Necessary and sufficient conditions for boundedness of multilinear fractional integrals with rough kernels on Morrey type spaces

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1 Shi et al. Journal of Ineualities and Alications 206) 206:43 DOI 0.86/s y RESEARCH Oen Access Necessary and sufficient conditions for boundedness of multilinear fractional integrals with rough kernels on Morrey tye saces Yanlong Shi, Zengyan Si2, Xiangxing Tao3* and Yafeng Shi4,5 * Corresondence: xxtao@hotmail.com 3 Faculty of Science, Zhejiang University of Science & Technology, Hangzhou, Zhejiang 30023, P.R. China Full list of author information is available at the end of the article Abstract In this article, we stu necessary and sufficient conditions on the arameters of the boundedness on Morrey saces and modified Morrey saces for T,α and M,α, which are a multilinear fractional integral and a multilinear fractional maximal oerator with rough kernel, resectively. Our results extend some known results significantly. MSC: 42B20; 42B25; 42B35 Keywords: multilinear maximal oerator; multilinear fractional integrals; rough kernel; Morrey saces; modified Morrey saces Introduction Suose that is homogeneous of degree zero on Rn and Ls Sn ) with < s, where Sn denotes the unit shere of Rn. Moreover, m will denote an integer, θj j =,..., m) will be fixed, distinct, and nonzero real numbers, and < α < n. We denote f = f,..., fm ), then the multilinear fractional integral oerator on Rn is given by the formula T,α fx) = Rn m y) fj x θj y), y n α j= and the multilinear fractional maximal oerator M,α is given by M,α fx) = r> rn α m y) fj x θj y). j= If α =, then M M, is the multilinear maximal oerator. When m = and, if let θ =, T,α will be the Riesz otential oerator Iα [, ] given by Iα f x) = Rn f x y). y n α 206 Shi et al. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License htt://creativecommons.org/licenses/by/4.0/), which ermits unrestricted use, distribution, and reroduction in any medium, rovided you give aroriate credit to the original authors) and the source, rovide a link to the Creative Commons license, and indicate if changes were made.

2 Shi et al. Journal of Ineualities and Alications 206) 206:43 Page 2 of 9 Sanne and Adams obtained two remarkable results on Morrey saces see Definition 2. in Section 2)forI α. Their results can be summarized as follows. Proosition. [3, 4] Sanne, but ublished by Peetre) Let 0<α < n, < n/α, 0 λ < n α,/ =/ α/n, and μ/ = λ/. Then for >,the oerator I α is bounded from L,λ R n ) to L,μ R n ) and for =,I α is bounded from L,λ R n ) to WL,μ R n ). Proosition.2 [5, 6] Let 0<α < n, < n/α,0 λ < n α. i) If >, then the condition / / = α/n λ) is necessary and sufficient for the boundedness of the oerator I α from L,λ R n ) to L,λ R n ). ii) If =, then the condition / = α/n λ) is necessary and sufficient for the boundedness of the oerator I α from L,λ R n ) to WL,λ R n ). If λ = 0, then the statement of Proositions. and.2 reduces to the well-known Har- Littlewood-Sobolev ineuality. On the other hand, in 20, Guliyev et al. [6] foundthis ineuality in modified Morrey saces see Definition 2.2 in Section 2) wasalsovalidand roved the following. Proosition.3 [6] Let 0<α < n, < n/α,0 λ < n α. i) If >, then the condition α/n / / α/n λ) is necessary and sufficient for the boundedness of the oerator I α from L,λ to L,λ. ii) If =, then the condition α/n / α/n λ) is necessary and sufficient for the boundedness of the oerator I α from L,λ to W L,λ. When m 2and, Grafakos [7] studied Lebesgue boundedness of T,α.Recently, Gunawan [8] extended Grafakos result to Morrey saces and rovided a multi-version for the sufficiency of conclusion i) in Proosition.2. Proosition.4 [8] Let 0<α < n, be the harmonic mean of,..., m >,< < n/α, 0 λ < n α, / / = α/n λ), then the oerator T,α is bounded from L,λ R n ) L m,λ R n ) to L,λ R n ). When m 2and L s S n ), Ding and Lu [9] studied the L L m boundedness for T,α. After this work above, a natural uestion is: what roerties does the oerator T,α have on Morrey and modified Morrey saces? We give answers as follows. Theorem. Let 0<α < n, L s S n ) with <s, s = s/s ), be the harmonic mean of,..., m >,0 λ < n α, < n/α and satisfy λ m = λ j for 0 λ j < n..) j i) If > s, then the condition / / = α/n λ) is necessary and sufficient for the boundedness of the oerator T,α from L,λ R n ) L m,λ m R n ) to L,λ R n ). ii) If = s, then the condition /s / = α/n λ) is necessary and sufficient for the boundedness of the oerator T,α from L,λ R n ) L m,λ m R n ) to WL,λ R n ). Moreover, similar conclusions hold for M,α.

3 Shi et al. Journal of Ineualities and Alications 206) 206:43 Page 3 of 9 Theorem.2 Let α,, s, j, λ j,, and λ be as in Theorem.. i) If > s, then the condition α/n / / α/n λ) is necessary and sufficient for the boundedness of the oerator T,α from L,λ R n ) L m,λ m R n ) to L,λ R n ). ii) If = s, then the condition α/n /s / α/n λ) is necessary and sufficient for the boundedness of the oerator T,α from L,λ R n ) L m,λ m R n ) to W L,λ R n ). Moreover, similar estimates hold for M,α. Remark. Note that Theorems. and.2 covers Proositions.2 and.3,resectively. Also, the case λ = λ = = λ m and reduces to Proosition.4; thecaseλ = λ = = λ m = 0 gives the result of Ding and Lu [9]onLebesguesaces. We observe that, in Theorems. and.2, the boundedness in the limiting case = n λ)/α remains oen. In fact, when = n/α i.e. λ =0),DingandLu[9] foundm,α is bounded from L L m to L, but this corresonding result for T,α in this case does not hold. Our next goal is to extend Ding and Lu s result to the case 0 λ < n α,as the continuation of Theorems. and.2. Theorem.3 Let 0<α < n,0 λ < n α, L s S n ) with <s, be the harmonic mean of,..., m >and satisfy.). i) If =n λ)/α s, then the oerator M,α is bounded from L,λ R n ) L m,λ m R n ) to L R n ). ii) If s n λ)/α n/α, then the oerator M,α is bounded from L,λ R n ) L m,λ m R n ) to L R n ). Finally we shall describe the organization of this aer. In the following section, we will stu the boundedness of maximal oerator M on Morrey and modified Morrey saces. The last section we will devote to the boundedness of T,α and M,α and to showing the roof of Theorems.,.2 and.3. Throughout this aer, we assume the letter C always remains to denote a ositive constantthatmayvaryateachoccurrencebutisindeendentoftheessentialvariables. 2 Boundedness of maximal oerator M In this art, we investigate the boundedness of maximal oerator M see Section ) on Morrey and modified Morrey saces defined by the following definitions. Definition 2. [3 5, 0] Let <,0 λ n.wedenotebyl,λ = L,λ R n )themorrey sace, and by WL,λ = WL,λ R n ) the weak Morrey sace, as the set of locally integrable functions f x), x R n, with the finite norms f L,λ R n ) = f WL,λ R n ) = resectively. r f y), { y Bx, t): f y) > r },

4 Shi et al. Journal of Ineualities and Alications 206) 206:43 Page 4 of 9 Definition 2.2 [6] Let <,0 λ n,[t] = min{, t}.wedenoteby L,λ = L,λ R n ) the modified Morrey sace, and by W L,λ = W L,λ R n ) the weak modified Morrey sace, as the set of locally integrable functions f x), x R n, with the finite norms f L,λ R n ) = f W L,λ R n ) = resectively. r f y), { y Bx, t): f y) > r }, It is easy to see that L,0 R n )= L,0 R n )=L R n ), WL,0 R n )=W L,0 R n )=WL R n ). If λ <0orλ > n, then L,λ R n )=L,λ R n )= where is the set of all functions euivalent to 0 on R n. In addition, from [6], we know L,λ R n) L,λ R n) L R n), max { f L,λ, f L } f L,λ. Recall the definition of M,asasecialcasewhenm =, andθ =,M is the Har-Littlewood maximal oerator M. In994, Nakai[] obtained the boundedness of M on Morrey saces, later Guliyev [6] studied the oerator M on modified Morrey saces and get a result arallel to Nakai s result. Lemma 2. [] Let < and 0 λ < n. Then for >,M is bounded from L,λ to L,λ and for =,M is bounded from L,λ to WL,λ. Lemma 2.2 [6] Let < and 0 λ < n. Then for >,M is bounded from L,λ to L,λ and for =,M is bounded from L,λ to W L,λ. When m 2and L s S n ), we find M also has the same roerties by roviding the following multi-version of Lemmas 2. and 2.2. Theorem 2.3 Let L s S n ) with <s,0 λ < n, be the harmonic mean of,..., m >, s and satisfy.). i) If > s, there exists a ositive constant C such that M f L,λ f j L j,λ j. ii) If = s, there exists a ositive constant C such that M f WL,λ f j L j,λ j. Theorem 2.4 Let L s S n ) with <s,0 λ < n, be the harmonic mean of,..., m >, s and satisfy.).

5 Shi et al. Journal of Ineualities and Alications 206) 206:43 Page 5 of 9 i) If > s, there exists a ositive constant C such that m M f L,λ f j L j,λ j. ii) If = s, there exists a ositive constant C such that m M f W L,λ f j L j,λ j. Here,wegiveonlytheroofofTheorem2.4 and omit the roof of Theorem 2.3 due to the similarity. Proof Since L s S n )withs >, Hölder s ineuality yields y) r n r n = r n = C r n f j x θ j y) f j x θ j y) /s s y) /s s f j x θ j y) /s r s ξ) s z n dξ dz S n r n f j x θ j y) s [ M s fj j / ) x) ] /s j, which imlies a ointwise estimate M fx) /s fj x θ j y) s j / ) /s j 0 [ M s fj j / ) x) ] /s j. 2.) i) If > s,by2.) and the Hölder ineuality, we get for all x R n and t >0. M fy) [t] λ j [ M s fj j / ) y) ] 2 /s j /s [ M fj s j / ) y) ] /s ) /j

6 Shi et al. Journal of Ineualities and Alications 206) 206:43 Page 6 of 9 Taking the th root of both sides and alying Lemma 2.2 with /s >andthefact f j s j / L /s,λ j,weget M f L,λ = = C = C [t] λ j M fy) / M f s j /) /s j j f s j / /s j j L /s,λ j f j L j,λ j, L /s,λ j [ M fj s j / ) y) ] /s /j which is the desired ineuality. ii) If = s, for any β >0,letε 0 = β, ε m =andε, ε 2,...,ε m > 0 be arbitrary which will be chosen later. From the ointwise estimate 2.), we get { y Bx, t): M fy) > β } Let us now take ε, ε 2,...,ε m >0suchthat ε j = [ m f j L j,λ j ] s / j, j =,2,...,m. ε j β s / j f j L j,λ j m { y Bx, t): [ M s f j / ) j y) ] /s j ε j > Then, alying Lemma 2.2 with /s =andthefactf j j L,λ j,weget { y Bx, t): M fy) > β } = C = C = C m {y Bx, t):m ) ) f j ε j } j j y)> [t] λ λ j)/ j ε j m [t] λ j m m = C λ λ j )/ j [t] ε j ε j εj ε j [ εj m β β ε j ) j f j j L j,λ j ) ] j f j L j,λ j f j L j,λ j ) f j L j,λ j. ) j fj j,λ L j ) s [t] λ λ j)/ j ε j }.

7 Shi et al. Journal of Ineualities and Alications 206) 206:43 Page 7 of 9 Hence, we obtain the following ineuality: M f W L,λ = β β>0 f j L j,λ j. { y Bx, t): M fy) > β } This is the conclusion ii) of Theorem Boundedness of T,α and M,α The resent section consists of two arts which are about the bounded estimates on Morrey and modified saces for the multilinear fractional integral oerator T,α and the multilinear fractional maximal oerator M,α,resectively. 3. Boundedness on Morrey saces In this art, we will rove Theorem.. Let us begin with a reuisite Hedberg s tye estimates, which lays a key role in roving Theorem.. Lemma 3. Let 0<α < n, L s S n ) with <s, be the harmonic mean of,..., m >,0 λ < n α, s < n/α and satisfy.), then there exists a ositive constant C such that T,α fx) [ M fx) ] α/n λ) f j α/n λ) L j,λ j. Proof For any δ > 0, we slit the integral into two arts: ) y) T,α fx)= + y <δ y δ y n α For Ax, δ), we have Ax, δ) y <δ y) y n α f j x θ j y) =: Ax, δ)+sx, δ). fj x θ j y) 2 i δ y <2 i δ 2 i δ ) α n y) y n α y <2 i δ fj x θ j y) y) 2 i δ ) α n 2 i δ ) n M fx) 2 n α δ α M fx) δ α M fx). 2 iα f j x θ j y)

8 Shi et al. Journal of Ineualities and Alications 206) 206:43 Page 8 of 9 Recalling the conditions of Lemma 3., wecansees <n λ)/α, which imlies α <n λ)/ n λ)/s,thenweget n αs > n n λ)s / n n λ)=λ. InordertoestimateSx, δ), we choose a real number σ such that n αs > σ > n n λ)s / λ. One can then see from the choice of σ that n n α σ /s ) s <0 3.) and n σ )/s n λ)/ <0. 3.2) Then, using the Hölder ineuality, we obtain Sx, δ) y δ y δ y) y n α σ /s y σ /s =: E σ δ) F σ x, δ). fj x θ j y) y) s ) /s y n α σ /s )s y δ y σ fj x θ j y) s /s For E σ δ), by the fact 3.), we obtain E σ δ)= ξ) /s s r n n α σ /s )s dξ dr = Cδ α n σ )/s. S n For F σ x, δ), we have F σ x, δ) δ 2 i δ y <2 i+ δ y σ f j x θ j y) s /s 2 i δ ) σ /s f j x θ j y) s y <2i+δ /s. If > s, alying Hölder s ineuality and the fact 3.2), we have F σ x, δ) 2 i δ ) σ /s y <2i+δ 2 i δ ) n σ )/s n/ ) /s / y <2 i+ δ y <2 i+ δ fj x θ j y) f j x θ j y) / /

9 Shi et al. Journal of Ineualities and Alications 206) 206:43 Page 9 of 9 2 i δ ) n σ )/s n λ)/ 2 i+ δ) λ 2 i δ ) n σ )/s n λ)/ m 2 i+ δ) λ j 2 i δ ) n σ )/s n λ)/ m f j L j,λ j δ n σ )/s n λ)/ f j L j,λ j. y <2 i+ δ y <2 i+ δ If = s, using the Hölder ineuality and the fac < σ,weget F σ x, δ) 2 i δ ) σ /s fj x θ j y) s y <2i+δ 2 i δ ) σ /s +λ/s 2 i+ δ) λ 2 i δ ) λ σ )/s m 2 i+ δ) λ j 2 i δ ) λ σ )/s m f j L j,λ j m λ σ )/s δ f j L j,λ j = Cδ n σ )/s n λ)/ Hence, for every s,wehave Sx, δ) δ α n λ)/ f j L j,λ j. f j L j,λ j. Thus T,α fx) δ α M fx)+δ α n λ)/ Now take [ M δ = fx) ) ] /n λ), m f j L j,λ j y <2 i+ δ y <2 i+ δ f j x θ j y) /s / fj x θ j y) /j j fj x θ j y) s fj x θ j y) j f j L j,λ j ), δ >0. /s /j and then we get the conclusion of Lemma 3..

10 Shi et al. Journal of Ineualities and Alications 206) 206:43 Page 0 of 9 Now we are rea to rove Theorem.. Proof of Theorem. Firstofall,wewilldevoteoureffortstotheroofofi). Sufficiency. By Lemma 3. and the conclusion i) of Theorem 2.3,wehave T,α fy) / ) f j / L j,λ M fy) ) / j f j / L j,λ j f j L j,λ j. f j / L j,λ j Taking the remum for x R n and t > 0, we will get the desired conclusion. Necessity. Suose that T,α is bounded from L,λ L m,λ m to L,λ.Definef ɛ x)= f ɛx),...,f m ɛx)) for ɛ > 0. Then it is easy to show that Thus T,α f ɛ y)=ɛ α T,α fɛy). 3.3) T,α f ɛ L,λ = ɛ α = ɛ α n/ T,α fɛy) / Bɛx,ɛt) = ɛ α n/+λ/ ɛt) λ = ɛ α n λ)/ T,α f L,λ. T,α fy) / Bɛx,ɛt) T,α fy) / Since T,α is bounded from L,λ L m,λ m to L,λ,wehave T,α f L,λ = ɛ α+n λ)/ T,α f ɛ L,λ ɛ α+n λ)/ = Cɛ α+n λ)/ = Cɛ α+n λ)/ fj ɛ ) L j,λ j = Cɛ α+n λ)/ where C is indeendent of ɛ. ɛ n/ j ɛ λ j n)/ j j m = Cɛ α+n λ)/ n λ)/ f j L j,λ j, j f j ɛy) /j j Bɛx,ɛt) ɛt) λ j fj y) /j j Bɛx,ɛt) fj y) /j j

11 Shi et al. Journal of Ineualities and Alications 206) 206:43 Page of 9 If / </ + α/n λ), then for all f L,λ L m,λ m,wehave T,α f L,λ =0as ɛ 0. If / >/ + α/n λ), then for all f L,λ L m,λ m,wehave T,α f L,λ =0as ɛ. Therefore we get / =/ + α/n λ). We roceed to rove ii). Sufficiency. For any β > 0, alying Lemma 3. and the conclusion ii) of Theorem 2.3,weget T,α f WL,λ { = β y Bx, t): T β>0 fy) > β } { β β>0 y Bx, t): CM fy) ) s / β>0 β>0 β β / f j s / L j,λ j > β } / [ { β /s }] / y Bx, t):m fy)> C m f j s / L j,λ j [ { y Bx, t):m fy)> [ m f j s / ) /s ] L β j,λ m s / j f j β>0 β L j,λ j [ m f j s / L β j,λ j β>0 β f j L j,λ j. f j s / L j,λ j ] β C m f j s / L j,λ j ) /s }] s s Thus, we comlete the sufficiency of ii). Necessity. Let T,α be bounded from L,λ L m,λ m to WL,λ.Becausewehave 3.3)forf ɛ x)=f ɛx),...,f m ɛx)) with ɛ >0,thenweobtain T,α f ɛ WL,λ = = r r = ɛ n/ r = ɛ α n/+λ/ rɛ α {y : T,α f ɛ y) >r} {y : T,α fɛy) >rɛ α } = ɛ α n λ)/ T,α f WL,λ. {y Bɛx,ɛt): T,α fy) >rɛ α } ɛt) λ {y Bɛx,ɛt): T,α fy) >rɛ α }

12 Shi et al. Journal of Ineualities and Alications 206) 206:43 Page 2 of 9 Since T,α is bounded from L,λ L m,λ m to WL,λ,wehave T,α f WL,λ = ɛ α+n λ)/ T,α f ɛ WL,λ ɛ α+n λ)/ ɛ α+n λ)/ n λ)/ where C is indeendent of ɛ. f j L j,λ j, f j ɛ ) L j,λ j If / </ + α/n λ), then for all f L,λ L m,λ,wehave T,α f WL,λ =0as ɛ 0. If / >/ + α/n λ), then for all f L,λ L m,λ,wehave T,α f WL,λ =0as ɛ. Conseuently, we get / =/ + α/n λ). Next, we rove conclusions i) and ii) hold for M,α. By the same arguments as above we get the necessity art and the sufficiency art follows from the conclusion of T,α and the following lemma. Lemma 3.2 [9] Suose that 0<α < n, L s S n ) with <s. Then M,α f)x) α,n T,α f ) x), where f = f,..., f m ). Then the roof of Theorem. is comleted. As an alication of Theorem., we get Sanne tye estimates, which can be seen a multi-version of Proosition.. Corollary 3. Let α,, s, j, λ j,, and λ be as in Theorem.,/ =/ α/n, μ/ = λ/. i) If > s, then T,α is bounded from L,λ R n ) L m,λ m R n ) to L,μ R n ). ii) If = s, then T,α is bounded from L,λ R n ) L m,λ m R n ) to WL,μ R n ). Moreover, similar estimates hold for M,α. Proof From Lemma 3.2, we only need to show the boundedness of T,α. First, we choose t to satisfy n μ)/ =n λ)/t,thenweget /t =n μ)/n λ)=/ α/n λ)</ α/n =/. Then Hölder s ineuality imlies L t,λ R n ) L,μ R n )andwl t,λ R n ) WL,μ R n ). In fact, there exists a constant C >0suchthat T,α f L,u T,α f L t,λ and T,α f WL,u T,α f WL t,λ.

13 Shi et al. Journal of Ineualities and Alications 206) 206:43 Page 3 of 9 Then, by Theorem.,wehave T,α f L,u T,α f L t,μ f j L j,λ j for > s and T,α f WL,u T,α f WL t,μ f j L j,λ j for = s. Thus, the roof of Corollary 3. is comleted. As an another alication, by Hölder s ineuality, we obtain an Olsen s ineuality as in the following corollary, which is a multi-version of the results in considered by Olsen in [2] in the stu of the Schrödinger euation with erturbed otentials W. Corollary 3.2 Let α,, s, j, λ j,, and λ be as in Theorem. and let W L n λ)/α,λ. If > s and / / = α/n λ), then there exists a ositive constant C such that W T,α f L,λ R n ) W L n λ)/α,λ R n ) Moreover, similar estimates hold for M,α. f j L j,λ. j R n ) 3.2 Boundedness on modified Morrey saces This art we will devote to the boundedness on modified Morrey saces and show the roof Theorem.2 and.3. With the same arguments on Morrey saces, we also begin with a reuisite Hedberg s tye estimates on modified Morrey saces. Lemma 3.3 Let 0<α < n, L s S n ) with <s, be the harmonic mean of,..., m >,0 λ < n α, s < n/α and satisfy.), then there exists a ositive constant C such that T,α fx) M fx) ) / f j / L j,λ j. Proof For any δ > 0, we do the same decomoosition of T,α as in the roof of Lemma 3., then we only need to estimate F σ x, δ). We also choose the same σ during the roof of Lemma 3.,thenweget n σ )/s n/ n σ )/s n λ)/ <0. 3.4) If > s, by the Hölder ineuality and the fact 3.4), we obtain F σ x, δ) 2 i δ ) σ /s fj x θ j y) s y <2i+δ 2 i δ ) n σ )/s n/ y <2 i+ δ /s f j x θ j y) /

14 Shi et al. Journal of Ineualities and Alications 206) 206:43 Page 4 of 9 2 i δ ) n σ )/s n/[ 2 i+ δ ] λ/ 2 i δ ) n σ )/s n/[ 2 i+ δ ] λ/ 2 i δ ) n σ )/s n/[ 2 i+ δ ] λ/ [2 i+ δ] λ [2 i+ δ] λ j f j L j,λ j = C f j L j,λ j 2 i δ ) n σ )/s n/[ 2 i+ δ ] λ/ f j L j,λ j f j L j,λ j f j L j,λ j δ n σ )/s n/ [2δ] λ/ [log2 2δ ] y <2 i+ δ y <2 i+ δ f j x θ j y) 2 i δ ) n σ )/s n/, ifδ /2, 2 i δ ) n σ )/s n λ)/ + i=[log 2 2δ ]+ / fj x θ j y) /j j 2 i δ ) n σ )/s n/, if0<δ < /2 { δ n σ )/s n/, ifδ /2, δ n σ )/s n λ)/ + δ n σ )/s n/, if0<δ < /2 { δ n σ )/s n/, ifδ /2, δ n σ )/s n λ)/, if0<δ < /2 f j L j,λ j. If = s, using the Hölder ineuality and the fact 0 λ < σ,weget F σ x, δ) 2 i δ ) σ /s fj x θ j y) s y <2i+δ 2 i δ ) σ /s [ 2 i+ δ ] λ/s [2 i+ δ] λ 2 i δ ) σ /s [ 2 i+ δ ] λ/s [2 i+ δ] λ j f j L j,λ j 2 i δ ) σ /s [ 2 i+ δ ] λ/s f j L j,λ j f j L j,λ j f j L j,λ j [log2 2δ ] /s y <2 i+ δ y <2 i+ δ fj x θ j y) s 2 i δ ) σ /s, ifδ /2, 2 i δ ) λ σ )/s + i=[log 2 2δ ]+ /s f j x θ j y) /j j 2 i δ ) σ /s, if0<δ < /2 { δ σ /s, ifδ /2, δ λ σ )/s + δ σ /s, if0<δ < /2 { δ σ /s, ifδ /2, δ λ σ )/s, if0<δ < /2

15 Shi et al. Journal of Ineualities and Alications 206) 206:43 Page 5 of 9 δ σ /s [2δ] λ/ f j L j,λ j δ n σ )/s n/ [2δ] λ/ f j L j,λ j. Then, combining with the estimates E σ δ) δ α n σ )/s,wehave Thus Sx, δ) δ α n/ [2δ] λ/ f j L j,λ j, forevery s. T,α fx) δ α M fx)+δ α n/ [2δ] λ/ { min δ α M fx)+δ α n/ δ α M fx)+δ α n λ)/ Minimizing with resect to δ,at and we have [ M δ = fx) ) m f j L j,λ j [ M δ = fx) ) m f j L j,λ j ] /n ] /n λ) T,α fx) { M fx) min m f j L j,λ j M fx) ) / ) f j L j,λ j f j L j,λ j, } f j L j,λ j. α f j / L j,λ j. n M fx), m f j L j,λ j α } n λ m f j L j,λ j This is the conclusion of Lemma 3.3. Now we give the roof of Theorem.2. Proof of Theorem.2 Similarly to the roof of sufficiency in Theorem., bythebound- edness of M in Theorem 2.4, wewillgetthesufficiency.now,wegiveonlytheroofof necessity.

16 Shi et al. Journal of Ineualities and Alications 206) 206:43 Page 6 of 9 Let [ɛ],+ = max{, ɛ},by3.3), for f ɛ x)withɛ >0,weget T,α f ɛ L,λ = ɛ α = ɛ α n/ = ɛ α n/ [ɛt] t>0 [t] [ɛt] λ ) λ/ Bɛx,ɛt) T,α fɛy) / Bɛx,ɛt) T,α fy) / T,α fy) / = ɛ α n/ [ɛ] λ,+ T,αf L,λ and T,α f ɛ W L,λ = = r r = ɛ n/ r = ɛ α n/ t>0 rɛ α ) λ/ [ɛt] {y : T,α f ɛ y) >r} {y : T,α fɛy) >rɛ α } {y Bɛx,ɛt): T,α fy) >rɛ α } [t] { [ɛt] λ y Bɛx, ɛt): T,α fy) > rɛ α} = ɛ α n/ [ɛ] λ,+ T,αf W L,λ. i) Assume that T,α is bounded from L,λ L m,λ m to L,λ,weget T,α f L,λ = ɛα+n/ [ɛ] λ,+ T,αf ɛ L,λ ɛ α+n/ [ɛ] λ,+ = Cɛ α+n/ [ɛ] λ,+ = Cɛ α+n/ [ɛ] λ,+ ɛ α+n/ [ɛ] λ,+ fj ɛ ) L j,λ j [ɛt] λ j ɛ n/ j [t] λ j ɛ n/ [ɛt] j t>0 [t] Bɛx,ɛt) [t] λ j fj ɛy) /j j ) λj / j f j y) /j j Bɛx,ɛt) f j y) /j j

17 Shi et al. Journal of Ineualities and Alications 206) 206:43 Page 7 of 9 ɛ α+n/ n/ [ɛ] λ,+ [ɛ] λ,+ ɛ α+n/ n/ λ [ɛ] λ,+ f j L j,λ j f j L j,λ j, where C is indeendent of ɛ. When / </ + α/n, then for all f L,λ L m,λ m,wehave T,α f L,λ =0as ɛ 0. When / >/ + α/n λ), then for all f L,λ L m,λ m,wehave T,α f L,λ =0 as ɛ. Therefore we get α/n / / α/n λ). ii) Assume that T,α is bounded from L,λ L m,λ m to W L,λ,wehave λ T,α f W L,λ = ɛα+n/ [ɛ],+ T,αf ɛ W L,λ ɛα+n/ [ɛ] λ,+ ɛ α+n/ n/ λ λ [ɛ],+ f j L j,λ j, fj ɛ ) L j,λ j where C is indeendent of ɛ. When / </ + α/n, then for all f L,λ L m,λ,wehave T,α f W L,λ =0as ɛ 0. When / >/ + α/n λ), then for all f L,λ L m,λ,wehave T,α f W L,λ =0 as ɛ. Conseuently, we get α/n / / α/n λ). Next, we rove conclusions i) and ii) hold for M,α. By the same arguments as above we get the necessity art and the sufficiency art follows from the conclusion of T,α and Lemma [9]. This comletes the roof of Theorem.2. Finally we show the roof of Theorem.3. Proof of Theorem.3 By the Hölder ineuality, we have M,α fx)= r n α m y) fj x θ j y) ) / y) r n α y) r n α / m /s r α n/ y) s r n r α n/ m fj x θ j y) f j x θ j y) j /j. / fj x θ j y) /j j fj x θ j y) /j j

18 Shi et al. Journal of Ineualities and Alications 206) 206:43 Page 8 of 9 i) If =n λ)/α s,bythefact.), we obtain M,α fx) r α n λ)/ = C f j L j,λ j. r λ j fj x θ j y) /j j ii) If n λ)/α n/α,usingthefact.), we get M,α fx) r α n/ [r] λ/ r α n/ [r] λ/ = C [r] λ j f j L j,λ j { f j L j,λ j max r 0<r< f j L j,λ j. α n λ f j x θ j y) /j j, r α n/} r Therefore, we comlete the roof of Theorem.3. Finally, we would like to remark that our theorems generalize the relevant results in [3 5]. Cometing interests The authors declare that they have no cometing interests. Authors contributions All authors read and aroved the final manuscrit. Author details Zhejiang Pharmaceutical College, Ningbo, Zhejiang 3500, P.R. China. 2 School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, Henan , P.R. China. 3 Faculty of Science, Zhejiang University of Science & Technology, Hangzhou, Zhejiang 30023, P.R. China. 4 School of Science, Ningbo University of Technology, Ningbo, Zhejiang 352, P.R. China. 5 School of Statistics and Management, Shanghai University of Finance and Economics, Shanghai, , P.R. China. Acknowledgements The research of the first author was orted by the Soft Science Foundation of Ningbo City No. 205A000). The second author was orted by the National Natural Science Foundation of China No and No. 5069). The third author was orted by the National Natural Science Foundation of China No. 7306). Thanks are also given to the anonymous referees for careful reading and suggestions. Received: 8 March 205 Acceted: 6 December 205 References. Lu, S, Ding, Y, Yan, D: Singular Integrals and Related Toics. World Scientific, Singaore 2006) 2. Stein, EM: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton 993) 3. Guliyev, V, Shukurov, P: On the boundedness of the fractional maximal oerator, Riesz otential and their commutators in generalized Morrey saces. Oer. Theory, Adv. Al. 229, ) 4. Peetre, J: On the theory of L,λ.J.Funct.Anal.4, ) 5. Adams, DR: A note on Riesz otentials. Duke Math. J. 42, ) 6. Guliyev, V, Hasnov, J, Zeren, Y: Necesary and sufficient conditions for the boundedness of the Riesz otential in modified Morrey saces. J. Math. Ineual. 5, )

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