ϕ inf {ν R 1 : ν ϕ is almos deceasing on l,. A.12 We say ha a coninuous funcion ϕ in, is in he class W, R 1 if is esicion o, 1 belongs o W, 1 and is esicion o 1, belongs o W 1,. Fo funcions in W, R 1 he noaion Z β,β (R 1 ) ( ) Z β, 1 Z β 1,, Z γ,γ R 1 Z γ, 1 Z γ 1, A.13

18 18 Jounal of Funcion Spaces and Applicaions has an obvious meaning noe ha in A.13 we use Z β 1, and Z γ 1,, no Ẑ β 1, and Ẑ γ 1,. In he case whee he indices coincide, ha is, β β : β, we will simply wie Z β R 1 and similaly fo Z γ R 1. We also denoe Φ β γ ( ) R 1 : Z β( ) ( ) R 1 Z γ R 1. A.14 Making use of Theoem A.3 fo Φ α, 1 and elaions A.11, one easily aives a β he following saemen. Lemma A.4. Le ϕ W R 1.Then ) ϕ Z β,β (R 1 m ( ϕ ) >β, ( ) m ϕ >β, A.15 ) ϕ Z γ,γ (R 1 M ( ϕ ) <γ, ( ) M ϕ <γ. A.16 Acknowledgmen The hid auho hanks Luleå Univesiy of Technology fo financial suppo fo eseach visis in June of 212. Refeences 1 D. E. Edmunds, V. Kokilashvili, and A. Meskhi, Bounded and Compac Inegal Opeaos, vol. 543, Kluwe Academic, Dodech, The Nehelands, V. Kokilashvili, A. Meskhi, and L.-E. Pesson, Weighed Nom Inequaliies fo Inegal Tansfoms wih Poduc Kenels, Nova Science, New Yok, NY, USA, A. Kufne, L. Maliganda, and L.-E. Pesson, The Hady Inequaliy Abou Is Hisoy and Some Relaed Resuls, Pilsen, Czech, A. Kufne and L.-E. Pesson, Weighed Inequaliies of Hady Type, Wold Scienific, Rive Edge, NJ, USA, D. Lukkassen, A. Medell, L.-E. Pesson, and N. Samko, Hady and singula opeaos in weighed genealized Moey spaces wih applicaions o singula inegal equaions, Mahemaical Mehods in he Applied Sciences, vol. 35, no. 11, pp , H. Tiebel, Enopy and appoximaion numbes of limiing embeddings an appoach via Hady inequaliies and quadaic foms, Jounal of Appoximaion Theoy, vol. 164, no. 1, pp , L.-E. Pesson and N. Samko, Weighed Hady and poenial opeaos in he genealized Moey spaces, Jounal of Mahemaical Analysis and Applicaions, vol. 377, no. 2, pp , N. Samko, Weighed Hady and singula opeaos in Moey spaces, Jounal of Mahemaical Analysis and Applicaions, vol. 35, no. 1, pp , N. Samko, Weighed Hady and poenial opeaos in Moey spaces, Jounal of Funcion Spaces and Applicaions, vol. 212, Aicle ID , 21 pages, M. Giaquina, Muliple Inegals in he Calculus of Vaiaions and Nonlinea Ellipic Sysems, Pinceon Univesiy Pess, Pinceon, NJ, USA, A.Kufne,O.John,andS.Fučík, Funcion Spaces, Noodhoff Inenaional, Leyden, Mass, USA, M. E. Taylo, Tools fo PDE: Pseudodiffeenial Opeaos, Paadiffeenial Opeaos, and Laye Poenials, vol. 81 of Mahemaical Suveys and Monogaphs, Ameican Mahemaical Sociey, Povidence, RI, USA, H. Rafeio, N. Samko, and S. Samko, Moey-Campanao spaces: an oveview, in Opeao Theoy, Pseudo-Diffeenial Equaions, and Mahemaical Physics, Bikhäuse Mahemaics.

19 Jounal of Funcion Spaces and Applicaions G. T. Dzhumakaeva and K. Zh. Nauyzbaev, Lebesgue-Moey spaces, Izvesiya Akademii Nauk Kazakhskoĭ SSR. Seiya Fiziko-Maemaicheskaya, no. 5, pp. 7 12, 1982 Russian. 15 C. T. Zoko, Moey space, Poceedings of he Ameican Mahemaical Sociey, vol. 98, no. 4, pp , V. Guliyev, Inegal opeaos on funcion spaces on homogeneous goups and on domains in R n [Ph.D. hesis], Seklov Mahemaical Insiue, Moscow, Russia, V. Guliyev, Funcion Spaces, Inegal Opeaos and Two Weighed Inequaliies on Homogeneous Goups, Some applicaions, Baku, Azebaijan, V. I. Buenkov, H. V. Guliyev, and V. S. Guliyev, On boundedness of he facional maximal opeao fom complemenay Moey-ype spaces o Moey-ype spaces, in The Ineacion of Analysis and Geomey, vol. 424 of Conempoay Mahemaics, pp , Ameican Mahemaical Sociey, Povidence, RI, USA, V. Guliyev, J. Hasanov, and S. Samko, Maximal, poenial and singula opeaos in he local complemenay vaiable exponen Moey ype spaces, Jounal of Mahemaical Analysis and Applicaions. In pess. 2 N. K. Bai and S. B. Sečkin, Bes appoximaions and diffeenial popeies of wo conjugae funcions, Poceedings of he Moscow Mahemaical Sociey, vol. 5, pp , 1956 Russian. 21 N. K. Kaapeyans and N. Samko, Weighed heoems on facional inegals in he genealized Hölde spaces H ω ρ via indices m ω and M ω, Facional Calculus & Applied Analysis, vol. 7, no. 4, pp , S. G. Keĭn,Ju.I.Peunin,andE.M.Semënov, Inepolaion of Linea Opeaos, Nauka, Moscow, Russia, L. Maliganda, Indices and inepolaion, Disseaiones Mahemaicae, vol. 234, p. 49, W. Mauszewska and W. Olicz, On some classes of funcions wih egad o hei odes of gowh, Sudia Mahemaica, vol. 26, pp , L.-E. Pesson, N. Samko, and P. Wall, Quasi-monoone weigh funcions and hei chaaceisics and applicaions, Mahemaical Inequaliies and Applicaions, vol. 12, no. 3, pp , N. Samko, Singula inegal opeaos in weighed spaces wih genealized Hölde condiion, Poceedings of A. Razmadze Mahemaical Insiue, vol. 12, pp , N. Samko, On non-equilibaed almos monoonic funcions of he Zygmund-Bay-Sechkin class, Real Analysis Exchange, vol. 3, no. 2, pp , 24/ L. Maliganda, Olicz Spaces and Inepolaion, Depaameno de Maemáica, Univesidade Esadual de Campinas, Campinas, Bazil, N. G. Samko, S. G. Samko, and B. G. Vakulov, Weighed Sobolev heoem in Lebesgue spaces wih vaiable exponen, Jounal of Mahemaical Analysis and Applicaions, vol. 335, no. 1, pp , 27.