Sublinear operators with rough kernel generated by Calderón-Zygmund operators and their commutators on generalized local Morrey spaces

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1 Balakishiyev et al. Jounal of Inequalities and Applications :61 DOI /s y R E S E A R C H Open Access Sublinea opeatos with ough kenel geneated by Caldeón-Zygmund opeatos and thei commutatos on genealized local Moey spaces Aydin S Balakishiyev 1,VagifSGuliyev 2,3, Feit Gubuz 4 and Ayhan Sebetci 4* * Coespondence: sebetci@ankaa.edu.t 4 Depatment of Mathematics, Ankaa Univesity, Ankaa, Tukey Full list of autho infomation is available at the end of the aticle Abstact In this pape, we will stu the boundedness of a lage class of sublinea opeatos with ough kenel T on the genealized local Moey spaces LM {x 0} p,ϕ,fos p, p 1o p < s,whee L s S n 1 withs > 1 ae homogeneous of degee zeo. In the case when b LC {x 0} p,λ is a local Campanato spaces, 1 < p <,andt,b be is a sublinea commutato opeato, we find the sufficient conditions on the pai ϕ 1, ϕ 2 which ensues the boundedness of the opeato T,b fom one genealized local Moey space LM {x 0} to anothe LM {x 0}. In all cases the conditions fo the boundedness of T ae given in tems of Zygmund-type integal inequalities on ϕ 1, ϕ 2, which do not make any assumptions on the monotonicity of ϕ 1, ϕ 2 in. Conditions of these theoems ae satisfied by many impotant opeatos in analysis, in paticula pseudo-diffeential opeatos, Littlewood-Paley opeatos, Macinkiewicz opeatos, and Bochne-Riesz opeatos. MSC: 42B20; 42B25; 42B35 Keywods: sublinea opeato; Caldeón-Zygmund opeato; ough kenel; genealized local Moey space; commutato; local Campanato space 1 Intoduction Fo x R n and >0,letBx, denote the open ball centeed at x of adius, Bx, denote its complement and Bx, isthelebesguemeasueoftheballbx,. Suppose that S n 1 is the unit sphee in R n n 2 equipped with the nomalized Lebesgue measue dσ. Let L s S n 1 with1<s be homogeneous of degee zeo. Suppose that T epesents a linea o a sublinea opeato, which satisfies, fo any f L 1 R n withcompact suppot and x / supp f, T f x c 0 whee c 0 is independent of f and x. x y f y, 1.1 R n x y n 2015 Balakishiyev et al.; licensee Spinge. This is an Open Access aticle distibuted unde the tems of the Ceative Commons Attibution License which pemits unesticted use, distibution, and epoduction in any medium, povided the oiginal wok is popely cedited.

2 Balakishiyev et al. Jounal of Inequalities and Applications :61 Page 2 of 18 Fo a function b, suppose that the commutato opeato T,b epesentsalineaoa sublinea opeato, which satisfies, fo any f L 1 R n with compact suppot and x / supp f, T,b f x c 0 R n bx by x y x y n f y, 1.2 whee c 0 is independent of f and x. We point out that the condition 1.1inthecase 1was fist intoduced by Soia and Weiss in [1]. The condition 1.1 is satisfied by many inteesting opeatos in hamonic analysis, such as the Caldeón-Zygmund opeatos, Caleson maximal opeatos, Ha- Littlewood maximal opeatos, C Feffeman singula multiplies, R Feffeman singula integals, Ricci-Stein oscillatoy singula integals, the Bochne-Riesz means, and so on see [1, 2] fo details. Let L s S n 1 with1<s behomogeneous of degee zeo and satisfy the cancelation condition x dσ x =0, S n 1 whee x = x/ x fo any x 0. The homogeneous singula integal opeato T defined by x y T f x=p.v. f y R n x y n satisfies the condition 1.1. It is obvious that when 1, T is the singula integal opeato T. Theoem A [3] Suppose that 1 p <, L s S n 1, s >1,is homogeneous of degee zeo and has mean value zeo on S n 1. If s p, p 1o p < s, then the opeato T is bounded on L p R n. Also the opeato T is bounded fom L 1 R n to WL 1 R n. Let b be a locally integable function on R n, then we shall define the commutatos geneated by singula integal opeatos with ough kenels and b as follows: [ ]x y [b, T ]f x bxt f 1 x T bf x=p.v. bx by f y. R n x y n Theoem B [3] Suppose that L s S n 1, s >1,is homogeneous of degee zeo and has mean value zeo on S n 1. Let 1<p < and b BMOR n. If s pop< s, then the commutato opeato [b, T ] is bounded on L p R n. The classical Moey spaces M p,λ wee fist intoduced by Moey in [4] tostuthe local behavio of solutions to second ode elliptic patial diffeential equations. Fo the boundedness of the Ha-Littlewood maximal opeato, the factional integal opeato and the Caldeón-Zygmund singula integal opeato on these spaces, we efe the eades to [5 7]. Fo the popetiesand applications of classical Moey spaces, see [8 11]and efeences theein. The genealized Moey spaces M p,ϕ ae obtained by eplacing λ by a function ϕ in the definition of the Moey space. Duing the last decades vaious classi-

3 Balakishiyev et al. Jounal of Inequalities and Applications :61 Page 3 of 18 cal opeatos, such as maximal, singula, and potential opeatos, wee widely investigated in both in classical and genealized Moey spaces. In this pape, we pove the boundedness of the opeatos T fom one genealized local Moey space LM {x 0} to anothe LM {x 0},1<p <, andfomthespacelm {x 0} 1,ϕ 1 to the weak space WLM {x 0} 1,ϕ 2.Inthecaseb LC {x 0},wefinhesufficientconditionsonthepai ϕ 1, ϕ 2 which ensue the boundedness of the commutato opeatos [b, T ]fomlm {x 0} to LM {x 0},1<p <, 1 p = 1 p p 2. By A B we mean that A CB with some positive constant C independent of appopiate quantities. If A B and B A,wewiteA B and say that A and B ae equivalent. 2 Genealized local Moey spaces We find it convenient to define the genealized Moey spaces in the fom as follows. Definition 2.1 Let ϕx, be a positive measuable function on R n 0, and1 p <. WedenotebyM p,ϕ R n the genealized Moey space, the space of all functions f L loc p Rn with finite quasinom f Mp,ϕ = sup x R n,>0 ϕx, 1 f Lp Bx,. 2.1 The genealized Moey spaces M p,ϕ R n withnom2.1 intoducedbymizuhaain [12], which was late extended and studied by many authos see [13, 14]. Note that the genealized Moey spaces M p,ϕ R n with nomalized fom f Mp,ϕ = sup x R n,>0 ϕx, 1 Bx, 1 p f Lp Bx, 2.2 wee fist defined by Guliyev in [15]. Also, in [15], thee was defined the weak genealized Moey space WM p,ϕ WM p,ϕ R n of all functions f WL loc p Rn fowhich f WMp,ϕ = sup x R n,>0 ϕx, 1 Bx, 1 p f WLp Bx, <. Accoding to this definition, we ecove the Moey space M p,λ and weak Moey space WM p,λ unde the choice ϕx, = λ n p : M p,λ = M p,ϕ ϕx,= λ n p, WM p,λ = WM p,ϕ ϕx,= λ n p. Recall that in 1994 the doctoal thesis [16]byGuliyevseealso[17 20] intoduced the local Moey-type space LM pθ,w given by f LMpθ,w = w f Lp B0, Lθ 0, <, whee w is a positive measuable function defined on 0,. The main pupose of [16] also of[17 20] is to give some sufficient conditions fo the boundedness of factional integal opeatos and singula integal opeatos defined on homogeneous Lie goups in the local Moey-type space LM pθ,w. In a seies of papes by Buenkov, H Guliyev and

4 Balakishiyev et al. Jounal of Inequalities and Applications :61 Page 4 of 18 VGuliyev, etc. see[21 24], some necessay and sufficient conditions fo the boundedness of factional maximal opeatos, factional integal opeatos, and singula integal opeatos in local Moey-type spaces LM pθ,w wee given. Definition 2.2 Let ϕx, be a positive measuable function on R n 0, and1 p <. We denote by LM p,ϕ LM p,ϕ R n the genealized cental local Moey space, the space of all functions f L loc p Rn with finite quasinom f LMp,ϕ = sup ϕ0, 1 B0, 1 p f Lp B0,. >0 Also by WLM p,ϕ WLM p,ϕ R n we denote the weak genealized Moey space of all functions f WL loc p Rn fowhich f WLMp,ϕ = sup ϕ0, 1 B0, 1 p f WLp B0, <. >0 Paticulaly, if θ =, LM p,w = LM p,w, then the genealized cental Moey spaces LM p,ϕ ae the same spaces as the local Moey spaces LM pθ,w with w =ϕ0, 1 n/p. Note that f M p,ϕ if and only if f x x R n foms a bounded set in LM pϕ. Definition 2.3 Let ϕx, be a positive measuable function on R n 0, and1 p <. Fo any fixed x 0 R n we denote by LM {x 0} p,ϕ LM {x 0} p,ϕ R n thegenealizedlocalmoey space, the space of all functions f L loc p Rn with finite quasinom f LM p,ϕ = f x0 + LMp,ϕ. Also by WLM {x 0} p,ϕ WLM {x 0} p,ϕ R n we denote the weak genealized local Moey space of all functions f WL loc p Rn fowhich f WLM p,ϕ = f x0 + WLMp,ϕ <. Accoding to this definition, we ecove the local Moey space LM {x 0} p,λ Moey space WLM {x 0} p,λ LM {x 0} p,λ = LM{x 0} p,ϕ ϕx0,= λ n unde the choice ϕx 0, = λ n p :, p WLM{x 0} p,λ = WLM{x 0} p,ϕ ϕx0,= λ n. p and weak local Wiene [25, 26] looked fo a way to descibe the behavio of a function at the infinity. The conditions he consideed ae elated to appopiate weighted L q spaces. Beuling [27] extended this idea and defined a pai of dual Banach spaces A q and B q,whee1/q +1/q =1. To be pecise, A q is a Banach algeba with espect to the convolution, expessed as a union of cetain weighted L q spaces; the space B q is expessed as the intesection of the coesponding weighted L q spaces. Feichtinge [28] obseved that the space B q can be descibed by f Bq = sup 2 kn q f χ k Lq R n, 2.3 k 0

5 Balakishiyev et al. Jounal of Inequalities and Applications :61 Page 5 of 18 whee χ 0 is the chaacteistic function of the unit ball {x R n : x 1}, χ k is the chaacteistic function of the annulus {x R n :2 k 1 < x 2 k }, k =1,2,...Byduality,thespace A q R n, called the Beuling algeba now, can be descibed by f Aq = k=0 2 kn q f χ k Lq R n. 2.4 Let Ḃ q R n andȧ q R n be the homogeneous vesions of B q R n anda q R n bytaking k Z in 2.3and2.4 instead of k 0thee. If λ <0oλ > n,thenlm {x 0} p,λ Rn =,whee is the set of all functions equivalent to 0 on R n.notethatlm p,0 R n =L p R n andlm p,n R n =Ḃ p R n ; Ḃ p,μ = LM p,ϕ ϕ0,= μn, WḂ p,μ = WLM p,ϕ ϕ0,= μn. Alvaez et al. [29], in ode to stu the elationship between cental BMO spaces and Moey spaces, intoduced λ-centalboundedmeanoscillationspacesandcentalmoey spaces Ḃ p,μ R n LM p,n+npμ R n, μ [ 1 p,0].ifμ < 1 p o μ >0,thenḂ p,μ R n =.Note that Ḃ p, 1 R n =L p R n andḃ p,0 R n =Ḃ p R n. Also define the weak cental Moey spaces p WḂ p,μ R n WLM p,n+npμ R n. Inspied by this, we conside the boundedness of singula integal opeato with ough kenel on genealized local Moey spaces and give the cental bounded mean oscillation estimates fo thei commutatos. 3 Sublinea opeatos with ough kenel geneated by Caldeón-Zygmund opeatos in the spaces LM {x 0} p,ϕ In this section we ae going to use the following statement on the boundedness of the weighted Ha opeato: H w gt:= t gsws ds, 0<t <, whee w is a fixed function non-negative and measuable on 0,. The following theoem was poved in [30, 31]. Theoem 3.1 Let v 1, v 2, and w be positive almost eveywhee and measuable functions on 0,. The inequality ess sup t>0 v 2 th w gt C ess sup v 1 tgt 3.1 t>0 holds fo some C >0fo all non-negative and non-deceasing g on 0, if and only if ws ds B := ess sup v 2 t t>0 t ess sup s<τ< v 1 τ <. Moeove, the value C = B isthe bestconstantfo3.1. The following statement, containing the esults obtained in [12, 13]was poved in[3].

6 Balakishiyev et al. Jounal of Inequalities and Applications :61 Page 6 of 18 Theoem 3.2 Suppose that L s S n 1, s >1,is homogeneous of degee zeo and has mean value zeo on S n 1. Let 1 s < p < and ϕx, satisfy the conditions c 1 ϕx, ϕx, t cϕx, 3.2 wheneve t, whee c 1 does not depend on t,, x R n, and ϕx, t p t Cϕx, p, 3.3 whee C does not depend on x and. Then the opeato T is bounded on M p,ϕ. The following statement, containing the esults obtained in [12, 13] was poved in[15, 16]seealso[17, 21 23, 32]. Theoem 3.3 Let 1 p < and ϕ 1, ϕ 2 satisfy the condition ϕ 1 0, t t Cϕ 20,, 3.4 whee C does not depend on. Then the opeato T is bounded fom LM p,ϕ1 to LM p,ϕ2 fo p >1and fom LM 1,ϕ1 to WLM 1,ϕ2 fo p =1. Coollay 3.4 Let 1 p < and ϕ 1, ϕ 2 satisfy the condition ϕ 1 x, t t Cϕ 2x,, 3.5 whee C does not depend on x and. Then the opeato T is bounded fom M p,ϕ1 to M p,ϕ2 fo p >1and fom M 1,ϕ1 to WM 1,ϕ2 fo p =1. The following statement, containing esults obtained in [15, 16], was poved in [30]. Theoem 3.5 Let x 0 R n,1 p <, and L s S n 1, s >1,be a homogeneous of degee zeo. Let also, fo s pop< s, the pai ϕ 1, ϕ 2 satisfy the condition ess inf t<τ< ϕ 1 x 0, ττ n p t n p +1 Cϕ 2 x 0,, 3.6 whee C does not depend on. Then the opeato T is bounded fom LM {x 0} to LM {x 0} fo p >1and fom LM {x 0} 1,ϕ 1 to WLM {x 0} 1,ϕ 2 fo p =1. Coollay 3.6 Let 1 p <, L s S n 1, s >1,be a homogeneous of degee zeo. Let also, fo s pop< s, the pai ϕ 1, ϕ 2 satisfy the condition ess inf t<τ< ϕ 1 x, ττ n p t n p +1 Cϕ 2 x,, 3.7 whee C does not depend on x and. Then the opeato T is bounded fom M p,ϕ1 to M p,ϕ2 fo p >1and fom M 1,ϕ1 to WM 1,ϕ2 fo p =1.

7 Balakishiyev et al. Jounal of Inequalities and Applications :61 Page 7 of 18 Lemma 3.7 Let x 0 R n,1 p <, T be a sublinea opeato satisfying condition 1.1 with L s S n 1, s >1,be a homogeneous of degee zeo, bounded on L p R n fo p >1,and bounded fom L 1 R n to WL 1 R n. If p >1and s p, then the inequality T f Lp Bx 0, n p f Lp Bx 0,tt n p 1 holds fo any ball Bx 0, and fo all f L loc p Rn. If p >1and p < s, then the inequality T f Lp Bx 0, n p n s f Lp Bx 0,tt n s n p 1 holds fo any ball Bx 0, and fo all f L loc p Rn. Moeove, fo s >1the inequality T f WL1 Bx 0, n t n 1 f L1 Bx 0,t 3.8 holds fo any ball Bx 0, and fo all f L loc 1 R n. Poof Let 1 < p < and s p. SetB = Bx 0, fo the ball centeed at x 0 and of adius. We epesent f as and have f = f 1 + f 2, f 1 y=fyχ 2B y, f 2 y=fyχ 2By, >0, 3.9 T f LB T f 1 Lp B + T f 2 Lp B. Since f 1 L p R n, T f 1 L p R n and fom the boundedness of T on L p R n it follows that T f 1 Lp B T f 1 Lp R n C f 1 Lp R n = C f Lp 2B, whee constant C > 0 is independent of f. Note that x Ls Bx 0,t = Bx x 0,t B0,t+ x x 0 whee c 0 =nv n 1/s and v n = B0, 1. y s 1 s y s 1 s t+ x x0 = n 1 d y s dσ y 0 S n 1 = c 0 Ls S n 1 B 0, t + x x 0 1 s, s

8 Balakishiyev et al. Jounal of Inequalities and Applications :61 Page 8 of 18 It is clea that x B, y 2Bimplies 1 2 x 0 y x y 3 2 x 0 y.weget T f 2 x 2 n c 1 2B By the Fubini theoem we have f y x y x 0 y n 2B 2B f y x y x 0 y n. 2B f y x y x 0 y t Bx 0,t x 0 y tn+1 f y x y f y x y t. n+1 Applying the Hölde inequality, we get f y x y f x 0 y n Lp Bx 0,t x Ls Bx 0,t Bx 0, t 1 p 1 1 s f Lp Bx 0,t B x0, t + x x 0 1 s Bx0, t 1 p 1 1 s f Lp Bx 0,t Bx 0, t 1 p 1 f Lp Bx 0,t Moeove, fo all p [1,, the inequality T f 2 Lp B n p is valid. Thus T f Lp B f Lp 2B + n p On the othe hand, Thus f Lp 2B n p f Lp 2B T f Lp B n p t n p +1 f Lp Bx 0,t t n 3.12 p +1 f Lp Bx 0,t t n. p +1 t n n p p +1 f Lp Bx 0,t t n. p +1 f Lp Bx 0,t t n p When 1 < p < s, by the Fubini theoem, the Minkowski inequality and 3.10, we get T f 2 Lp B B Bx 0,t Bx 0,t f y x y f y y Lp B p 1 p dx

9 Balakishiyev et al. Jounal of Inequalities and Applications :61 Page 9 of 18 Bx0, 1 p 1 s Bx 0, 1 p 1 s n p n s n p n s Bx 0,t Bx 0,t f L1 Bx 0,t Bx0, t 1 s f y y Ls B f y B x 0, + x 0 y 1 s f Lp Bx 0,tt n s n p Let p =1<s. Fom the weak 1, 1 boundedness of T and 3.13 it follows that T f 1 WL1 B T f 1 WL1 R n f 1 L1 R n = f L1 2B n f L1 Bx 0,t tn+1 Then fom 3.12and3.15wegettheinequality3.8. Theoem 3.8 Let x 0 R n,1 p <, T be a sublinea opeato satisfying condition 1.1 with L s S n 1, s >1,be a homogeneous of degee zeo. Suppose that the opeato T is bounded on L p R n fo p >1and bounded fom L 1 R n to WL 1 R n. Let also, fo s p, p 1,the pai ϕ 1, ϕ 2 satisfy the condition ess inf t<τ< ϕ 1 x 0, ττ n p t n p +1 Cϕ 2 x 0,, 3.16 and fo 1<p < sthepaiϕ 1, ϕ 2 satisfy the condition ess inf t<τ< ϕ 1 x 0, ττ n p t n p n s +1 Cϕ 2 x 0, n s, 3.17 whee C does not depend on. Then the opeato T is bounded fom LM {x 0} to LM {x 0}. Moeove, T f LM f LM. Also the opeato T is bounded fom LM {x 0} 1,ϕ 1 to WLM {x 0} 1,ϕ 2 and T f WLM 1,ϕ 2 f LM 1,ϕ 1. Poof Let 1 < p < and s p. By Lemma 3.7 and Theoem 3.1 with v 2 =ϕ 2 x 0, 1, v 1 =ϕ 1 x 0, 1 n p, g= f Lp Bx 0,,andw= n p 1 we have T f {x LM 0 } sup ϕ 2 x 0, 1 f Lp Bx 0,t >0 t n p +1 sup ϕ 1 x 0, 1 n p f Lp Bx 0, = f {x LM 0 }. >0

10 Balakishiyev et al. Jounal of Inequalities and Applications :61 Page 10 of 18 Let 1 < p < s. By Lemma 3.7 and Theoem 3.1 with v 2 =ϕ 2 x 0, 1, v 1 =ϕ 1 x 0, 1 n p + n s, g= f Lp Bx 0,,andw= n p + n s 1 we have T f {x LM 0 } sup ϕ 2 x 0, 1 n s Also fo p =1 >0 f Lp Bx 0,t t n p n s +1 sup ϕ 1 x 0, 1 n p f Lp Bx 0, = f {x LM 0 }. >0 T f {x WLM 0 } sup ϕ 2 x 0, 1 f L1 Bx 0,t 1,ϕ 2 >0 sup ϕ 1 x 0, 1 n f Lp Bx 0, = f {x LM 0 }. >0 1,ϕ 1 Coollay 3.9 Let x 0 R n,1 p <, T be a sublinea opeato satisfying condition 1.1, with L s S n 1, s >1,being a homogeneous of degee zeo and bounded on L p R n fo p >1,and bounded fom L 1 R n to WL 1 R n. Let also, fo s p, p 1,the pai ϕ 1, ϕ 2 satisfy the condition ess inf t<τ< ϕ 1 x, ττ n p t n p +1 Cϕ 2 x,, 3.18 and, fo 1<p < s, the pai ϕ 1, ϕ 2 satisfy the condition ess inf t<τ< ϕ 1 x, ττ n p t n p n s +1 Cϕ 2 x, n s, 3.19 whee C does not depend on x and. Then the opeato T is bounded fom M p,ϕ1 to M p,ϕ2. Moeove, T f Mp,ϕ2 f Mp,ϕ1. Also the opeato T is bounded fom M 1,ϕ1 to WM 1,ϕ2 and T f WM1,ϕ2 f M1,ϕ1. Coollay 3.10 Let 1 p < and ϕ 1, ϕ 2 satisfy condition Then the opeato Tis bounded fom LM {x 0} to LM {x 0} fo p >1and fom LM {x 0} 1,ϕ 1 to WLM {x 0} 1,ϕ 2. Let f L loc 1 R n. The ough Ha-Littlewood maximal function M is defined by M f x=sup t>0 1 x y f y. Bx, t Bx,t Then we can give the following coollay. Coollay 3.11 Let 1 p <, L s S n 1. Fo s p, p 1,the pai ϕ 1, ϕ 2 satisfies condition 3.16 and, fo 1<p < s, the pai ϕ 1, ϕ 2 satisfies condition Then the opeatos M and T ae bounded fom LM {x 0} to LM {x 0}, fo p >1,and fom LM {x 0} 1,ϕ 1 to WLM {x 0} 1,ϕ 2.

11 Balakishiyev et al. Jounal of Inequalities and Applications :61 Page 11 of 18 Coollay 3.12 Let 1 p <, L s S n 1. Fo s p, p 1,the pai ϕ 1, ϕ 2 satisfies condition 3.18 and, fo 1<p < s, the pai ϕ 1, ϕ 2 satisfies condition Then the opeatos M and T ae bounded fom M p,ϕ1 to M p,ϕ2 and fom M 1,ϕ1 to WM 1,ϕ2. Remak 3.13 Note that, in the case s =, Coollay 3.9 was poved in [33]. The condition 3.16in Theoem 3.8 is weake than condition 3.4in Theoem 3.3 see [33]. 4 Commutatos of linea opeatos with ough kenel geneated by Caldeón-Zygmund opeatos in the spaces LM {x 0} p,ϕ Let T be a linea opeato; fo a function b,wedefinethecommutato[b, T]by [b, T]f x =bxtf x Tbf x fo any suitable function f.let T be a Caldeón-Zygmund singula integal opeato. A well-known esult of Coifman et al. [34] states that the commutato [b, T]f = b Tf Tbf is bounded on L p R n, 1 < p <, ifandonlyifb BMOR n. The commutato of Caldeón-Zygmund opeatos plays an impotant ole in stuing the egulaity of solutions of elliptic patial diffeential equations of second ode see, fo example, [8 10, 35]. The definition of a local Campanato space is as follows. Definition 4.1 Let 1 q < and 0 λ < 1 n the LC {x 0} q,λ Rn local Campanato space, if whee Define f LC q,λ f Bx0, = 1 = sup >0 Bx 0, 1+λq 1 f y. Bx 0, Bx 0, Bx 0,. A function f Lloc q Rn is said to belong to 1/q f y fbx0 q, <, LC {x 0} q,λ Rn = { f L loc q Rn : f {x LC 0 } < }. q,λ In [36], Lu and Yang intoduced the cental BMO space CBMO q R n =LC {0} q,0 Rn. Note that BMOR n q>1 CBMO{x 0} q R n, 1 q <. ThespaceCBMO {x 0} q R n canbeegaded as a local vesion of BMOR n, the space of bounded mean oscillation, at the oigin. But they have quite diffeent popeties. The classical John-Nienbeg inequality shows that functions in BMOR n ae locally exponentially integable. This implies that, fo any 1 q <, the functions in BMOR n can be descibed by means of the condition: 1 1/q sup f y f B q <, B R n B B whee B denotes an abitay ball in R n.howeve,thespacecbmo {x 0} q R n dependsonq.if q 1 < q 2,thenCBMO {x 0} q 2 R n CBMO {x 0} R n. Theefoe, thee is no analogy of the famous q 1

12 Balakishiyev et al. Jounal of Inequalities and Applications :61 Page 12 of 18 John-Nienbeg inequality of BMOR n fothespacecbmo {x 0} q R n. One can imagine that the behavio of CBMO {x 0} q R n may be quite diffeent fom that of BMOR n. We will use the following statement on the boundedness of the weighted Ha opeato: Hw g:= 1+ln t gtwt, 0,, whee w is a weight. The following theoem was poved in [37]. Theoem 4.2 [37] Let v 1, v 2, and w be positive almost eveywhee and measuable functions on 0,. The inequality ess sup v 2 Hw g C ess sup v 1 g 4.1 >0 >0 holds, fo some C >0fo all non-negative and non-deceasing g on 0,, if and only if B := sup v 2 1+ln t wt <. 4.2 >0 sup t<s< v 1 s Moeove, the value C = Bisthebestconstantfo4.1. Remak 4.3 In itisassumehat0 =0. Lemma 4.4 Let b be a function in LC {x 0} q,λ Rn, 1 q <,0 λ < 1 n, and 1, 2 >0.Then 1 Bx 0, 1 1+λq Bx 0, 1 whee C >0is independent of b, 1, and 2. 1 by bbx0 q q, 2 C 1+ ln 1 b {x LC 0 }, q,λ 2 In [3] the following statement was poved fo the commutatos of singula integal opeatos with ough kenels, containing the esult in [12, 13]. Theoem 4.5 Suppose that L s S n 1, s >1,is homogeneous of degee zeo and b BMOR n. Let 1 s < p <, ϕx, satisfy the conditions 3.2 and 3.3. Then the opeato [b, T ] is bounded on M p,ϕ. Lemma 4.6 Let x 0 R n,1<p <, b LC {x 0} Rn, 1 p = 1 p p 2, and 0 λ < 1 n. Let also T be a linea opeato satisfying condition 1.1 with L s S n 1, s >1,be a homogeneous of degee zeo and bounded on L p R n fo 1<p <. Then, fo s p 1, the inequality [b, T ]f Lp Bx 0, b LC n p holds, fo any ball Bx 0, and fo all f L loc p 1 R n. 1+ln t t nλ p n 1 1 f Lp1 Bx 0,t

13 Balakishiyev et al. Jounal of Inequalities and Applications :61 Page 13 of 18 Also, fo p 1 < s, the inequality [b, T ]f Lp Bx 0, b LC n p n s holds, fo any ball Bx 0, and fo all f L loc p 1 R n. 1+ln t t nλ p n + n 1 s 1 f Lp1 Bx 0,t Poof Let 1 < p <, b LC {x 0} Rn, and 1 p = 1 p p 2.AsinthepoofofLemma3.7, we epesent the function f in the fom 3.9and have [b, T ]f x J 1 + J 2 + J 3 + J 4 = bx b B T f 1 x T b bb f1 x+ bx bb T f 2 x T b bb f2 x. Hence we get [b, T ]f Lp B J 1 Lp B + J 2 Lp B + J 3 Lp B + J 4 Lp B. Fom the boundedness of T on L p R n and Lemma 4.4 it follows that J 1 Lp B b b B T f 1 Lp R n b b Lp2 B R n T f 1 Lp1 R n C b {x LC 0 } p n +nλ 2 f 1 Lp1 R n = C b {x LC 0 } p n + 2 p n +nλ 1 f Lp1 2B b {x LC 0 } n p Fom Lemma 4.4 fo J 2 we have 1+ln t t 1 n p 1 t nλ n p 1 1 f Lp1 Bx 0,t. J 2 Lp B T b bb f1 Lp R n b b B f1 Lp R n b b B Lp2 R n f 1 Lp1 R n b {x LC 0 } p n + 2 p n +nλ 1 f Lp1 2B b {x LC 0 } n p t 1 n p 1 1+ln t t nλ p n 1 1 f Lp1 Bx 0,t. Fo J 3,itisknownthatx B, y 2B, which implies 1 2 x 0 y x y 3 2 x 0 y. When s p 1, by the Fubini theoem and 3.10, and applying the Hölde inequality, we have T f 2 x c 0 x y f y x 0 y n 2B < x 0 y <t x y f y t 1 n

14 Balakishiyev et al. Jounal of Inequalities and Applications :61 Page 14 of 18 Bx 0,t x y f y t 1 n f Lp1 Bx 0,t x Ls Bx Bx 0,t 0, t 1 p s t 1 n f Lp1 Bx 0,t B x0, t + x x 0 1 s Bx0, t 1 p s t n p 1 1 f Lp1 Bx 0,t. Hence, fom Lemma 4.4 we get J 3 Lp B = b b B T f 2 Lp R n b b B Lp R n t n p 1 1 f Lp1 Bx 0,t b b Lp2 B R n p n 1 t p n 1 1 f Lp1 Bx 0,t b {x LC 0 } n p +nλ b {x LC 0 } n p 1+ln t 1+ln t t n p 1 1 f Lp1 Bx 0,t t nλ n p 1 1 f Lp1 Bx 0,t. When p 1 < s, by the Fubini theoem, the Minkowski inequality, 3.10 and fom Lemma 4.4,weget J 3 Lp B B Bx 0,t Bx 0,t Bx 0,t f y bx bb x y f y b b B y Lp B p 1 p dx f y b bblp2 B y Lp1 B b {x LC 0 } p n +nλ 2 B p s b LC b {x LC 0 } n p n s n p n s +nλ Bx 0,t f y y Ls B f L1 Bx 0,t B x0, t + x 0 y 1 s 1+ln t t nλ+ n s n p 1 1 f Lp1 Bx 0,t. 4.3 Fo x B, by the Fubini theoem, applying the Hölde inequality, and fom Lemma 4.4 we have T b bb f2 x by b B x y f y 2B x y n by bb x y f y x 0 y n 2B

15 Balakishiyev et al. Jounal of Inequalities and Applications :61 Page 15 of 18 < x 0 y <t by b Bx0,t Bx 0,t + + by bb x y f y b Bx0, b Bx0,t x y f y Bx 0,t x y f y b b Bx0,t f Lp Bx 0,t y Ls Bx Bx 0,t 0, t 1 p 1 1 s b Bx0, b Bx0,t f Lp1 Bx 0,t b bbx0,t Lp2 Bx 0,t f L p1 Bx 0,tt 1 p n 1 + b LC b LC Then fo J 4 we have 1+ln t 1+ln t y Ls Bx0, t 1 p s t n 1 Bx 0,t t nλ n p 1 1 f Lp1 Bx 0,t t nλ n p 1 1 f Lp1 Bx 0,t. J 4 Lp B T b bb f2lp R n b {x LC 0 } n p 1+ln t t nλ p n 1 1 f Lp1 Bx 0,t. When p 1 < s,bythefubinitheoem,3.10, and the Minkowski inequality, we get T f 2 Lp B B B 1 p 1 s n p n s n p n s Bx 0,t Bx 0,t f y x y f y y Lp B Bx 0,t p 1 p dx f y y Ls B f L1 Bx 0,t B x0, t + x 0 y 1 s t n s n p 1 1 f Lp1 Bx 0,t. 4.4 Now combining all the above estimates, we end the poof of Lemma 4.6. The following theoem is tue. Theoem 4.7 Suppose that x 0 R n,1<p <, T is a linea opeato satisfying condition 1.1 with L s S n 1, s >1,is homogeneous of degee zeo and bounded on L p R n. Let b LC {x 0} Rn, 1 p = 1 p p 2,0 λ < 1 n. Let also, fo s p 1, the pai ϕ 1, ϕ 2 satisfy the condition 1+ln t ess inft<τ< ϕ 1 x 0, ττ n p t n Cϕ p +1 nλ 2 x 0,, 4.5

16 Balakishiyev et al. Jounal of Inequalities and Applications :61 Page 16 of 18 and, fo p 1 < s, the pai ϕ 1, ϕ 2 satisfy the condition 1+ln t ess inft<τ< ϕ 1 x, ττ n p t n p n Cϕ s +1 2 x, n s, 4.6 whee C does not depend on. Then the opeato [b, T ] is bounded fom LM {x 0} to LM {x 0}. Moeove, [b, T ]f LM b LC f LM. Poof The statement of Theoem 4.7 follows by Lemma 4.6 and Theoem 4.2 in the same manneasinthepoofoftheoem3.8. Coollay 4.8 Suppose that x 0 R n, L s S n 1 with s >1,is homogeneous of degee zeo. Let 1<p <, b LC {x 0} Rn, 1 p = 1 p p 2, and 0 λ < 1 n. Let also, fo s p 1, the pai ϕ 1, ϕ 2 satisfy the condition 4.5, and, fo p < s, the pai ϕ 1, ϕ 2 satisfy the condition 4.6. Then the opeato [b, T ] is bounded fom LM {x 0} to LM {x 0}. Coollay 4.9 Let T be a linea opeato satisfying condition 1.1 with L s S n 1, s >1, being homogeneous of degee zeo and bounded on L p R n. Suppose 1<p < and b BMOR n. Let also, fo s p, the pai ϕ 1, ϕ 2 satisfy the condition 1+ln t ess inft<τ< ϕ 1 x, ττ n p t n Cϕ p +1 2 x,, 4.7 and, fo p < s, the pai ϕ 1, ϕ 2 satisfy the condition 1+ln t ess inft<τ< ϕ 1 x, ττ n p t n p n Cϕ s +1 2 x, n s, 4.8 whee C does not depend on x and. Then the opeato [b, T ] is bounded fom M p,ϕ1 to M p,ϕ2. Moeove, [b, T ]f Mp,ϕ2 b BMO f Mp,ϕ1. Coollay 4.10 Suppose that L s S n 1 with s >1,is homogeneous of degee zeo. Let 1<p < and b BMOR n. Let also, fo s p, the pai ϕ 1, ϕ 2 satisfy the condition 4.7 and, fo p < s, the pai ϕ 1, ϕ 2 satisfy the condition 4.8. Then the opeato [b, T ] is bounded fom M p,ϕ1 to M p,ϕ2. Remak 4.11 Note that the boundedness of sublinea opeatos with ough kenel and its commutato on the genealized cental local Moey spaces LM p,ϕ wee studied in [38]. Also, in the case s = Coollay 4.8 was poved in [30]andCoollay4.10 in [33]. Competing inteests The authos declae that they have no competing inteests.

17 Balakishiyev et al. Jounal of Inequalities and Applications :61 Page 17 of 18 Authos contibutions All authos contibuted equally to the witing of this pape. All authos ead and appoved the final manuscipt. Autho details 1 Depatment of Mathematical Analysis, Baku State Univesity, Baku, Azebaijan. 2 Institute of Mathematics and Mechanics of NAS of Azebaijan, Baku, Azebaijan. 3 Depatment of Mathematics, Ahi Evan Univesity, Kisehi, Tukey. 4 Depatment of Mathematics, Ankaa Univesity, Ankaa, Tukey. Acknowledgements The authos would like to expess thei gatitude to the efeees fo thei vey valuable comments and suggestions. The eseach of V Guliyev was patially suppoted by the gant of Science Development Foundation unde the Pesident of the Republic of Azebaijan, Gant EIF /10/1 and by the gant of Ahi Evan Univesity Scientific Reseach Pojects PYO.FEN Received: 20 Septembe 2014 Accepted: 29 Januay 2015 Refeences 1. Soia, F, Weiss, G: A emak on singula integals and powe weights. Indiana Univ. Math. J. 43, Lu, G, Lu, S, Yang, D: Singula integals and commutatos on homogeneous goups. Anal. Math. 28, Ding, Y, Yang, D, Zhou, Z: Boundedness of sublinea opeatos and commutatos on L p,ω R n. Yokohama Math. J. 46, Moey, CB: On the solutions of quasi-linea elliptic patial diffeential equations. Tans. Am. Math. Soc. 43, Adams, DR: A note on Riesz potentials. Duke Math. J. 42, Chiaenza, F, Fasca, M: Moey spaces and Ha-Littlewood maximal function. Rend. Mat. 7, Peete, J: On the theoy of M p,λ.j.funct.anal.4, Chiaenza, F, Fasca, M, Longo, P: Inteio W 2,p -estimates fo nondivegence elliptic equations with discontinuous coefficients. Ric. Mat. 40, Chiaenza, F, Fasca, M, Longo, P: W 2,p -Solvability of Diichlet poblem fo nondivegence elliptic equations with VMO coefficients. Tans. Am. Math. Soc. 336, Di Fazio, G, Ragusa, MA: Inteio estimates in Moey spaces fo stong solutions to nondivegence fom equations with discontinuous coefficients. J. Funct. Anal. 112, Di Fazio, G, Palagachev, DK, Ragusa, MA: Global Moey egulaity of stong solutions to the Diichlet poblem fo elliptic equations with discontinuous coefficients. J. Funct. Anal. 166, Mizuhaa, T: Boundedness of some classical opeatos on genealized Moey spaces. In: Igai, S ed. Hamonic Analysis. ICM 90 Satellite Poceedings, pp Spinge, Tokyo Nakai, E: Ha-Littlewood maximal opeato, singula integal opeatos and Riesz potentials on genealized Moey spaces. Math. Nach. 166, Samko, N: Weighted Ha and singula opeatos in Moey spaces. J. Math. Anal. Appl. 3501, Guliyev, VS: Boundedness of the maximal, potential and singula opeatos in the genealized Moey spaces. J. Inequal. Appl. 2009, Aticle ID Guliyev, VS: Integal opeatos on function spaces on the homogeneous goups and on domains in R n.docto s degee dissetation, Mat. Inst. Steklov, Moscow pp. in Russian 17. Guliyev, VS: Function spaces, Integal Opeatos and Two Weighted Inequalities on Homogeneous Goups. Some Applications, Cashioglu, Baku pp. in Russian 18. Guliyev, VS: Some popeties of the anisotopic Riesz-Bessel potential. Anal. Math. 26, Guliyev, VS, Mustafayev, RC: Integal opeatos of potential type in spaces of homogeneous type. Dokl. Akad. Nauk, Ross. Akad. Nauk 354, Guliyev, VS, Mustafayev, RC: Factional integals in spaces of functions defined on spaces of homogeneous type. Anal. Math. 24, Buenkov, VI, Guliyev, HV, Guliyev, VS: Necessay and sufficient conditions fo boundedness of the factional maximal opeatos in the local Moey-type spaces. J. Comput. Appl. Math. 2081, Buenkov, VI, Guliyev, VS: Necessay and sufficient conditions fo the boundedness of the Riesz potential in local Moey-type spaces. Potential Anal. 303, Buenkov, V, Gogatishvili, A, Guliyev, VS, Mustafayev, R: Boundedness of the Riesz potential in local Moey-type spaces. Potential Anal. 351, Buenkov, V, Guliyev, VS, Sebetci, A, Taaykova, TV: Necessay and sufficient conditions fo the boundedness of genuine singula integal opeatos in local Moey type spaces. Euasian Math. J. 1, Wiene, N: Genealized hamonic analysis. Acta Math. 55, Wiene, N: Taubeian theoems. Ann. Math. 33, Beuling, A: Constuction and analysis of some convolution algebas. Ann. Inst. Fouie Genoble 14, Feichtinge, H: An elementay appoach to Wiene s thid Taubeian theoem on Euclidean n-space. In: Symposia Mathematica Cotona, Sympos. Math., vol. 29. Academic Pess, New Yok Alvaez, J, Guzman-Patida, M, Lakey, J: Spaces of bounded λ-cental mean oscillation, Moey spaces, and λ-cental Caleson measues. Collect. Math. 51, Guliyev, VS: Local genealized Moey spaces and singula integals with ough kenel. Azeb. J. Math. 32, Guliyev, VS: Genealized local Moey spaces and factional integal opeatos with ough kenel. J. Math. Sci. N.Y. 1932, Akbulut, A, Guliyev, VS, Mustafayev, R: Boundedness of the maximal opeato and singula integal opeato in genealized Moey spaces. Math. Bohem. 1371, Guliyev, VS, Aliyev, SS, Kaaman, T, Shukuov, PS: Boundedness of sublinea opeatos and commutatos on genealized Moey space. Integal Equ. Ope. Theoy 713,

18 Balakishiyev et al. Jounal of Inequalities and Applications :61 Page 18 of Coifman, R, Rochbeg, R, Weiss, G: Factoization theoems fo Ha spaces in seveal vaiables. Ann. Math. 1032, Ragusa, MA: Cauchy-Diichlet poblem associated to divegence fom paabolic equations. Commun. Contemp. Math. 63, Lu, SZ, Yang, DC: The cental BMO spaces and Littlewood-Paley opeatos. Appox. Theoy Appl. 11, Guliyev, VS: Genealized weighted Moey spaces and highe ode commutatos of sublinea opeatos. Euasian Math. J. 33, Fan, Y: Boundedness of sublinea opeatos and thei commutatos on genealized cental Moey spaces. J. Inequal. Appl. 2013, Aticle ID