Eleconic Jounal of Diffeenial Equaions, Vol. 208 (208, No. 0, pp. 24. ISSN: 072-669. URL: hp://ejde.mah.xsae.edu o hp://ejde.mah.un.edu GLOBAL REGULARITY IN ORLICZ-MORREY SPACES OF SOLUTIONS TO NONDIVERGENCE ELLIPTIC EQUATIONS WITH VMO COEFFICIENTS VAGIF S. GULIYEV, AYSEL A. AHMADLI, MEHRIBAN N. OMAROVA, LUBOMIRA SOFTOVA Communicaed by Viceniu D. Radulescu Absac. We show coninuiy in genealized Olicz-Moey spaces M Φ,ϕ (R n of sublinea inegal opeaos geneaed by Caldeón-Zygmund opeao and hei commuaos wih BMO funcions. The obained esimaes ae used o sudy global egulaiy of he soluion of he Diichle poblem fo linea unifomly ellipic opeao L = P n i,j= aij (xd ij wih disconinuous coefficiens. We show ha Lu M Φ,ϕ implies he second-ode deivaives belong o M Φ,ϕ.. Inoducion The classical Moey spaces L p,λ ae oiginally inoduced in [37] o sudy he local behavio of soluions o ellipic paial diffeenial equaions. In fac, he bee inclusion beween he Moey and he Hölde spaces pemis o obain highe egulaiy of he soluions o diffeen ellipic and paabolic bounday poblems. Recall ha fo a bounded domain Ω R n saisfying he cone popey, he space L p,λ wih p < consiss of all funcions f L p (Ω such ha ( /p f Lp,λ (Ω = sup B λ f(y dy p <, B Ω whee B anges ove all balls in R n ceneed in some poin x Ω and of adius > 0. Fo he popeies and applicaions of he classical Moey spaces, we efe he eades o [7, 37, 4, 43] and he efeences hee. Chiaenza and Fasca [8] showed he boundedness of he Hady-Lilewood maximal opeao in L p,λ (R n ha allows hem o pove coninuiy of facional and classical Caldeón-Zygmund opeaos in hese spaces. Recall ha inegal opeaos of ha kind appea in he epesenaion fomulae of he soluions of ellipic/paabolic equaions and sysems. Thus he coninuiy of he Caldeón-Zygmund inegals implies egulaiy of he soluions in he coesponding spaces. Mizuhaa[36] gave a genealizaion of hese spaces consideing a weigh funcion ω(x, : R n R R insead of λ. He sudied also a coninuiy in L p,ω of some classical inegal opeaos. Lae Nakai 200 Mahemaics Subjec Classificaion. 35J25, 35B40, 42B20, 42B35, 46E30. Key wods and phases. Genealized Olicz-Moey spaces; Caldeón-Zygmund inegals; commuaos; VMO; ellipic equaions; Diichle poblem. c 208 Texas Sae Univesiy. Submied Sepembe, 207. Published May 0, 208.
2 V. S. GULIYEV, A. A. AHMADLI, M. N. OMAROVA, L. SOFTOVA EJDE-208/0 exended he esuls of Chiaenza and Fasca in L p,ω imposing ceain inegal and doubling condiions on ω (see [38]. Taking a weigh ω = ϕ p n he condiions of Mizuhaa-Nakai become ϕ(x, p d C ϕ(x, p, C ϕ(x, C, 2, ϕ(x, whee he consans do no depend on, and x R n. In seies of woks, he fis auho sudies he coninuiy in genealized Moey spaces of sublinea opeaos geneaed by vaious inegal opeaos as Caldeón- Zygmund, Riesz poenal and ohes (see [8, 9, 2]. The following heoem obained in [8] exends he esuls of Nakai in Moey-ype spaces wih weigh ω = ϕ n (fo he definiion of he spaces see 3 Theoem. ([8, 9]. Le p < and (ϕ, ϕ 2 saisfy he condiion ϕ (x, d Cϕ 2(x,, (. whee C does no depend on x and. Then he maximal opeao M and he Caldeón-Zygmund inegal opeaos K ae bounded fom M p,ϕ o M p,ϕ2 fo p > and fom M,ϕ o he weak space W M,ϕ2. Lae his esul was exended on spaces wih weake condiion on he weigh pai (ϕ, ϕ 2 (see [2], see also [, 2, 3]. Fo moe ecen esuls on boundedness and coninuiy of singula inegal opeaos in genealized Moey and new funcional spaces and hei applicaion in he diffeenial equaions heoy see [2, 4, 5, 5, 6, 20, 25, 26, 35, 40, 42, 44, 48, 49, 5] and he efeences hee. Thoughou his pape he following noaion will be used: D i u = u/ x i, Du = (D u,..., D n u means he gadien of u, D ij u = 2 u/ x i x j, D 2 u = {D ij u} n ij= is he Hessian maix of u, B = B(x 0, = {x R n : x x 0 < }, B c = R n \ B, 2B = B(x 0, 2, S n is a uni sphee in R n, Ω R n is a domain and Ω = Ω B (x, x Ω, R n = {x R n : x = (x, x n, x R n, x n > 0}, B B (x 0, = B(x 0, R n, 2B = B (x 0, 2 whee x 0 = (x, 0. The sandad summaion convenion on epeaed uppe and lowe indices is adoped. The lee C is used fo vaious posiive consans and may change fom one occuence o anohe. In his pape, we shall use he symbol A B o indicae ha hee exiss a univesal posiive consan C, independen of all impoan paamees, such ha A CB. A B means ha A B and B A. 2. Peliminaies on Olicz and Olicz-Moey spaces Definiion 2.. A funcion Φ : [0, ] [0, ] is called a Young funcion if Φ is convex, lef-coninuous, lim 0 Φ( = Φ(0 = 0 and lim Φ( = Φ( =. Fom he convexiy and Φ(0 = 0 i follows ha any Young funcion is inceasing. If hee exiss s (0, such ha Φ(s =, hen Φ( = fo s. We say ha Φ 2, if fo any a >, hee exiss a consan C a > 0 such ha Φ(a C a Φ( fo all > 0. A Young funcion Φ is said o saisfy he 2 -condiion, denoed also by Φ 2, if Φ( Φ(k, 0, 2k
EJDE-208/0 GLOBAL REGULARITY IN ORLICZ-MORREY SPACES 3 fo some k >. The funcion Φ( = saisfies he 2 -condiion bu does no saisfy he 2 -condiion. If < p <, hen Φ( = p saisfies boh he condiions. The funcion Φ( = e saisfies he 2 -condiion bu does no saisfy he 2 -condiion. The following wo indices ϕ( q Φ = inf >0 Φ(, p ϕ( Φ = sup >0 Φ( of Φ, whee ϕ( is he igh-coninuous deivaive of Φ, ae well known in he heoy of Olicz spaces. As is well known, p Φ < Φ 2, and he funcion Φ is sicly convex if and only if q Φ >. If 0 < q Φ p Φ <, hen Φ( Φ( is inceasing and q Φ is deceasing on (0,. p Φ Lemma 2.2 ([29, Lemma.3.2]. Le Φ 2. Then hee exis p > and b > such ha Φ( 2 p b Φ( 2 p fo 0 < < 2. Lemma 2.3 ([47, Poposiion 62.20]. Le Φ be a Young funcion wih canonical epesenaion Φ( = 0 ϕ(sds, 0. ( Assume ha Φ 2. Moe pecisely Φ(2 AΦ( fo some A 2. If p > log 2 A, hen ϕ(s Φ( s p ds p, > 0. (2 Assume ha Φ 2. Then 0 ϕ(s s ds Φ(, > 0. Recall ha a funcion Φ is said o be quasiconvex if hee exis a convex funcion ω and a consan c > 0 such ha ω( Φ( cω(c, [0,. Le Y be he se of all Young funcions Φ such ha 0 < Φ( < fo 0 < <. (2. If Φ Y, hen Φ is absoluely coninuous on evey closed ineval in [0, and bijecive fom [0, o iself. Definiion 2.4. Fo a Young funcion Φ, he se L Φ (R n = { f L loc (R n : Φ(k f(x dx < fo some k > 0 } R n is called Olicz space. The space L loc Φ (Rn endowed wih he naual opology is defined as he se of all funcions f such ha fχ B L Φ (R n fo all balls B R n.
4 V. S. GULIYEV, A. A. AHMADLI, M. N. OMAROVA, L. SOFTOVA EJDE-208/0 Noe ha L Φ (R n is a Banach space wih espec o he nom f LΦ = inf { ( f(x λ > 0 : dx }, λ R n Φ see, fo example [45, Secion 3, Theoem 0], so ha ( f(x dx. f LΦ R n Φ Fo a measuable se Ω R n, a measuable funcion f and > 0, le m(ω, f, = {x Ω : f(x > }. In he case Ω = R n, we sholy denoe i by m(f,. Definiion 2.5. The weak Olicz space is defined by he nom W L Φ (R n = {f L loc(r n : f W LΦ < } f W LΦ { = inf λ > 0 : sup >0 Φ(m ( f } λ,. Fo Young funcions Φ and Ψ, we wie Φ Ψ if hee exiss a consan C such ha Φ(C Ψ( Φ(C fo all 0. If Φ Ψ, hen L Φ (R n = L Ψ (R n wih equivalen noms. We noe ha, fo Young funcions Φ and Ψ, if hee exis C, R such ha Φ(C Ψ( Φ(C hen Φ Ψ. Fo a Young funcion Φ and 0 s, le Φ (s = inf{ 0 : Φ( > s} fo (0, R (R,, (inf =. If Φ Y, hen Φ is he usual invese funcion of Φ. We noe ha Φ(Φ ( Φ (Φ( fo 0 <. Fo a Young funcion Φ, he complemenay funcion Φ( is defined by { sup{s Φ(s : s [0, }, [0, Φ( =, =. (2.2 The complemenay funcion Φ is also a Young funcion and Φ = Φ. If Φ( =, hen Φ( = 0 fo 0 and Φ( = fo >. If < p <, /p /p = and Φ( = p /p, hen Φ( = p /p. If Φ( = e, hen Φ( = ( log(. Remak 2.6. Noe ha Φ 2 if and only if Φ 2. Also, if Φ is a Young funcion, hen Φ 2 if and only if Φ γ be quasiconvex fo some γ (0, (see, fo example [29, p. 5]. I is known ha Φ ( Φ ( 2 fo 0. (2.3 The following analogue of he Hölde inequaliy is known.
EJDE-208/0 GLOBAL REGULARITY IN ORLICZ-MORREY SPACES 5 Theoem 2.7 ([50]. Fo a Young funcion Φ and is complemenay funcion Φ, he following inequaliy is valid fg L(R n 2 f LΦ g L e Φ. Noe ha Young funcions saisfy he popey Φ(α αφ( (2.4 fo all 0 < α < and 0 <, which is a consequence of he convexiy: Φ(α = Φ(α ( α0 αφ( ( αφ(0 = αφ(. Lemma 2.8 ([3, 34]. Le Φ be a Young funcion and B a ball in R n. Then χ B W LΦ(R n = χ B LΦ(R n = Φ ( B. In he nex secions whee we pove ou main esimaes, we use he following lemma, which follows fom Theoem 2.7 and Lemma 2.8. Lemma 2.9. Fo a Young funcion Φ and B = B(x,, we have f L(B 2 B Φ ( B f LΦ(B. Definiion 2.0. Le ϕ(x, be a posiive measuable funcion on R n (0, and Φ any Young funcion. We denoe by M Φ,ϕ (R n he genealized Olicz-Moey space, he space of all funcions f L loc Φ (Rn wih finie quasinom f MΦ,ϕ = sup ϕ(x, Φ ( B(x, f LΦ(B(x,. x R n,>0 Also by W M Φ,ϕ (R n we denoe he weak genealized Olicz-Moey space of all funcions f W L loc Φ (Rn fo which f W MΦ,ϕ = sup ϕ(x, Φ ( B(x, f W LΦ(B(x, <, x R n,>0 whee W L Φ (B(x, denoes he weak L Φ -space of measuable funcions f fo which f W LΦ(B(x, fχ B(x, W LΦ(R n. Accoding o his definiion, we ecove he spaces M p,ϕ and W M p,ϕ unde he choice Φ( = p : M p,ϕ = M Φ,ϕ Φ(= p, W M Φ,λ = W M Φ,ϕ Φ(= p. 3. Definiions and saemen of he poblem In he pesen secion we give he definiions of he funcional spaces o which he coefficiens and he daa of he poblem belong. The domain Ω R n supposed o be bounded wih Ω C,. Definiion 3.. Le ϕ : Ω R R be a measuable funcion and p <. The genealized Olicz-Moey space M Φ,ϕ (Ω consiss of all f L loc Φ (Ω f MΦ,ϕ(Ω = sup ϕ(x, Φ ( B(x, f LΦ(Ω B(x, x Ω,>0 Fo any bounded domain Ω we define M Φ,ϕ (Ω aking f L Φ (Ω and Ω insead of B(x, in he nom above.
6 V. S. GULIYEV, A. A. AHMADLI, M. N. OMAROVA, L. SOFTOVA EJDE-208/0 The genealized Sobolev-Olicz-Moey space W 2,Φ,ϕ (Ω consiss of all Sobolev funcions u W 2,Φ (Ω wih disibuional deivaives D s u M Φ,ϕ (Ω, endowed wih he nom u W2,Φ,ϕ(Ω = D s f MΦ,ϕ(Ω. 0 s 2 The space W 2,Φ,ϕ (Ω W,Φ 0 (Ω consiss of all funcions u W 2,Φ(Ω W,Φ 0 (Ω wih D s u M Φ,ϕ (Ω, and is endowed by he same nom. Recall ha W,Φ 0 (Ω is he closue of C0 (Ω wih espec o he nom in W,Φ. Definiion 3.2. Le ϕ : Ω R R be a measuable funcion, he genealized weak Moey space W M Φ,ϕ (Ω consiss of all measuable funcions such ha f W M Φ,ϕ(Ω = sup ϕ(x, Φ ( B(x, f W LΦ(Ω B(x,, x Ω,>0 whee W L Φ (Ω B(x, denoes he weak L Φ -space of measuable funcions f fo which f W LΦ(B(x, fχ Ω B(x, W LΦ(R n. Fo a bounded domain Ω we define he space W M Φ,ϕ (Ω aking f W L Φ (Ω. Definiion 3.3. Le a L loc (R n and a B = B B a(ydy is he mean inegal of a. We say ha a BMO (bounded mean oscillaion, [3] if a = sup sup a(y a B dy <. R>0 B, R B B The quaniy a is a nom in BMO modulo consan funcion unde which BMO is a Banach space; a V MO (vanishing mean oscillaion, [46] if a BMO and lim γ a(r = lim R 0 R 0 sup B, R B B a(y a B dy = 0. The quaniy γ a (R is called V MO-modulus of a. Fo any bounded domain Ω R n we define BMO(Ω and V MO(Ω aking a L (Ω and Ω insead of B in he definiion above. Accoding o [, 32], having a funcion a BMO(Ω o V MO(Ω i is possible o exend i in he whole R n peseving is BMO-nom o V MO-modulus, especively. In he following we use his popey wihou explici efeences. Any bounded unifomly coninuous funcion f BU C wih modulus of coninuiy ω f ( is also V MO and γ f ( ω f (. Besides ha, BMO and V MO conain also disconinuous funcions and he following example shows he inclusion W,n (R n V MO BMO. Example 3.4. f α (x = log x α V MO fo any α (0, ; f α W,n (R n fo α (0, /n, f α / W,n (R n fo α [ /n, ; f(x = log x BMO\V MO; sin f α (x V MO L (R n. In he Secions 4, 6 and 7 we sudy coninuiy in he spaces M Φ,ϕ of ceain sublinea inegals and hei commuaos wih BM O funcions. These esuls unified wihe known esimaes in L p (R n pemi o obain coninuiy of he Caldeón- Zygmund opeaos in M p,ϕ (R n ha is shown in 8. The las secion is dedicaed
EJDE-208/0 GLOBAL REGULARITY IN ORLICZ-MORREY SPACES 7 o he Diichle poblem fo a linea unifomly ellipic opeao wih V MO coefficiens. This poblem is fisly sudied by Chiaenza, Fasca and Longo. In hei pionee woks [9], [0] hey pove unique song solvabiliy of Lu a ij (xd ij u = f(x a.a. x Ω, u W 2,p (Ω W 0,p(Ω, p (, (3. poviding such way he classical heoy on opeaos wih coninuous coefficiens o hose wih disconinuous ones. Lae hei esuls ae exended in he Sobolev- Moey spaces W 2,p,λ (Ω W,p(Ω, 0 λ (, n (see [5], [6]. In he pesen wok we show ha Lu M Φ,ϕ (Ω implies he same egulaiy of he second ode deivaives D ij u. The weigh ϕ(x, saisfies an inegal condiion weake han (.. 4. Sublinea opeaos and commuaos geneaed by singula inegals in he space M Φ,ϕ (R n In his secion we pesen esuls obained in [27] concening coninuiy of sublinea opeaos geneaed by singula inegals as Caldeón-Zygmund. Le T be a sublinea opeao such ha fo any f L (R n wih compac suppo and x / suppf holds whee C is independen of f. T f(x C R n f(y dy, (4. x y n Theoem 4.. Le Φ any Young funcion, ϕ, ϕ 2 : R n R R be measuable funcions such ha fo any x R n and fo any > 0, ( ϕ (x, s ess inf <s< Φ ( s n Φ ( n d Cϕ 2(x, (4.2 and T be sublinea opeao saisfying (4.. (i If T bounded on L Φ (R n, hen T is bounded fom M Φ,ϕ (R n o M Φ,ϕ2 (R n and T f MΦ,ϕ2 (R n C f MΦ,ϕ (R n. (ii If T bounded fom L Φ (R n o W L Φ (R n, hen i is bounded fom M Φ,ϕ (R n o W M Φ,ϕ2 (R n and T f W MΦ,ϕ2 (R n C f MΦ,ϕ (R n wih consans independen of f. Noe ha condiion (4.2 is weake han he one in Theoem.. condiion (. holds hen ( ϕ (x, s ess inf <s< Φ ( s n Φ ( n d ϕ (x, d Indeed, if ha implies (4.2. We give also wo examples of admissible pais of funcions. Example 4.2. Fo β (0, n conside he weigh funcions ϕ ( = β ( { Φ ( n π } sin max,, 2β ϕ2 ( = Φ ( n.
8 V. S. GULIYEV, A. A. AHMADLI, M. N. OMAROVA, L. SOFTOVA EJDE-208/0 If (0, hen ess inf <s< ϕ (x,s ( = 0 and Φ s n ( ϕ (x, s ess inf <s< Φ ( s n Φ ( n d = Hence he pai (ϕ, ϕ 2 saisfies (4.2 bu no (.. Example 4.3. Fo β (0, n conside he funcions ϕ ( = They saisfy condiion (4.2 bu no (.. 0 (0, ( β Φ Cϕ 2 (. β χ (, (Φ ( n, ϕ 2( = β Φ ( n. n (, Conside now he commuao T a f = T [a, f] = at f T (af such ha fo any f L Φ (R n wih a compac suppo and x / suppf holds f(y T a f(x C a(x a(y dy, (4.3 R x y n n whee C is independen of f and x. Suppose in addiion ha T a is bounded in L Φ (R n saisfying he esimae T a f LΦ(R n C a f LΦ(R n. Then he following esul holds (see [4, 27]. Theoem 4.4. Le Φ any Young funcion, a BMO, ϕ, ϕ 2 : R n R R be measuable funcions such ha fo any x R n and fo any > 0, ( ln ( ϕ (x, s ess inf <s< Φ ( s n Φ ( n d Cϕ 2(x,, (4.4 whee C does no depend on x and. Suppose T a be a sublinea opeao saisfying (4.3 and bounded on L Φ (R n. Then he opeao T a is bounded fom M Φ,ϕ o M Φ,ϕ2 T a f MΦ,ϕ2 (R n C a f MΦ,ϕ (R n. 5. Nonsingula inegal opeaos in he Olicz space L Φ (R n The following heoem was poved in [0]. Theoem 5.. Le x R n and Kf(x = R n f(y x y n dy, x = (x, x n. (5. Then hee exiss a consan C independen of f, such ha Kf Lp(R n C p f Lp(R n, < p <, Kf W L(R n C f L(R n. Theoem 5.2. Le Φ be a Young funcion and K be a nonsingula inegal opeao, defined by (5.. If Φ 2 2, hen he opeao K is bounded on L Φ (R n and if Φ 2, hen he opeao K is bounded fom L Φ (R n o W L Φ (R n.
EJDE-208/0 GLOBAL REGULARITY IN ORLICZ-MORREY SPACES 9 Poof. Fis we pove ha fo Φ 2 he nonsingula inegal opeao K is bounded fom L Φ (R n o W L Φ (R n. We ake f L Φ (R n saisfying f LΦ =. Fix λ > 0 and define f = χ { f >λ} f and f 2 = χ { f λ} f. Then f = f f 2. We have { Kf > λ} { Kf > λ/2} { Kf 2 > λ/2}, Φ(λ { Kf > λ} Φ(λ{ Kf > λ/2} Φ(λ { Kf 2 > λ/2}. We know ha fom he weak (, boundedness and L p, p > boundedness of K, { K(χ { f >λ} f > λ} f, λ { f >λ} { K(χ { f λ} f > λ} λ p f p. { f λ} Since f W L (R n and Φ(λ λ inceasing we have Φ(λ { x R n : Kf (x > λ } Φ(λ f (x dx 2 λ R n = Φ(λ λ f(x Φ( f(x dx R n f(x = Φ( f(x dx. By Lemma 2.2 and f 2 L p (R n we have Φ(λ { x R n : Kf 2 (x > λ } Φ(λ 2 λ p = Φ(λ Thus we obain {x R n : Kf(x > λ} = R n λ p R n R n {x R n : f(x >λ} f(x dx R n f 2 (x p dx {x R n : f(x λ} f(x p dx f(x p Φ( f(x f(x p dx Φ( f(x dx. C Φ( f(x dx Φ(λ R n ( Φ. λ C f LΦ Since LΦ nom is homogeneous his inequaliy is ue fo evey f L Φ (R n. Now poved ha fo Φ 2 2 he nonsingula inegal opeao K is bounded in L Φ (R n. As befoe we use disibuion funcions. R n ( Kf(x Φ dx = Λ Λ 0 ϕ ( λ {x R n Λ : Kf(x > λ} dλ
0 V. S. GULIYEV, A. A. AHMADLI, M. N. OMAROVA, L. SOFTOVA EJDE-208/0 = 2 Λ 0 ϕ ( 2λ {x R n Λ : Kf(x > 2λ} dλ. Wha is diffeen fom he esimae fo he maximal opeao is he poin ha K is no L (R n bounded. Le p > be sufficienly lage. Then {x R n : Kf(x > 2λ} {x R n : K(χ { f >λ} f(x > λ} {x R n : K(χ { f λ} f(x > λ}. By he weak (, boundedness and L p -boundedness of K (see Theoem 5. we have {x R n : K(χ { f >λ} f(x > λ} λ {x R n : f(x >λ} f(x dx, {x R n : K(χ { f λ} f(x > λ} λ p {x R n : f(x λ} f(x p dx. Using he same calculaion used fo he maximal opeao woks fo he fis em, ( 2λ ϕ { Λ Λ K(χ ( c f { f >λ} f > λ} dλ Φ. (5.2 Λ 0 Fo he second em a simila compuaion sill woks, bu we use ha Φ 2, ( 2λ ϕ { Λ 0 Λ K(χ { f λ} f(x > λ} dλ ( 2λ ( dλ ϕ f(x p dx Λ 0 Λ {x R n : f(x λ} λ p f(x p( ( 2λ dλ ϕ Λ Λ λ p dx. R n f(x R n Using Lemma 2.3 (, we have ( 2λ ϕ { Λ 0 Λ K(χ { f λ} f(x > λ} dλ ( 2 f(x ( c f(x Φ dx Φ dx. Λ Λ R n R n (5.3 Thus, puing ogehe (5.2 and (5.3, we obain ( Kf(x ( c0 f(x Φ dx Φ dx. R n Λ R n Λ Again we shall label he consan we wan o disinguish fom ohe less impoan consans. As befoe, if we se Λ = c 2 f LΦ (R n, hen we obain R n ( Kf(x Φ dx. Λ Hence he opeao nom of T is less han c 2.
EJDE-208/0 GLOBAL REGULARITY IN ORLICZ-MORREY SPACES 6. Sublinea opeaos geneaed by nonsingula inegal opeaos in he space M Φ,ϕ (R n We use he following saemen on he boundedness of he weighed Hady opeao H wg( := g(sw(sds, 0 < <, whee w is a weigh. The following heoem was poved in [22, 23] and in he case w = in [6]. Theoem 6.. Le v, v 2 and w be weighs on (0, and v ( be bounded ouside a neighbohood of he oigin. The inequaliy sup >0 v 2 (Hwg( C sup v (g( (6. >0 holds fo some C > 0 fo all non-negaive and non-deceasing g on (0, if and only if w(sds B := sup v 2 ( <. (6.2 >0 sup s<τ< v (τ Moeove, he value C = B is he bes consan fo (6.. Remak 6.2. In (6. and (6.2 i is assumed ha = 0 and 0 = 0. Fo any x R n define x = (x, x n and ecall ha x 0 = (x, 0. Le T be a sublinea opeao such ha fo any f L (R n wih a compac suppo holds T f(y f(x C dy. (6.3 x y n Lemma 6.3. Le Φ any Young funcion, f L loc Φ (Rn, be such ha R n and T be a sublinea opeao saisfying (6.3. (i If T bounded on L Φ (R n, hen T f LΦ(B (x 0, f LΦ(B (x 0,Φ ( n d < (6.4 C Φ ( n (ii If T bounded fom L Φ (R n on W L Φ (R n, hen T f W LΦ(B (x 0, C Φ ( n 2 whee he consans ae independen of x 0, and f. 2 f LΦ(B (x 0,Φ ( n d. (6.5 f LΦ(B (x 0,Φ ( n d, (6.6 Poof. (i Denoe B = B (x 0,, B = B (x 0, and fo any f L loc Φ (Rn wie f = f f 2 wih f = fχ 2B and f 2 = fχ (2B. Because of he (Φ, Φ-boundedness c of he opeao T (see Theoem 5.2 and f L Φ (R n we have T f LΦ(B T f LΦ(R n C f LΦ(R n = C f LΦ(2B. I is easy o see ha fo abiay poins x B and y (2B c i holds 2 x0 y x y 3 2 x0 y. (6.7
2 V. S. GULIYEV, A. A. AHMADLI, M. N. OMAROVA, L. SOFTOVA EJDE-208/0 Applying (6.3 and he Fubuni heoem o T f 2 we obain T f 2 (y f 2 (x C R n x y n dy f(y C (2B x 0 y n dy C f(y c (2B c ( d C f(y dy C 2 2 ( 2 x 0 y < B d f(y dy n. n Applying Hölde s inequaliy (Lemma 2.9, we obain f(y (2B x 0 dy f y n LΦ(B c LΦ e (B Diec calculaions give = 2 2 2 T f 2 LΦ(B Φ ( n x 0 y d n f LΦ(B Φ ( B f LΦ(B Φ ( n d. 2 d n f LΦ(B Φ ( n d and he above esimae holds fo all f L Φ (R n saisfying (6.4. Thus T f LΦ(B f L Φ(2B Φ ( n On he ohe hand, C f LΦ(2B = Φ ( n f L Φ(2B C Φ ( n 2 2 d n (6.8 (6.9 f LΦ(B Φ ( n d. (6.0 2 Φ ( n d f LΦ(B Φ ( n d (6. which ogehe wih (6.0 gives (6.5. (ii Le now f L Φ (R n, he weak (Φ, Φ-boundedness of T (see Theoem 5.2 implies T f W LΦ(B T f W LΦ(R n C f LΦ(R n = C f LΦ(2B. Esimae (6.6 follows by (6.8. Theoem 6.4. Le Φ any Young funcion, ϕ, ϕ 2 : R n R R be measuable funcions saisfying (4.2 and T be a sublinea opeao saisfying (6.3. (i If T bounded in L Φ (R n hen i is bounded fom M Φ,ϕ (R n in M Φ,ϕ2 (R n and T f MΦ,ϕ2 (R n C f MΦ,ϕ (R n. (6.2
EJDE-208/0 GLOBAL REGULARITY IN ORLICZ-MORREY SPACES 3 (ii If T bounded fom L Φ (R n o W L Φ (R n hen i is bounded fom M Φ,ϕ (R n o W M Φ,ϕ2 (R n and T f MΦ,ϕ2 (R n C f W MΦ,ϕ (R n wih consans independen of f. Poof. Le T be bounded in L Φ (R n. Then by Lemma 6.3 we have T f MΦ,ϕ2 (R n sup x 0, >0 ϕ 2 (x 0, Applying he Theoem 6. o he above inegal wih f LΦ(B (x 0,Φ ( n d. w( = Φ ( n, v 2 (x 0, = ϕ 2 (x 0,, v (x 0, = ϕ (x 0, Φ ( n, H wg(x 0, = g(x 0, = f LΦ(B (x 0,, f LΦ(B (x 0,w(d, whee condiion (6.2 is equivalen o (4.2, we obain T f MΦ,ϕ2 (R n sup ϕ (x 0, Φ ( n f LΦ(B (x 0, = f MΦ,ϕ (R n x R n, >0. The case p = is eaed in he same manne using (6.6 and (6.2, T f W M,ϕ2 (R n sup x 0, >0 ϕ 2 (x 0, f LΦ(B (x 0,Φ ( n d = sup x 0, >0 ϕ (x 0, Φ ( n f LΦ(B (x 0, = f MΦ,ϕ (R n. 7. Commuaos of sublinea opeaos geneaed by nonsingula inegals in he space M Φ,ϕ (R n Fo a funcion a BMO and sublinea opeao T saisfying (6.3 define he commuao T a = [a, T ]f = a T f T (af. Suppose ha fo any f L (R n wih compac suppo and x / supp f, i holds T f(y a f(x C a(x a(y dy, (7. x y n R n wih a consan independen of f and x. Suppose in addiion ha T a is bounded in L Φ (R n saisfying T a f LΦ(R n C a f LΦ(R n. Ou aim is o show boundedness of T a in M Φ,ϕ (R n. Fo his goal we ecall some well known popeies of he BM O funcions. Lemma 7. (John-Nienbeg lemma [3]. Le a BMO and p (,. Then fo any ball B i holds ( /p a(y a B dy p C(p a. (7.2 B B
4 V. S. GULIYEV, A. A. AHMADLI, M. N. OMAROVA, L. SOFTOVA EJDE-208/0 Definiion 7.2. A Young funcion Φ is said o be of uppe ype p (esp. lowe ype p fo some p [0,, if hee exiss a posiive consan C such ha, fo all [, (esp. [0, ] and s [0,, Φ(s C p Φ(s. Remak 7.3. We know ha if Φ is lowe ype p 0 and uppe ype p wih < p 0 p <, hen Φ 2 2. Convesely if Φ 2 2, hen Φ is lowe ype p 0 and uppe ype p wih < p 0 p < (see [29]. Befoe poving he main heoems, we need he following lemma. Lemma 7.4 ([30]. Le b BMO(R n. Then hee is a consan C > 0 such ha b B b B C b ln fo 0 < 2 <, whee C is independen of b, x,, and. In he following lemma which was poved in [24] we povide a genealizaion of he popey (7.2, fom L p -noms o Olicz noms. Lemma 7.5. Le b BMO and Φ be a Young funcion. Le Φ is lowe ype p 0 and uppe ype p wih p 0 p <, hen b sup Φ ( n b( b B(x, LΦ(B(x,. x R n,>0 Fo he vaiable exponen Lebesgue space L p( Lemma 7.5 was poved in [28]. Fo a Young funcion Φ, le a Φ := Φ ( inf (0, Φ(, b Φ ( Φ := sup (0, Φ(. Remak 7.6. I is known ha Φ 2 2 if and only if < a Φ b Φ < (See, fo example [33]. Remak 7.7. Remaks 7.6 and Remak 7.3 show ha a Young funcion Φ is lowe ype p 0 and uppe ype p wih < p 0 p < if and only if < a Φ b Φ <. To esimae he commuao we shall employ he same idea which we used in he poof of Lemma 6.3. Lemma 7.8. Le Φ be a Young funcion wih Φ 2 2, a BMO and T a be a bounded opeao in L Φ (R n saisfying (7.. Suppose ha fo all f L loc Φ (Rn and > 0 holds ( ln f LΦ(B (x0, Φ ( n d <. (7.3 Then T a f LΦ(B a Φ ( n 2 ( ln f LΦ(B (x 0, Φ ( n d. Poof. Decompose f as f = fχ 2B fχ (2B c = f f 2. Fom he boundedness of T a in L Φ (R n i follows ha T a f LΦ(B T a f LΦ(R n a f LΦ(R n = a f LΦ(2B.
EJDE-208/0 GLOBAL REGULARITY IN ORLICZ-MORREY SPACES 5 On he ohe hand, because of (6.7, we can wie T ( ( a f 2 LΦ(B a(x a(y f(y pdx /p B (2B x 0 y n dy c ( ( a(y a B f(y pdx /p B (2B x 0 y n dy c ( ( a(x a B f(y pdx /p (2B x 0 y n dy = I I 2. c We esimae I as follows I = = B Φ ( n Φ ( n Φ ( n Φ ( n a(y a B f(y (2B x 0 y n dy c a(y a B (2B f(y c x 0 y a(y a B f(y dy 2 2 2 x 0 y B a(y a B f(y dy d n. Applying Hölde s inequaliy, Lemma 7. and (7.4, we obain ( I Φ ( n a(y a B f(y dy d n Φ ( n ( Φ ( n Φ ( n 2 B 2 2 a Φ ( n To esimae I 2 noe ha d dy n a B a B f(y dy d B n a( a B f LΦ(B L Φ e (B 2 a B a B 2 ( ln I 2 = a( a B LΦ(B By Lemma 7. and (6.8 we obain a f(y I 2 Φ ( n (2B x 0 y n dy a c Φ ( n Summing I and I 2 we obain ha fo all p (,, T ( a f 2 LΦ(B ln Finally, a Φ ( n T a f LΦ(B a f LΦ(2B a Φ ( n 2 d n d n f LΦ(B Φ ( n d f LΦ(B Φ ( n d. (2B c f(y x 0 y n dy. 2 2 f LΦ(B Φ ( n d. f LΦ(B Φ ( n d. ( ln f LΦ(B Φ ( n d
6 V. S. GULIYEV, A. A. AHMADLI, M. N. OMAROVA, L. SOFTOVA EJDE-208/0 and he saemen follows by (6.. Theoem 7.9. Le Φ be a Young funcion wih Φ 2 2, a BMO and ϕ, ϕ 2 : R n R R be measuable funcions saisfying (4.4. Suppose T a be a sublinea opeao bounded on L Φ (R n and saisfying (7.. Then T a is bounded fom M Φ,ϕ (R n o M Φ,ϕ2 (R n and wih a consan independen of f. T a f MΦ,ϕ2 (R n C a f MΦ,ϕ (R n (7.4 The saemen of he heoem follows by Lemma 7.8 and Theoem 6. in he same manne as he poof of Theoem 6.4. 8. Singula and nonsingula inegal opeaos in he spaces M Φ,ϕ In his secion we deal wih Caldeón-Zygmund ype inegals and hei commuaos wih BM O funcions. We sa wih he definiion of he coesponding kenel. Definiion 8.. A measuable funcion K(x, ξ : R n R n \ {0} R is called a vaiable Caldeón-Zygmund kenel if: (i K(x, is a Caldeón-Zygmund kenel fo almos all x R n : (a K(x, C (R n \ {0}, (b K(x, µξ = µ n K(x, ξ fo all µ > 0, (c S n K(x, ξdσ ξ = 0, S n K(x, ξ dσ ξ <, (ii max β 2n D β ξ K(x, ξ L (R n S n = M < independenly of x. The singula inegals Kf(x = P. V. K(x, x yf(ydy, R n C[a, f](x = P. V. K(x, x yf(y[a(x a(y]dy R n = a(xkf(x K(af(x ae bounded in L Φ (R n (see [39], moeove which implies Kf(x C R n K(x, ξ ξ n K ( x, ξ M ξ n ξ f(y a(x a(y f(y dy, C[a, f](x C x y n R x y n dy n and hence he validiy of all esuls fom 4. Le us noe ha any measuable funcion ϕ : R n R R saisfying he condiion (4.4 saisfies also (4.2 wih ϕ ϕ 2 ϕ. Hence he following esuls hold as a simple applicaion of he esimaes fom 4. Theoem 8.2. Le Φ be a Young funcion wih Φ 2 2 and ϕ : R n R R be measuable funcion such ha fo all x R n and > 0 ( ln ( ϕ (x, s ess inf <s< Φ ( s n Φ ( n d Cϕ(x,. (8.
EJDE-208/0 GLOBAL REGULARITY IN ORLICZ-MORREY SPACES 7 Then fo any f M Φ,ϕ (R n and a BMO hee exis consans depending on n, p, ϕ and he kenel such ha Kf MΦ,ϕ(R n C f MΦ,ϕ(R n, C[a, f] MΦ,ϕ(R n C a f MΦ,ϕ(R n. The above heoem follows fom (6.2 and (7.4. Example 8.3. The weigh ϕ( = β Φ ( n, 0 < β < n saisfies condiion (8.. Example 8.4. The weigh ϕ( = β Φ ( n ln m (e, m, 0 < β < n saisfies condiion (8. and he space M Φ,ϕ does no coincide wih any Moey space. Since we aim a sudying egulaiy popeies of he soluion of he Diichle poblem (3. we need of some addiional local esuls. Coollay 8.5. Le Ω R n, Ω C,, a BMO(Ω and f M Φ,ϕ (Ω wih Φ and ϕ as in Theoem 8.2. Then Kf MΦ,ϕ(Ω C f MΦ,ϕ(Ω C[a, f] MΦ,ϕ(Ω C a f MΦ,ϕ(Ω (8.2 wih C = C(n, p, ϕ, Ω, K. Coollay 8.6. Le Φ and ϕ be as in Theoem 8.2 and a V MO wih V MOmodulus γ a. Then fo any ε > 0 hee exiss a posiive numbe ρ 0 = ρ 0 (ε, γ a such ha fo any ball B wih a adius (0, ρ 0 and all f M Φ,ϕ (B holds wih C = C(n, p, ϕ, Ω, K. C[a, f] MΦ,ϕ(B Cε f M Φ,ϕ(B, (8.3 To obain he above esimaes i suffices o exend K(x, and f( as zeo ouside Ω (see [9, Theoem 2.] fo deails. Recall ha he exension of a keeps is BMO nom o V MO-modulus accoding o [, 32]. Fo any x, y R n, x = (x, x n define he genealized eflecion T (x; y as T (x; y = x 2x n a n (y a nn (y T (x = T (x; x : R n R n, whee a n is he las ow of he coefficiens maix a. consans C, C 2 depending on n and Λ, such ha C x y T (x y C 2 x y, x, y R n. Then hee exis posiive Fo any f M Φ,ϕ (R n and a BMO conside he nonsingula inegal opeaos Kf(x = K(x, T (x yf(ydy, C[a, f](x = a(xkf(x K(af(x. R n The kenel K(x, T (x y : R n R n R is no singula and veifies he condiions (i(b and (ii fom Definiion 8.. Moeove which implies Kf(x C K(x, T (x y M T (x y n C x y n, R n f(y x y dy, C[a, f](x C a(x a(y f(y R n x y dy. The following esimaes ae simple consequence of he esuls in 6 and 7.
8 V. S. GULIYEV, A. A. AHMADLI, M. N. OMAROVA, L. SOFTOVA EJDE-208/0 Theoem 8.7. Le Φ be a Young funcion wih Φ 2 2, a BMO(R n and ϕ be measuable funcion saisfying (8.. Then he opeaos Kf and C[a, f] ae coninuous in M Φ,ϕ and fo all f M Φ,ϕ (R n holds Kf MΦ,ϕ(R n C f MΦ,ϕ(R n, C[a, f] MΦ,ϕ(R n C a f MΦ,ϕ(R n (8.4 wih a consan dependen on known quaniies only. Coollay 8.8. Le Φ and ϕ be as in Theoem 8.7 and a V MO wih a V MOmodulus γ a. Then fo any ε > 0 hee exiss a posiive numbe ρ 0 = ρ 0 (ε, γ a such ha fo any ball B wih a adius (0, ρ 0 and all f M Φ,ϕ (B holds whee C is independen of ε, f and. C[a, f] MΦ,ϕ(B Cε f M Φ,ϕ(B, (8.5 The poof of he above coollay is as ha of [9, Theoem 2.3]. 9. Diichle poblem We conside he Diichle poblem fo second ode linea equaions subjec o he following condiions: Lu := a ij (xd ij u = f(x a.a. x Ω, u W 2,Φ,ϕ (Ω W 0,Φ(Ω (H Unifom ellipiciy of L: hee exiss a consan Λ > 0, such ha Λ ξ 2 a ij (xξ i ξ j Λ ξ 2 a ij (x = a ji (x i, j n. a.a. x Ω, ξ R n (9. This assumpion implies immediaely essenial boundedness of he coefficiens a ij L (Ω. (H2 Regulaiy of he daa: a ij V MO(Ω and f M Φ,ϕ (Ω wih < p < and ϕ : Ω R R measuable. Theoem 9. (Ineio esimae. Le u W2,Φ loc (Ω and L be a linea unifomly ellipic opeao wih V MO coefficiens such ha Lu MΦ,ϕ loc (Ω wih Φ 2 2 and ϕ saisfying (8.. Then D ij u M Φ,ϕ (Ω fo any Ω Ω Ω and D 2 u MΦ,ϕ(Ω C ( u MΦ,ϕ(Ω Lu MΦ,ϕ(Ω, (9.2 whee he consan depends on known quaniies and dis (Ω, Ω. Poof. Take an abiay poin x supp u and a ball B (x Ω, choose a poin x 0 B (x and fix he coefficiens of L in x 0. Conside he consan coefficiens opeao L 0 = a ij (x 0 D ij. Fom he classical heoy we know ha a soluion v C0 (B (x of L 0 v = (L 0 Lv Lv can be pesened as Newonian ype poenial v(x = Γ 0 (x y[(l 0 Lv(y Lv(y]dy, B
EJDE-208/0 GLOBAL REGULARITY IN ORLICZ-MORREY SPACES 9 whee Γ 0 (x y = Γ(x 0, x y is he fundamenal soluion of L 0. Taking D ij v and unfeezing he coefficiens we obain fo all i, j =,..., n (cf. [9] D ij v(x = P. V. Γ ij (x, x y[lv(y ( a hk (x a hk (y D hk v(y]dy B Lv(x Γ j (x, yy i dσ y (9.3 S n = K ij Lv(x C ij [a hk, D hk v](x Lv(x Γ j (x; yy i dσ y. S n Hee Γ ij (x, ξ sand fo he deivaives D ξiξ j Γ(x, ξ. The known popeies of he fundamenal soluion imply ha Γ ij (x, ξ ae vaiable Caldeón-Zygmund kenels in he sense of Definiion 8.. The epesenaion fomula (9.3 sill holds fo any v W 2,p (B W,p(B 0 because of he appoximaion popeies of he Sobolev funcions wih C0 funcions. In view of (8.2, (8.3 and (9.3 fo each ε > 0 hee exiss 0 (ε such ha fo any < 0 (ε i holds D 2 v Φ,ϕ; C ( ε D 2 v Φ,ϕ; Lv Φ,ϕ;, Φ,ϕ; := MΦ,ϕ(B. Choosing ε (and hence also! small enough we can move he nom of D 2 v on he lef-hand side ha gives D 2 v Φ,ϕ; C Lv Φ,ϕ;. (9.4 Define a cu-off funcion η(x such ha fo θ (0,, θ = θ(3 θ/2 > θ and s = 0,, 2 we have { x B θ η(x = η(x C0 (B, D s η C[θ( θ] s. 0 x B θ Applying (9.4 o v(x = η(xu(x W 2,Φ,ϕ (B W 0,Φ (B we obain D 2 u Φ,ϕ;θ C Lv Φ,ϕ;θ ( C Lu Φ,ϕ;θ Du Φ,ϕ;θ u Φ,ϕ;θ θ( θ [θ( θ] 2. Define he weighed semi-nom Θ s = sup 0<θ< [ ] s D θ( θ s u Φ,ϕ;θ, s = 0,, 2. Because of he choice of θ we have θ( θ 2θ ( θ. Thus, afe sandad ansfomaions and aking he supemum wih espec o θ (0, he las inequaliy ewies as Θ 2 C ( 2 Lu Φ,ϕ; Θ Θ 0. (9.5 Lemma 9.2 (Inepolaion inequaliy. Thee exiss a consan C independen of such ha Θ εθ 2 C ε Θ 0 fo any ε (0, 2. Poof. By simple scaling agumens we obain in M Φ,ϕ (R n an inepolaion inequaliy analogous o [7, Theoem 7.28] Du Φ,ϕ; δ D 2 u Φ,ϕ; C δ u Φ,ϕ;, δ (0,.
20 V. S. GULIYEV, A. A. AHMADLI, M. N. OMAROVA, L. SOFTOVA EJDE-208/0 We can always find some θ 0 (0, such ha Θ 2[θ 0 ( θ 0 ] Du Φ,ϕ;θ0 ( 2[θ 0 ( θ 0 ] δ D 2 u Φ,ϕ;θ0 C δ u Φ,ϕ;θ 0. The asseion follows choosing δ = ε 2 [θ 0( θ 0 ] < θ 0 fo any ε (0, 2. Inepolaing Θ in (9.5, we obain 2 4 D2 u Φ,ϕ;/2 Θ 2 C ( 2 Lu Φ,ϕ; u Φ,ϕ; and hence he Caccioppoli-ype esimae ( D 2 u Φ,ϕ;/2 C Lu Φ,ϕ; 2 u Φ,ϕ;. (9.6 Le v = {v ij } n ij= [L Φ,ω(B ] n2 be abiay funcion maix. Define he opeaos S ijhk (v hk (x = C ij [a hk, v hk ](x i, j, h, k =,..., n. Because of he V MO popeies of a ij s we can choose so small ha n i,j,h,k= S ijhk <. (9.7 Now fo a given u W 2,Φ (B W,Φ 0 (B wih Lu M Φ,ϕ (B define H ij (x = K ij Lu(x Lu(x Γ j (x; yy i dσ y S n and (8.2 implies H ij M Φ,ϕ (B. Define he opeao W by he seing { n ( Wv = Sijhk v hk H ij (x } n h,k= ij= : [ M Φ,ϕ (B ] n 2 [ M Φ,ϕ (B ] n 2. By (9.7 he opeao W is a conacion mapping and hee exiss a unique fixed poin ṽ = {ṽ ij } n ij= [M Φ,ϕ(B ] n2 of W such ha Wṽ = ṽ. On he ohe hand i follows fom he epesenaion fomula (9.3 ha also D 2 u = {D ij u} n ij= is a fixed poin of W. Hence D 2 u ṽ, ha is D ij u M Φ,ϕ (B and in addiion (9.6 holds. The ineio esimae (9.2 follows fom (9.6 by a finie coveing of Ω wih balls B /2, < dis(ω, Ω. To pove a local bounday esimae fo he nom of D ij u we define he space W γ0 2,Φ (B as a closue of C γ0 = {u C0 (B(x 0, : u(x = 0 fo x n 0} wih espec o he nom of W 2,p. Theoem 9.3 (Bounday esimae. Le u W γ0 2,Φ (B and suppose ha Lu M Φ,ϕ (B wih Φ 2 2 and ϕ saisfying (8.. Then D ij u M Φ,ϕ (B and fo each ε > 0 hee exiss 0 (ε such ha D ij u Φ,ϕ;B C Lu Φ,ϕ;B, (0, 0. (9.8
EJDE-208/0 GLOBAL REGULARITY IN ORLICZ-MORREY SPACES 2 Poof. Fo u W γ0 2,Φ (B he bounday epesenaion fomula holds (see [0] D ij u(x = P. V. Γ ij (x, x ylu(ydy B P. V. Γ ij (x, x y [ a hk (x a hk (y ] D hk u(ydy (9.9 B Lu(x Γ j (x, yy i dσ y I ij (x, i, j =,..., n, S n whee we have se i, j =,..., n, I ij (x = Γ ij (x, T (x ylu(ydy B Γ ij (x, T (x y [ a hk (x a hk (y ] D hk u(ydy I in (x = I ni (x = Γ il (x, T (x y(d n T (x l B B {[ a hk (x a hk (y ] D hk u(y Lu(y } dy, i =,..., n, I nn (x = Γ ls (x, T (x y(d n T (x l (D n T (x s B {[ a hk (x a hk (y ] D hk u(y Lu(y } dy, whee D n T (x = ( (D n T (x,..., (D n T (x n = T (e n, x. Applying esimaes (8.4 and (8.5, aking ino accoun he V MO popeies of he coefficiens a ij s, i is possible o choose 0 so small ha D ij u p,ϕ;b C Lu p,ϕ;b fo each < 0. Fo an abiay funcion maix w = {w ij } n ij= [M Φ,ϕ(B ] n2 define S ijhk (w hk (x = C ij [a hk, w hk ](x, i, j, h, l =,..., n, S ijhk (w hk (x = C ij [a hk, w hk ](x, i, j =,..., n ; h, k =,..., n, S inhk (w hk (x = C il [a hk, w hk ](D n T (x l, i, h, k =,..., n, S nnhk (w hk (x = C ls [a hk, w hk ](x(d n T (x l (D n T (x s, h, k =,..., n. Because of (8.3 and (8.5 we can ake so small ha n i,j,h,k= Now, given u W γ0 2,p (B wih Lu M Φ,ϕ (B we se S ijhk S ijhk <. (9.0 H ij (x = K ij Lu(x K ij Lu(x K il Lu(x(D n T (x l K ls Lu(x(D n T (x l (D n T (x s Lu(x Γ j (x, yy i dσ y S n
22 V. S. GULIYEV, A. A. AHMADLI, M. N. OMAROVA, L. SOFTOVA EJDE-208/0 and he Theoems 8.2 and 8.7 imply H ij M Φ,ϕ (B. Define he opeao { n ( Uw = Sijhk (w hk S ijhk (w hk H n ij (x} h,k= By (9.0 i is a conacion mapping in [ M Φ,ϕ (B ] n 2 and hee is unique fixed poin w = { w ij } n ij= such ha U w = w. On he ohe hand, i follows fom he epesenaion fomula (9.9 ha also D 2 u = {D ij u} n ij= is a fixed poin of U. Hence D 2 u w, D ij u M Φ,ϕ (B and he esimae (9.8 holds. Theoem 9.4. Le Φ be a Young funcion wih Φ 2 2 and L be unifomly ellipic opeao saisfying condiions H and H 2. Then fo any funcion f M Φ,ϕ (Ω he unique soluion of he poblem (9. has second deivaives in M Φ,ϕ (Ω. Moeove D 2 u MΦ,ϕ(Ω C ( u MΦ,ϕ(Ω f MΦ,ϕ(Ω (9. and he consan C depends on known quaniies only. Poof. Since M Φ,ϕ (Ω L Φ (Ω, poblem (9. is uniquely solvable in he Sobolev space W 2,Φ (Ω W,Φ 0 (Ω accoding o [0]. By local flaeing of he bounday, coveing wih semi-balls, aking a paiion of uniy subodinaed o ha coveing and applying of esimae (9.8 we obain a bounday a pioi esimae ha unified wih (9.2 ensues validiy of (9.. Acknowledgmens. The auhos ae gaeful o Pofesso Viceniu Radulescu fo his valuable commens. V. S. Guliyev and M. Omaova wee paially suppoed by he s Azebaijan- Russia Join Gan Compeiion (Ageemen numbe No. 8-5-06005, and by a gan fom he Pesidium of Azebaijan Naional Academy of Science 205. Refeences [] P. Acquisapace; On BMO egulaiy fo linea ellipic sysems, Ann. Ma. Pua. Appl., 6 (992, 23 270. [2] A. Akbulu, V. S. Guliyev, R. Musafayev; On he boundedness of he maximal opeao and singula inegal opeaos in genealized Moey spaces, Mah. Bohem. 37 ( (202, 27-43. [3] C. Benne, R. Shapley; Inepolaion of opeaos, Academic Pess, Boson, 988. [4] G. Bonanno, G. Molica Bisci, V. Radulescu; Infiniely many soluions fo a class of nonlinea eigenvalue poblem in Olicz-Sobolev spaces, C. R. Mah. Acad. Sci. Pais, 349 (5-6 (20, 263-268. [5] G. Bonanno, G. Molica Bisci, V. D. Radulescu; Quasilinea ellipic non-homogeneous Diichle poblems hough Olicz-Sobolev spaces, Nonlinea Anal., 75 (2 (202, 444-4456. [6] V. Buenkov, A. Gogaishvili, V. S. Guliyev, R. Musafayev; Boundedness of he facional maximal opeao in local Moey-ype spaces, Complex Va. Ellipic Equ., 55 (8-0 (200, 739-758. [7] S. Campanao; Sisemi elliici in foma divegenza. Regolaiá all ineno, Quadeni di Scuola Nom. Sup. Pisa (980. [8] F. Chiaenza, M. Fasca; Moey spaces and Hady-Lilewood maximal funcion, Rend Ma., 7 (987, 273-279. [9] F. Chiaenza, M. Fasca, P. Longo; Ineio W 2,p -esimaes fo nondivegence ellipic equaions wih disconinuous coefficiens, Riceche Ma., 40 (99, 49-68. [0] F. Chiaenza, M. Fasca, P. Longo; W 2,p -solvabiliy of Diichle poblem fo nondivegence ellipic equaions wih VMO coefficiens, Tans. Ame. Mah. Soc. 336 (993, 84-853. ij=.
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24 V. S. GULIYEV, A. A. AHMADLI, M. N. OMAROVA, L. SOFTOVA EJDE-208/0 [36] T. Mizuhaa; Boundedness of some classical opeaos on genealized Moey spaces, Hamonic Anal., Poc. Conf., Sendai/Jap. 990, ICM-90 Saell. Conf. Poc., (99 83-89. [37] C. B. Moey; On he soluions of quasi-linea ellipic paial diffeenial equaions, Tans. Ame. Mah. Soc., 43 (938, 26-66. [38] E. Nakai; Hady-Lilewood maximal opeao, singula inegal opeaos and he Riesz poenials on genealized Moey spaces, Mah. Nach., 66 (994, 95-03. [39] E. Nakai; Caldeón-Zygmund opeaos on Olicz-Moey spaces and modula inequaliies, Banach and funcion spaces II, 393-40, Yokohama Publ., Yokohama, 2008. [40] D. Palagachev, L. Sofova; Fine egulaiy fo ellipic sysems wih disconinuous ingediens, J. Ach. Mah., 86 (2 (2006, 45-53. [4] J. Peee; On he heoy of L p,λ, J. Func. Anal. 4 (969, 7-87. [42] L. E. Pesson, N. Samko; Weighed Hady and poenial opeaos in he genealized Moey spaces, J. Mah. Anal. Appl., 377 (2 (20, 792-806. [43] L. C. Piccinini; Inclusioni a spazi di Moey, Boll. Unione Ma. Ial., IV. Se. 2 (969, 95-99. [44] V. D. Radulescu, D. D. Repovs; Paial diffeenial equaions wih vaiable exponens. Vaiaional mehods and qualiaive analysis, Monogaphs and Reseach Noes in Mahemaics. CRC Pess, Boca Raon, FL, 205. [45] M. M. Rao, Z. D. Ren; Theoy of Olicz Spaces, M. Dekke, Inc., New Yok, 99. [46] D. Saason; On funcions of vanishes mean oscillaion, Tans. Ame. Mah. Soc., 207 (975, 39-405. [47] Y. Sawano; A Handbook of Hamonic Analysis, Tokyo, 20. [48] L. G. Sofova; Singula inegals and commuaos in genealized Moey spaces, Aca Mah. Sin., Engl. Se., 22 (2006, 757-766. [49] L. G. Sofova; The Diichle poblem fo ellipic equaions wih VMO coefficiens in genealized Moey spaces, Advances in hamonic analysis and opeao heoy, 37-386, Ope. Theoy Adv. Appl., 229, Bikhäuse/Spinge Basel AG, Basel, 203. [50] G. Weiss; A noe on Olicz spaces, Pougal Mah., 5 (956, 35-47. [5] J. Xiao; A new pespecive on he Riesz poenial, Adv. Nonlinea Anal., 6 (207, no. 3, 37-326. Vagif S. Guliyev Ahi Evan Univesiy, Depamen of Mahemaics, 4000 Kisehi, Tukey. S.M. Nikol skii Insiue of Mahemaics a RUDN Univesiy, Moscow, 798, Russia. Insiue of Mahemaics and Mechanics, Az 4 Baku, Azebaijan E-mail addess: vagif@guliyev.com Aysel A. Ahmadli Dumlupina Univesiy, Depamen of Mahemaics, 4000 Kyahya, Tukey E-mail addess: aysel.ahmadli@gmail.com Mehiban N. Omaova Baku Sae Univesiy, AZ4 Baku, Azebaijan. Insiue of Mahemaics and Mechanics, Az 4 Baku, Azebaijan E-mail addess: mehibanomaova@yahoo.com Lubomia G. Sofova Depamen of Mahemaics, Univesiy of Saleno, Fisciano, Ialy E-mail addess: lsofova@unisa.i