BOUNDEDNESS OF THE MAXIMAL, POTENTIAL AND SINGULAR OPERATORS IN THE GENERALIZED VARIABLE EXPONENT MORREY SPACES

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1 MATH. SCAND. 107 (2010), BOUNDEDNESS OF THE MAXIMAL, POTENTIAL AND SINGULAR OPERATORS IN THE GENERALIZED VARIABLE EXPONENT MORREY SPACES VAGIF S. GULIYEV, JAVANSHIR J. HASANOV and STEFAN G. SAMKO Absrac We consider generalized Morrey spaces M p( ),ω ( ) wih variable exponen and a general funcion ω(x,r) defining he Morrey-ype norm. In case of bounded ses R n we prove he boundedness of he Hardy-Lilewood maximal operaor and Calderon-Zygmund singular operaors wih sandard kernel, in such spaces. We also prove a Sobolev-Adams ype M p( ),ω ( ) M q( ),ω ( )-heorem for he poenial operaors I α( ), also of variable order. The condiions for he boundedness are given i erms of Zygmund-ype inegral inequaliies on ω(x,r), which do no assume any assumpion on monooniciy of ω(x,r) in r. 1. Inroducion In he sudy of local properies of soluions o parial differenial equaions, ogeher wih weighed Lebesgue spaces, Morrey spaces L p,λ ( ) play an imporan role, see [14], [25]. Inroduced by C. Morrey [27] in 1938, hey are defined by he norm f L p,λ := sup r λ p f L p (B(x,r)), x,r>0 where 0 λ<n,1 p<. As is known, las wo decades here is an increasing ineres o he sudy of variable exponen spaces and operaors wih variable parameers in such spaces, we refer for insance o he surveying papers [12], [20], [22], [38], on he progress in his field, including opics of Harmonic Analysis and Operaor Theory, see also references herein. Variable exponen Morrey spaces L p( ),λ( ) ( ), were inroduced and sudied in [2] and [29] in he Euclidean seing and in [21] in he seing of meric measure spaces, in case of bounded ses. In [2] here was proved he boundedness of he maximal operaor in variable exponen Morrey spaces L p( ),λ( ) ( ) under he log-condiion on p( ) and λ( ) and for poenial operaors, under he same log-condiion and he assumpions inf x α(x) > 0, Received 2 July 2009, in revised form 27 Sepember 2009.

2 286 v. s. guliyev, j. j. hasanov and s. g. samko sup x [λ(x) + α(x)] <n, here was proved a Sobolev ype L p( ),λ( ) L q( ),λ( ) -heorem. In he case of consan α, here was also proved a boundedness heorem in he limiing case = n λ(x), when he poenial operaor α I α acs from L p( ),λ( ) ino BMO. In [29] he maximal operaor and poenial operaors were considered in a somewha more general space, bu under more resricive condiions on. P. Häsö in [18] used his new local-o-global approach o exend he resul of [2] on he maximal operaor o he case of he whole space R n. In [21] here was proved he boundedness of he maximal operaor and he singular inegral operaor in variable exponen Morrey spaces L p( ),λ( ) in he general seing of meric measure spaces. In he case of consan p and λ, he resuls on he boundedness of poenial operaors and classical Calderon- Zygmund singular operaors go back o [1] and [32], respecively, while he boundedness of he maximal operaor in he Euclidean seing was proved in [9]; for furher resuls in he case of consan p and λ see for insance [5] [8]. We inroduce he generalized variable exponen Morrey spaces M p( ),ω ( ) over an open se R n. Generalized Morrey spaces of such a kind in he case of consan p were sudied in [4], [13], [26], [28], [30], [31]. Wihin he frameworks of he spaces M p( ),ω ( ), over bounded ses R n we consider he Hardy-Lilewood maximal operaor Mf (x) = sup B(x,r) 1 f(y) dy r>0 B(x,r) poenial ype operaors I α(x) f(x)= he fracional maximal operaor M α(x) f(x)= sup r>0 x y α(x) n f(y)dy, 0 <α(x)<n, B(x,r) α(x) n 1 f(y) dy, B(x,r) 0 α(x)<n of variable order α(x) and Calderon-Zygmund ype singular operaor Tf (x) = K(x,y)f(y)dy, where K(x,y) is a sandard singular kernel, ha is, a coninuous funcion defined on {(x, y) : x = y} and saisfying he esimaes K(x,y) C x y n for all x = y,

3 boundedness of some operaors in generalized morrey spaces 287 y z σ K(x,y) K(x,z) C, σ > 0, if x y > 2 y z, x y n+σ x ξ σ K(x,y) K(ξ,y) C, σ > 0, if x y > 2 x ξ. x y n+σ We find he condiion on he funcion ω(x,r) for he boundedness of he maximal operaor M and he singular inegral operaors T in generalized Morrey space M p( ),ω ( ) wih variable under he log-condiion on p( ). For poenial operaors, under he same log-condiion and he assumpions inf α(x) > 0, x sup α(x) < n x we also find he condiion on ω(x,r) for he validiy of a Sobolev-Adams ype M p( ),ω ( ) M q( ),ω ( )-heorem, which recovers he known resul for he case of he classical Morrey spaces wih variable exponens, when ω(x,r) = r λ(x) n and hen 1 q(x) = 1 α(x) n λ(x). The paper is organized as follows. In Secion 2 we provide necessary preliminaries on variable exponen Lebesgue and Morrey spaces. In Secion 3 we inroduce he generalized Morrey spaces wih variable exponens and recall some facs known for generalized Morrey spaces wih consan p. In Secion 4 we deal wih he maximal operaor, while poenial operaors are sudied in Secion 5. In Secion 6 we rea Calderon-Zygmund singular operaors. The main resuls are given in Theorems 4.2, 5.2, 5.5, 6.2. We emphasize ha he resuls we obain for generalized Morrey spaces are new even in he case when is consan, because we do no impose any monooniciy ype condiion on ω(x,r). The advance in his paper is based on he usage of he approach developed in [15], [16] for consan p, and presened for variable in Theorems 4.1, 5.4, 6.1, and on he esimae of Lemma 2.5. Noaion. R n is he n-dimensional Euclidean space, R n is an open se, l = diam ; χ E (x) is he characerisic funcion of a se E R n ; B(x,r) ={y R n : x y <r}, B(x,r) = B(x,r) ; by c, C, c 1,c 2, ec., we denoe various absolue posiive consans, which may have differen values even in he same line. 2. Preliminaries on variable exponen Lebesgue and Morrey spaces Le p( ) be a measurable funcion on wih values in [1, ). An open se is assumed o be bounded hroughou he whole paper. We suppose ha (2.1) 1 <p p + <,

4 288 v. s. guliyev, j. j. hasanov and s. g. samko where p := ess inf x > 1, p + := ess sup x <. By L p( ) ( ) we denoe he space of all measurable funcions f(x) on such ha I p( ) (f ) = f(x) dx <. Equipped wih he norm f p( ) = inf { ( ) } f η>0:i p( ) 1, η his is a Banach funcion space. By p ( ) =, x, we denoe he 1 conjugae exponen. The Hölder inequaliy is valid in he form ( 1 f(x) g(x) dx + 1 ) p p f p( ) g p ( ). For he basics on variable exponen Lebesgue spaces we refer o [39], [24]. Definiion 2.1. By W L( ) (weak Lipschiz) we denoe he class of funcions defined on saisfying he log-condiion (2.2) p(y) A ln x y, x y 1, x,y, 2 where A = A(p) > 0 does no depend on x,y. Theorem 2.2 ([10]). Le R n be an open bounded se and p W L( ) saisfy condiion (2.1). Then he maximal operaor M is bounded in L p( ) ( ). The following heorem for bounded ses, bu for variable α(x), was proved in [37] under he condiion ha he maximal operaor is bounded in L p( ) ( ), which became an uncondiional resul afer he resul of Diening [10] on maximal operaors. Theorem 2.3. Le R n be bounded, p, α W L( ) saisfy assumpion (2.1) and he condiions (2.3) inf α(x) > 0, sup x α(x) < n. x Then he operaor I α( ) is bounded from L p( ) ( ) o L q( ) ( ) wih (2.4) 1 q(x) = 1 α(x) n.

5 boundedness of some operaors in generalized morrey spaces 289 Singular operaors wihin he framework of he spaces wih variable exponens were sudied in [11]. From Theorem 4.8 and Remark 4.6 of [11] and he known resuls on he boundedness of he maximal operaor, we have he following saemen, which is formulaed below for our goals for a bounded, bu valid for an arbirary open se under he corresponding condiion in a infiniy. Theorem 2.4 ([11]). Le R n be a bounded open se and p W L( ) saisfy condiion (2.1). Then he singular inegral operaor T is bounded in L p( ) ( ). We will also make use of he esimae provided by he following lemma (see [36], Corollary o Lemma 3.22). Lemma 2.5. Le be a bounded domain and p saisfy he assumpion 1 p p + < and condiion (2.2). Le also sup ν(x) < and inf[n + ν(x)] > 0. Then (2.5) x y ν(x) χ B(x,r) (y) p(y) Cr ν(x)+ n, x, 0 <r<l= diam, where C does no depend on x and r. Remark 2.6. ( I may ) be shown ha he consan C in (2.5) may be esimaed as C = C 0 l n 1 p 1 p+, where C 0 does no depend on. Le λ(x) be a measurable funcion on wih values in [0,n]. The variable Morrey space L p( ),λ( ) ( ) is defined as he se of inegrable funcions f on wih he finie norm f L p( ),λ( ) ( ) = sup x,>0 λ(x) fχ B(x,) L p( ) ( ). The following saemens are known. Theorem 2.7 ([2]). Le be bounded and p W L( ) saisfy condiion (2.1) and le a measurable funcion λ saisfy he condiions 0 λ(x), sup λ(x) < n. x Then he maximal operaor M is bounded in L p( ),λ( ) ( ). Theorem 2.7 was exended o unbounded domains in [18]. Noe ha he boundedness of he maximal operaor in Morrey spaces wih variable was sudied in [21] in he more general seing of quasimeric measure spaces.

6 290 v. s. guliyev, j. j. hasanov and s. g. samko Theorem 2.8 ([2]). Le be bounded, p, α, λ W L( ) and p saisfy condiion (2.1). Le also λ(x) 0 and (2.6) inf α(x) > 0, sup x [λ(x) + α(x)] <n. x Then he operaor I α( ) is bounded from L p( ),λ( ) ( ) o L q( ),μ( ) ( ), where (2.7) 1 q(x) = 1 α(x) n and μ(x) q(x) = λ(x). Theorem 2.9 ([2]). Le be bounded, p, α, λ W L( ) and p saisfy condiion (2.1). Le also λ(x) 0 and condiions (2.6) hold. Then he operaor I α( ) is bounded from L p( ),λ( ) ( ) o L q( ),λ( ) ( ), where (2.8) 1 q(x) = 1 α(x) n λ(x). Theorem 2.10 ([2]). Le be bounded and p, α, λ W L( ) saisfy condiions (2.1) and he condiions inf α(x) > 0, x λ(x) + α(x) = n hold. Then he operaor M α( ) is bounded from L p( ),λ( ) ( ) o L ( ). 3. Variable exponen generalized Morrey spaces Everywhere in he sequel he funcions ω(x,r), ω 1 (x, r) and ω 2 (x, r) used in he body of he paper, are non-negaive measurable funcion on (0,l), l = diam. We find i convenien o define he generalized Morrey spaces in he form as follows. Definiion 3.1. Le 1 p <. The generalized Morrey space M p( ),ω ( ) is defined by he norm f M p( ), = sup x,r>0 ω(x,r) f L p( ) ( B(x,r)). According o his definiion, we recover he space L p( ),λ( ) ( ) under he choice ω(x,r) = r λ(x) n : L p( ),λ( ) ( ) = M p( ),ω( ) ( ). ω(x,r)=r λ(x) n

7 boundedness of some operaors in generalized morrey spaces 291 Everywhere in he sequel we assume ha (3.1) inf x,r>0 ω(x,r) > 0 which makes he space M p( ),ω ( ) nonrivial. Noe ha when p is consan, in he case of w(x, r) cons > 0, we have he space L ( ) Preliminaries on Morrey spaces wih consan exponens p In [15], [16], [28] and [30] here were obained sufficien condiions on funcions ω 1 and ω 2 for he boundedness of he singular operaor T from M p,ω 1 (R n ) o M p,ω 2 (R n ). In [30] he following condiion was imposed on w(x, r): (3.2) c 1 ω(x,r) ω(x,) c ω(x, r) whenever r 2r, where c( 1) does no depend on,r and x R n, joinly wih he condiion: (3.3) ω(x,) p d C ω(x, r) p r for he maximal or singular operaor and he condiion (3.4) αp ω(x,) p d Cr αp ω(x,r) p. r for poenial and fracional maximal operaors, where C(> 0) does no depend on r and x R n. Remark 3.2. Noe ha he righ-hand side inequaliy in (3.2) may be omied: i follows from he lef-hand-side one and (3.3), which we show in Secion 7. Remark 3.3. The lef-hand side inequaliy in (3.2) is saisfied for any nonnegaive funcion w(x, r) such ha here exiss a number a R 1 such ha he funcion r a w(x, r) is almos increasing in r uniformly in x: a w(x, ) cr a w(x, r) for all 0 < r< where c 1 does no depend on x,r,. Noe ha inegral condiions of ype (3.3) afer he paper [3] of 1956 are ofen referred o as Bary-Sechkin or Zygmund-Bary-Sechkin condiions, see also [17]. The classes of almos monoonic funcions saisfying such inegral condiions were laer sudied in a number of papers, see [19], [33], [34] and references herein, where he characerizaion of inegral inequaliies of

8 292 v. s. guliyev, j. j. hasanov and s. g. samko such a kind was given in erms of cerain lower and upper indices known as Mauszewska-Orlicz indices. Noe ha in he cied papers he inegral inequaliies were sudied as r 0. Such inequaliies are also of ineres when hey allow o impose differen condiions as r 0 and r ; such a case was deal wih in [35], [23]. In [30] he following saemens were proved. Theorem 3.4 ([30]). Le 1 < p < and ω(x,r) saisfy condiions (3.2) (3.3). Then he operaors M and T are bounded in M p,ω (R n ). Theorem 3.5 ([30]). Le 1 <p<, 0 <α< n, and ω(x,) saisfy p condiions (3.2) and (3.4). Then he operaors M α and I α are bounded from M p,ω (R n ) o M q,ω (R n ) wih 1 q = 1 p α n. The following saemen, conaining he resuls in [28], [30] was proved in [15] (see also [16]). Noe ha Theorems 3.6 and 3.7 do no impose condiion (3.2). Theorem 3.6 ([15]). Le 1 <p< and ω 1 (x, r), ω 2 (x, r) be posiive measurable funcions saisfying he condiion r ω 1 (x, ) d c 1 ω 2 (x, r). wih c 1 > 0 no depending on x R n and >0. Then he operaors M and T are bounded from M p,ω 1( ) (R n ) o M p,ω 2( ) (R n ). Theorem 3.7 ([15]). Le 0 <α<n, 1 <p<, q = 1 p α n and ω 1 (x, r), ω 2 (x, r) be posiive measurable funcions saisfying he condiion 1 r α ω 1 (x, ) d c 1 r α ω 2 (x, r). Then he operaors M α and I α are bounded from M p,ω 1( ) (R n ) o M q,ω 2( ) (R n ). 4. The maximal operaor in he spaces M p( ),ω( ) ( ) Theorem 4.1. Le be bounded and p W L( ) saisfy condiion (2.1). Then (4.1) Mf L p( ) ( B(x,)) C n 1 f L p( ) ( B(x,r)) dr, 0 << l 2 for every f L p( ) ( ), where C does no depend on f, x and.

9 boundedness of some operaors in generalized morrey spaces 293 Proof. We represen f as (4.2) f = f 1 + f 2, f 1 (y) = f(y)χ B(x,2) (y), f 2 (y) = f(y)χ \ B(x,2) (y), > 0, and have Mf L p( ) ( B(x,)) Mf 1 L p( ) ( B(x,)) + Mf 2 L p( ) ( B(x,)). By Theorem 2.2 we obain (4.3) Mf 1 L p( ) ( B(x,)) Mf 1 L p( ) ( ) C f 1 L p( ) ( ) = C f L p( ) ( B(x,2)), where C does no depend on f. From (4.3) we obain (4.4) Mf 1 L p( ) ( B(x,)) C n C n 2 1 f L p( ) ( B(x,r)) dr 1 f L p( ) ( B(x,r)) dr easily obained from he fac ha f L p( ) ( B(x,2)) is non-decreasing in, so ha f L p( ) ( B(x,2)) on he righ-hand side of (4.3) is dominaed by he righ-hand side of (4.4). Noe ha his complicaion of esimae in comparison wih (4.3) is done because he erm Mf 2 will be esimaed below in a similar form, see (4.6). To esimae Mf 2, we firs prove he following auxiliary inequaliy (4.5) x y n f(y) dy \ B(x,) C s n 1 f L p( ) ( B(x,s)) ds, To his end, we choose β> n p and proceed as follows x y n f(y) dy \ B(x,) ( ) β x y n+β f(y) s β 1 ds dy \ B(x,) x y ( = β s β 1 x y n+β f(y) dy C {y :2 x y s} 0 <<l. ) ds s β 1 f L p( ) ( B(x,s)) x y n+β L p ( ) ( B(x,s)) ds.

10 294 v. s. guliyev, j. j. hasanov and s. g. samko We hen make use of Lemma 2.5 and obain (4.5). For z B(x,) we ge Then by (4.5) Mf 2 (z) = sup B(z,r) 1 r>0 C sup r 2 C sup r 2 C Mf 2 (z) C \ B(x,2) C ( \ B(x,2)) B(z,r) ( \ B(x,2)) B(z,r) 2 f 2 (y) dy B(z,r) x y n f(y) dy. y z n f(y) dy x y n f(y) dy s n 1 f L p( ) ( B(x,s)) ds, s n 1 f L p( ) ( B(x,s)) ds, where C does no depend on x,r. Thus, he funcion Mf 2 (z), wih fixed x and, is dominaed by he expression no depending on z. Then (4.6) Mf 2 L p( ) ( B(x,)) C s n 1 f L p( ) ( B(x,s)) ds 1 L p( ) ( B(x,)). Since 1 L p( ) ( B(x,)) C n by Lemma 2.5, we hen obain (4.1) from (4.4) and (4.6). The following heorem exends Theorem 2.7 o he case of generalized Morrey spaces M p( ),ω ( ). Theorem 4.2. Le R n be an open bounded se and p W L( ) saisfy assumpion (2.1) and he funcion ω 1 (x, r) and ω 2 (x, r) saisfy he condiion (4.7) ω 1 (x, ) d Cω 2 (x, r), r where C does no depend on x and. Then he maximal operaor M is bounded from he space M p( ),ω 1 ( ) o he space M p( ),ω 2 ( ). Proof. Le f M p( ),ω 1 ( ).Wehave Mf M p( ),ω 2 ( ) = sup x, (0,l) ω 1 n 2 (x, ) Mf L p( ) ( B(x,)).

11 boundedness of some operaors in generalized morrey spaces 295 The esimaion is obvious for l 2 l in view of (3.1). For Mf M p( ),ω 2 ( ) = by Theorem 4.1 we obain Hence Mf M p( ),ω 2 ( ) C sup ω 1 n 2 (x, ) Mf L p( ) ( B(x,)) x, (0, 2) l sup ω 1 x,0< l Mf M p( ),ω 2 ( ) C f M p( ),ω 1 ( ) C f M p( ),ω 1 ( ) by (4.7), which complees he proof. 2 (x, ) sup x, (0,l) 1 f L p( ) ( B(x,r)) dr. 1 l ω 1 (x, r) dr ω 2 (x, ) r In he following corollary we recover, from Theorem 4.2, a resul obained in [2] in he case ω 1 (x, r) = ω 2 (x, r) = r λ(x) n. Corollary 4.3. Le R n be bounded,λ(x) 0 and sup x λ(x) < n and p WL( ) saisfy condiion (2.1). Then he maximal operaor M is bounded in he space L p( ),λ( ) ( ). Proof. I suffices o observe ha he funcion ω 1 (x, r) = ω 2 (x, r) = r λ(x) n defining he space L p( ),λ( ) ( ), saisfies condiion (4.7) under he assumpion sup x λ(x) < n. 5. Riesz poenial operaor in he spaces M p( ),ω( ) ( ) In his secion we exend Theorem 3.7 o he variable exponen seing. We give wo versions of such an exension, one being a generalizaion of Spanne s resul for poenial operaors, anoher exending he corresponding Adams resul. Noe ha Theorems 5.1 and 5.2 in he case of consan exponens p and λ were proved in [15] (see also [16]) Spanne ype resul Theorem 5.1. Le p, α W L( ) saisfy condiion (2.1) and le α(x), q(x) saisfy he condiions in (2.3) and (2.4). Then (5.1) I α( ) f L q( ) ( B(x,)) C n q(x) q(x) 1 f L p( ) ( B(x,r)) dr, 0 < l 2,

12 296 v. s. guliyev, j. j. hasanov and s. g. samko where is an arbirary number in ( 0, 2) l and C does no depend on f, x and. Proof. As in he proof of Theorem 4.1, we represen funcion f in form (4.2) and have I α( ) f(x)= I α( ) f 1 (x) + I α( ) f 2 (x). By Theorem 2.3 we obain I α( ) f 1 Lq( ) ( B(x,)) I α( ) f 1 Lq( ) ( ) C f 1 Lp( ) ( ) = C f Lp( ) ( B(x,2)). Then I α( ) f 1 Lq( ) ( B(x,)) C f L p( ) (B( x,2)), where he consan C is independen of f. Taking ino accoun ha we ge f Lp( ) ( B(x,2)) C n q(x) (5.2) I α( ) f 1 Lq( ) ( B(x,)) C n q(x) 2 q(x) 1 f Lp( ) ( B(x,r)) dr, 2 q(x) 1 f Lp( ) ( B(x,r)) dr. When x z, z y 2, we have 1 2 z y x y 3 z y, 2 and herefore I α( ) f 2 Lq( ) ( B(x,)) z y α(y) n f(y)dy C \ B(x,2) \ B(x,2) We choose β> n and obain q(x) x y α(x) n f(y) dy \ B(x,2) = β = β C C \ B(x,2) s β 1 ( L q( ) ( B(x,)) x y α(x) n f(y) dy χ B(x,) L q( ) ( ). ( ) x y α(x) n+β f(y) s β 1 ds dy x y {y :2 x y s} x y α(x) n+β f(y) dy s β 1 f Lp( ) ( B(x,s)) x y α(x) n+β Lp ( ) ( B(x,s)) ds s α(x) n 1 f Lp( ) ( B(x,s)) ds. ) ds

13 boundedness of some operaors in generalized morrey spaces 297 Therefore I α( ) f 2 Lq( ) ( B(x,)) C n q(x) which ogeher wih (5.2) yields (5.1). 2 s n q(x) 1 f Lp( ) ( B(x,s)) ds, Theorem 5.2. Le R n be an open bounded se and p, q W L( ) saisfy assumpion (2.1), α(x), q(x) saisfy he condiions in (2.3), (2.4) and he funcions ω 1 (x, r) and ω 2 (x, r) fulfill he condiion (5.3) r α(x) ω 1 (x, ) d Cω 2 (x, r), where C does no depend on x and r. Then he operaors M α( ) and I α( ) are bounded from M p( ),ω 1( ) ( ) o M q( ),ω 2( ) ( ). Proof. Le f M p( ),ω ( ). As usual, when esimaing he norm (5.4) I α( ) n q(x) f M q( ),ω 2 ( ) = sup x, >0 ω 2 (x, ) I α( ) fχ B(x,) L q( ) ( ), i suffices o consider only he values ( 0, l 2), hanks o condiion (3.1). We esimae I α( ) fχ B(x,) L q( ) ( ) in (5.4) by means of Theorem 5.1 and obain I α( ) f M q( ),ω 2 ( ) C sup x,>0 1 ω 2 (x, ) C f M p( ),ω 1 ( ) sup x,>0 I remains o make use of condiion (5.3). q(x) 1 f L p( ) ( B(x,r)) dr 1 l r α(x) ω 1 (x, r) dr. ω 2 (x, ) r In he following corollary we recover he resul obained in [2]. Corollary 5.3. Le be bounded, p, α, λ W L( ) and p saisfy condiion (2.1). Le also λ(x) 0 and he condiions (2.3), (2.4), (2.6), (2.7) be fulfilled. Then he operaor I α( ) is bounded from L p( ),λ( ) ( ) o L q( ),μ( ) ( ). Proof. I suffices o observe ha he funcion ω 1 (x, r) = r λ(x) n, ω 2 (x, r) = defining he spaces L p( ),λ( ) ( ) and L q( ),μ( ) ( ), saisfies condiions r μ(x) n q(x) (4.7) and (5.3) under assumpion (2.3) and he choice in (2.4), (2.7) for q(x).

14 298 v. s. guliyev, j. j. hasanov and s. g. samko 5.2. Adams ype resul Theorem 5.4. Le p W L( ) saisfy condiion (2.1) and le α(x) saisfy he condiions in (2.3). Then (5.5) I α( ) f(x) C α(x) Mf (x) + C r α(x) n 1 f L p( ) ( B(x,r)) dr, 0 < l 2, where is an arbirary number in ( 0, 2) l and C does no depend on f, x and. Proof. As in he proof of Theorem 4.1, we represen he funcion f in form (4.2) and have I α( ) f(x)= I α( ) f 1 (x) + I α( ) f 2 (x). For I α( ) f 1 (x), following Hedberg s rick (see for insance [37], p. 278, for he case of variable exponens), we obain I α( ) f 1 (x) C1 α(x) Mf (x). For I α( ) f 2 (x) we have I α( ) f 2 (x) x y α(x) n f(y) dy C \ B(x,2) \ B(x,2) f(y) dy r α(x) n 1 dr. x y Since x y rα(x) n 1 dr C 2l x y rα(x) n 1 dr, we obain I α( ) f 2 (x) C C 2l 2 ( 2< x y <r ) f(y) dy r α(x) n 1 dr f L p( ) ( B(x,r)) rα(x) n 1 dr, which proves (5.5). Theorem 5.5. Le p, α W L( ) saisfy assumpion (2.1), α(x) fulfill he condiions in (2.3) and le ω(x,) saisfy condiion (4.7) and he condiion (5.6) r α(x) 1 ω(x,)d Cr α(x) q(x), where q(x)>and C does no depend on x and r (0,l]. Then he operaors M α( ) and I α( ) are bounded from M p( ),ω( ) p( ) q( ),ω q( ) ( ) o M ( ). Proof. In view of he well known poinwise esimae M α( ) f(x) C(I α( ) f )(x), i suffices o rea only he case of he operaor I α( ).

15 boundedness of some operaors in generalized morrey spaces 299 Le f M p( ),ω( ) ( ). As in he proof of Theorem 4.2, when esimaing he norm I α( ) n q(x) f M q( ),ω = sup x, 0< l ω(x,) I α( ) fχ B(x,) L q( ) ( ), we may resric ourselves o he case of near he origin, 0 < l 2. By Theorem 5.4 we ge I α( ) f(x) Cr α(x) Mf (x) + C f M p( ),ω ( ) Making use of condiion (5.6), we obain r α(x) 1 (x,)d. I α( ) f(x) Cr α(x) Mf (x) + Cr α(x) q(x) f M p( ),ω ( ). ( ) q(x) f M p( ),ω α(x)q(x) We hen choose r = assuming ha f is no idenical 0. Mf (x) Hence, for every x, wehave I α( ) f(x) C(Mf (x)) q(x) f 1 q(x) M p( ),ω ( ). Hence he saemen of he heorem follows in view of he boundedness of he maximal operaor M in M p( ),ω ( ) provided by Theorem 4.2 in virue of condiion (4.7). Remark 5.6. Le ω(x,r) 1 (which may be supposed by (3.1)). For he exponen q(x), from (5.6) here follows he following bound 1 q(x) 1 α(x) m(x), [ ln m(x) = α(x) lim r 0 The corresponding exponen q(x) given by r α(x) 1 w(r, ) d ln r 1 (5.7) q(x) = 1 α(x) m(x), migh be called he Sobolev-Adams-ype exponen corresponding o he space M p( ),ω ( ). In paricular, for he Morrey space L p( ),λ ( ) (he case ω(x,r) = r λ(x) n ), from (5.7) we recoveradams exponen defined by 1 q(x) = 1 α(x) n λ(x). In he following corollary we recover he resul obained in [2]. ].

16 300 v. s. guliyev, j. j. hasanov and s. g. samko Corollary 5.7. Le R n be bounded, 0 λ(x) < n, p W L( ) saisfy condiion (2.1), and le (2.6), (2.8) be fulfilled. Then he operaors M α( ) and I α( ) are bounded from L p( ),λ( ) ( ) o L q( ),λ( ) ( ). Proof. I suffices o observe ha he funcions ω 1 (x, r) = r λ(x) n, ω 2 (x, r) = r λ(x) n q(x) defining he space L p( ),λ( ) ( ), saisfy condiions (4.7) and (5.6) under assumpion (2.6) and he choice of q(x) given in (2.8). 6. Singular operaors in he spaces M p( ),ω( ) ( ) Theorems 6.1 and 6.2 proved below, in he case of he consan exponen p were proved in [15] (see also [16]). The boundedness of singular operaors in Morrey spaces wih variable was sudied in [21] in he case where w(x, r) = r λ(x) n, bu in he more general seing of quasimeric measure spaces. Theorem 6.1. Le R n be an open bounded se, p W L( ) saisfy condiion (2.1) and f L p( ) ( ). Then (6.1) Tf L p( ) ( B(x,)) C n 1 f L p( ) ( B(x,r)) dr, 0 < l 2, where C does no depend on f and. Proof. We represen funcion f as in (4.2) and have Tf L p( ) ( B(x,)) Tf 1 L p( ) ( B(x,)) + Tf 2 L p( ) ( B(x,)). By Theorem 2.4 we obain Tf 1 L p( ) ( B(x,)) Tf 1 L p( ) ( ) C f 1 L p( ) ( ), so ha Tf 1 L p( ) ( B(x,)) C f L p( ) ( B(x,2)). Taking ino accoun he inequaliy we ge f L p( ) ( B(x,)) C n (6.2) Tf 1 L p( ) ( B(x,)) C n 2 1 f L p( ) ( B(x,r)) dr, 0 < l 2, 2 1 f L p( ) ( B(x,r)) dr. To esimae Tf 2 L p( ) ( B(x,)), we observe ha f(y) dy Tf 2 (z) C y z, n \B(x,2)

17 boundedness of some operaors in generalized morrey spaces 301 where z B(x,) and he inequaliies x z, z y 2 imply 1 2 z y x y 3 z y, and herefore 2 Tf 2 L p( ) ( B(x,)) C x y n f(y) dy χ B(x,) L p( ) ). \ B(x,2) Hence by esimae (2.5) (wih ν(x) 0) and inequaliy (4.5), we ge (6.3) Tf 2 L p( ) ( B(x,)) C n From (6.2) and (6.3) we arrive a (6.1). 2 1 f L p( ) ( B(x,r)) dr. Theorem 6.2. Le R n be an open bounded se, p W L( ) saisfy condiion (2.1) and ω 1 (x, ) and ω 2 (x, r) fulfill condiion (4.7). Then he singular inegral operaor T is bounded from he space M p( ),ω 1 ( ) o he space M p( ),ω 2 ( ). Proof. Le f M p( ),ω 1 ( ). As usual, when esimaing he norm n (6.4) Tf M p( ),ω 2 ( ) = sup x,>0 ω 2 (x, ) Tfχ B(x,) L p( ) ( ), i suffices o consider only he values ( 0, l 2), hanks o condiion (3.1). We esimae Tfχ B(x,) L p( ) ( ) in (6.4) by means of Theorem 6.1 and obain Tf M p( ),ω 2 ( ) C sup x,>0 1 ω 2 (x, ) C f M p( ),ω 1 ( ) I remains o make use of condiion (4.7). 7. Appendix sup x,>0 1 f L p( ) ( B(x,r)) dr 1 l ω 1 (x, r) dr. ω 2 (x, ) r Lemma 7.1. If c 1 ω(x,r) ω(x,) whenever 0 <r 2r, hen from (3.3) i follows ha he funcion w(x,r) r n is almos decreasing uniformly in x: w(x, r) 1 w(x, ) for all 0 r <, r n 2cCn n where c and C are he consan from (3.2) and (3.3) (and consequenly he righ-hand side inequaliy in (3.2) holds).

18 302 v. s. guliyev, j. j. hasanov and s. g. samko (7.1) Proof. From (3.3) we have w(x, r) r n 1 C from which (7.1) follows. 2 w(x, τ) w(x, ) 2 dτ τ n+1 cc dτ τ n+1 Acknowledgemens. The research of V. Guliyev and J. Hasanov was parially suppored by he gran of BGP II (projec ANSF Award / AZM BA-08). The research of S. Samko was suppored by Russian Federal Targeed Programme Scienific and Research-Educaional Personnel of Innovaive Russia for , projec N The auhors are hankful o he anonymous referee for he commens which improved he formulaions of Theorem 5.5 and Remark 5.6. REFERENCES 1. Adams, D. R., A noe on Riesz poenials, Duke Mah. J. 42 (1975), Almeida, A., Hasanov, J. J., Samko, S. G., Maximal and poenial operaors in variable exponen Morrey spaces, Georgian Mah. J. 15 (2008), Bary, N. K., and Sechkin, S. B., Bes approximaions and differenial properies of wo conjugae funcions (Russian), Trudy Moskov. Ma. Obšč. 5 (1956), Burenkov, V. I., Guliyev, H. V., Necessary and sufficien condiions for boundedness of he maximal operaor in he local Morrey-ype spaces, Sudia Mah. 163 (2004), Burenkov, V. I., Guliyev, V. S., Necessary and sufficien condiions for boundedness of he Riesz poenial in local Morrey-ype spaces, Poenial Anal. 31 (2009), Burenkov, V. I., Guliyev, H. V., and Guliyev, V. S., Necessary and sufficien condiions for he boundedness of he Riesz poenial in he local Morrey-ype spaces, Doklady Mah. 75 (2007), ; ranslaed from Doklady Akad. Nauk 412 (2007), Burenkov, V. I., Guliyev, H. V., and Guliyev, V. S., Necessary and sufficien condiions for boundedness of he fracional maximal operaor in local Morrey-ype spaces, J. Compu. Appl. Mah. 208 (2007), Burenkov, V. I., Guliyev, V. S., Serbeci, A., and Tararykova, T. V., Necessary and sufficien condiions for he boundedness of genuine singular inegral operaors in local Morrey-ype spaces (Russian), Doklady Akad. Nauk 422 (2008), Chiarenza, F., and Frasca, M., Morrey spaces and Hardy-Lilewood maximal funcion, Rend. Ma. Appl. (7) 7 (1987), Diening, L., Maximal funcion on generalized Lebesgue spaces L, Mah. Inequal. Appl. 7 (2004), Diening, L., and Růžička, M., Calderón-Zygmund operaors on generalized Lebesgue spaces L p( ) and problems relaed o fluid dynamics, J. Reine Angew. Mah. 563 (2003), Diening, L., Häsö, P., and Nekvinda, A., Open problems in variable exponen Lebesgue and Sobolev spaces, pp in: Funcion Spaces, Differenial Operaors and Nonlinear Analysis, Proc. Milovy 2004, Mah. Ins. Acad. Sci. Czech Republic, Praha Eridani, A., Gunawan, H., and Nakai, E., On generalized fracional inegral operaors, Sci. Mah. Jpn. 60 (2004), M. Giaquina, Muliple inegrals in he calculus of variaions and nonlinear ellipic sysems, Ann. of Mah. Sudies 105, Princeon Univ. Press, Princeon, NJ 1983.

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20 304 v. s. guliyev, j. j. hasanov and s. g. samko 38. Samko, S., On a progress in he heory of Lebesgue spaces wih variable exponen: maximal and singular operaors, Inegral Transform. Special Func, 16 (2005), Sharapudinov, I. I., The opology of he space L p() ([0, 1]), Ma. Zameki 26 (1979), INSTITUTE OF MATHEMATICS AND MECHANICS F.AGAEV STR. 9 AZ 1141 BAKU AZERBAIJAN and AHI EVRAN UNIVERSITY DEPARTMENT OF MATHEMATICS KIRSEHIR TURKEY vagif@guliyev.com INSTITUTE OF MATHEMATICS AND MECHANICS F.AGAEV STR. 9 AZ 1141 BAKU AZERBAIJAN hasanovjavanshir@yahoo.com.r UNIVERSIDADE DO ALGARVE UCEH CAMPUS DE GAMBELAS PT-8000 FARO PORTUGAL ssamko@ualg.p