Jump to ContentJump to Main Navigation
Show Summary Details
In This Section

Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

1 Issue per year


IMPACT FACTOR 2016 (Open Mathematics): 0.682
IMPACT FACTOR 2016 (Central European Journal of Mathematics): 0.489

CiteScore 2016: 0.62

SCImago Journal Rank (SJR) 2015: 0.521
Source Normalized Impact per Paper (SNIP) 2015: 1.233

Mathematical Citation Quotient (MCQ) 2015: 0.39

Open Access
Online
ISSN
2391-5455
See all formats and pricing
In This Section
Volume 14, Issue 1 (Jan 2016)

Issues

Fractional multilinear integrals with rough kernels on generalized weighted Morrey spaces

Ali Akbulut
  • Corresponding author
  • Department of Mathematics, Ahi Evran University, Kirsehir, Turkey
  • Email:
/ Amil Hasanov
  • Amil Hasanov, Gandja State University, Gandja, Azerbaijan
  • Email:
Published Online: 2016-12-17 | DOI: https://doi.org/10.1515/math-2016-0090

Abstract

In this paper, we study the boundedness of fractional multilinear integral operators with rough kernels TΩ,αA1,A2,,Ak, which is a generalization of the higher-order commutator of the rough fractional integral on the generalized weighted Morrey spaces Mp,ϕ (w). We find the sufficient conditions on the pair (ϕ1, ϕ2) with wAp,q which ensures the boundedness of the operators TΩ,αA1,A2,,Ak, from Mp,φ1wptoMp,φ2wq for 1 < p < q < ∞. In all cases the conditions for the boundedness of the operator TΩ,αA1,A2,,Ak, are given in terms of Zygmund-type integral inequalities on (ϕ1, ϕ2) and w, which do not assume any assumption on monotonicity of ϕ1 (x,r), ϕ2(x, r) in r.

Keywords: Fractional multilinear integral; Rough kernel; BMO; Generalized weighted Morrey space

MSC 2010: 42B20; 42B35; 47G10

1 Introduction and results

Multilinear harmonic analysis is an active area of research that is still developing. Multilinear operators appear also as technical tools in the study of linear singular integral (through the method of rotations), the analysis of nonlinear operators (through power series and similar expansions), and the resolution of many linear and nonlinear partial differential equations [4, 5, 15-17, 41].

It is well known that in 1967, Bajsanski and Coifman [3] proved the boundedness of the multilinear operator associated with the commutators of singular integrals considered by Calderon. In 1981, Cohen [9] studied the Lp boundedness of the multilinear integral operator TA defined by TAf(x)=p.v.RnΩ(xy)|xy|n+m1Rm(A;x,y)f(y)dy, where Ω is homogeneous of degree zero on ℝn with mean value zero on Sn-1. Moreover, Rm (A;x, y) denotes the m-th (m ≥ 2) remainder of the Taylor series of A at x about y; more precisely, Rm(A;x,y)=A(x)|γ|<m1γ!DγA(y)(xy)γ.

Using the method of good — λ inequality, in 1986, Cohen and Gosselin [10] proved that if Ω ∈ Lip1 (Sn-1) and DγABMO(ℝn), then TAfLpC|γ|=m1DγABMOfLp,1<p<, where the constant C > 0 is independent of f and A.

In 1994, for m = 2, Hofmann [32] proved that the multilinear operator TA is a bounded operator on Lp,w when Ω ∈ L(Sn-1) and wAp.

It is natural to ask whether the multilinear fractional integral operator with a rough kernel has the mapping properties similar to those of TΩA. The purpose of [12] is to study this problem. Let us give the definition of the multilinear fractional integral operator as follows: TΩ,αA1,A2,,Akf(x)=RnΩ(xy)|xy|nα+Nj=1kRmj(Aj;x,y)f(y)dy,

where 0<α<n,N=j=1k(mj1),min1jkmj2, Ω is homogeneous of degree zero and ΩLsSn1,s>1,RmjAj;x,y is as above.

When k = 1 and m = 1, then TΩ,αA is just the commutator of the fractional integral TΩ,αf with function A, TΩ,αAf(x)=RnΩ(xy)|xy|nα(A(x)A(y))f(y)dy.

When mj = 1 and Aj = A for j = 1,..., k, then TΩ,αA1,A2,,Ak is just the higher-order commutator TΩ,αA,kf given in [11], TΩ,αA,kf(x)=RnΩ(xy)|xy|nα(A(x)A(y))kf(y)dy.

When mj ≥ 2, TΩ,αA1,A2,,Ak f is a non-trivial generalization of the above commutator.

The classical Morrey spaces were originally introduced by Morrey in [36] to study the local behavior of solutions to second order elliptic partial differential equations. For the properties and applications of classical Morrey spaces, we refer the readers to [17, 18, 23, 36, 41]. Mizuhara [35] introduced generalized Morrey spaces. Later, in [23] Guliyev defined the generalized Morrey spaces Mp,ϕ with normalized norm. Recently, Komori and Shirai [34] considered the weighted Morrey spaces Lp,k (w) and studied the boundedness of some classical operators such as the Hardy-Littlewood maximal operator, the Calderón-Zygmund operator on these spaces. Also, Guliyev [24] introduced the generalized weighted Morrey spaces Mp,ϕ(w) and studied the boundedness of the classical operators and its commutators in these spaces Mp,ϕ(w), see also [24, 29, 33, 40]. In [24] the author gave a concept of generalized weighted Morrey space Mp,ϕ(w) which could be viewed as extension of both generalized Morrey space Mp,ϕ and weighted Morrey space Lp,K(w).

The weighted (Lp, Lq)-boundedness of such a commutator is given by Ding [13] and Lu in [14].

The following theorem was proved by Ding and Lu in [12].

Theorem 1.1: ([12]). Let 0<α<n,1/q=1/pα/n,1s<p<n/α,wsA(p/s,q/s) and let Ω be homogeneous of degree zero with Ω ∈ Ls(Sn-1). Moreover,for 1 ≤ jk, |γj| = mj - 1, mj 2 and DγjAjBMO(Rn). Then there exists a constant C, independent of Aj,1 ≤ jk and f, such that TΩ,αA1,A2,,AkfLq,wq(Rn)Cj=1k|γj|=mj1DγjAjfLp,wp(Rn).Here and in the sequel, we always denote by p' the conjugate index of any p > 1, that is 1=p + 1=p' = 1, and by C a constant which is independent of the main parameters and may vary from line to line. We define the generalized weighed Morrey spaces as follows.

Definition 1.2: Let 1 ≤ ρ <, ϕ be a positive measurable function on ℝ × (0, ∞) and w be non-negative measurable function onn. We denote by Mp,ϕ(w) the generalized weighted Morrey space, the space of all functions fLp,wlocRn with finite norm fMp,φ(w)=supxRn,r>0φ(x,r)1w(B(x,r))1pfLp,w(B(x,r)),where Lp,w(B(x, r)) denotes the weighted Lp-space of measurable functions f for which fLp,w(B(x,r))=B(x,r)|f(y)|pw(y)dy1p.Furthermore, by WMp,ϕ(w) we denote the weak generalized weighted Morrey space of all functions fWLp,wlocRn for which fWMp,φ,(w)=supxRn,r>0φ(x,r)1w(B(x,r))1pfWLp,w(B(x,r))<,where WLp,w(B(x, r)) denotes the weak Lp,w-space of measurable functions f for which fWLp,w(B(x,r))=supt>0t{yB(x,r):|f(y)|>t}w(y)dy1p.

Remark 1.3: (1) If w ≡ 1, then Mp,ϕ(1) = Mp,ϕ is the generalized Morrey space.(2) If φ(x,r)w(B(x,r))κ1p,thenMp,φ(w)=Lp,κ(w) is the weighted Morrey space.(3) If φ(x,r)v(B(x,r))κpw(B(x,r))1p,thenMp,φ(w)=Lp,κ(v,w) is the two weighted Morrey space.(4) If w1andφ(x,r)=rλnpwith0<λ<n,thenMp,φw=Lp,λRn is the classical Morrey space and WMρ,ϕ(w) = WLp(ℝn) is the weak Morrey space.(5) If φ(x,r)w(B(x,r))1p,thenMp,φ,(w)=Lp,w(Rn) is the weighted Lebesgue space.The commutators are useful in many nondivergence elliptic equations with discontinuous coefficients, [15-17, 26, 27, 41]. In the recent development of commutators, Pérez and Trujillo-González [42] generalized these multilinear commutators and proved the weighted Lebesgue estimates. Ye and Zhu in [45] obtained the boundedness of the multilinear commutators in weighted Morrey spaces Lp,k(w) for 1 < p < ∞ and 0 < κ < 1, where the symbol b belongs to bounded mean oscillation (BMO)n. Furthermore, the weighted weak type estimate of these operators in weighted Morrey spaces is Lp, k(w) for p = 1 and 0 < κ < 1. The following statement was proved by Guliyev in [24].

Theorem 1.4: ([24]). Let 0 < α < n, 1 < p < n/α and 1/q = 1/p — α/n, Ω ∈ L(𝕊n-1), wAp,q, A ∈ BMO(ℝn), and (ϕ1, ϕ2) satisfies the condition rlnke+tresssupt<s<φ1(x,s)wp(B(x,s))1pwq(B(x,t))1qdttCφ2(x,r),(1)where C does not depend on χ and r. Then the operator TΩ,αA,k is bounded from Mp,φ1wptoMq,φ2wq.It has been proved by many authors that most of the operators which are bounded on a weighted (unweighted) Lebesgue space are also bounded in an appropriate weighted (unweighted) Morrey space, see [8, 44]. As far as we know, there is no research regarding boundedness of the fractional multilinear integral operator on Morrey space. In this paper, we are going to prove that these results are valid for the rough fractional multilinear integral operator TΩ,αA1,A2,,Ak on generalized weighted Morrey space. Our main results can be formulated as follows.

Theorem 1.5: Let 0 < α < n, 1 ≤ s' < p < n/α and 1/q = 1/ p — α/n. Suppose that Ω is homogeneous of degree zero with Ω ∈ Ls(Sn-1) and (ϕ12) satisfy the condition (1). Let also, for 1 ≤ jk, |γj| = mj - 1, mj ≥ 2 and DγjAjBMORn. Suppose wsAps,qs, then the operator TΩ,αA1,A2,,Ak is bounded from Mp,φ1wptoMq,φ2wq. Moreover, then there is a constant C > 0 independent of f and A1, A2, ... , Ak such that TΩ,αA1,A2,,AkfMq,φ2(wq)Cj=1k|γj|=mj1DγjAjfMp,φ1(wp).In the case mj = 1 and Aj = A for j = 1,..., k from the Theorem 1.5 we get the Theorem 1.4. Also, in the case ω ≡ 1 we get the following corollary, which was proved in [1].

Corollary 1.6: ([1]). Let 0 < a < n, 1 ≤ s' < p < n/α and 1/q = 1/p — α/n. Suppose that Ω is homogeneous of degree zero with Ω ∈ Ls(Sn-1), and (ϕ1, ϕ2) satisfy the condition rlnke+tressinft<τ<φ1(x,τ)τnptnqdttC0φ2(x,r),where C0 does not depend on x and r. Let also, for 1 ≤ jk, |γj| = mj - 1, mj ≥ 2 and DγjAjBMORn. Then the operator TΩ,αA1,A2,,Ak is bounded from Mp,φ1RntoMq,φ2Rn. Moreover, there is a constant C > 0 independent of f such that TΩ,αA1,A2,,AkfMq,φ2Cj=1k|γj|=mj1DγjAjfMp,φ1.

Example 1.7: Let φ1(x,t)=(wq(B(x,t)))kp(wp(B(x,t)))1p,φ2(x,t)=(wq(B(x,t)))kp1q,0<k<p/q and wq A(ℝn). Then (ϕ1, ϕ2) satisfies the condition (1).In fact, from (2.8) in Section 2 we have constant δ > 0 such that wq(B(x,2jr))C2δjwq(B(x,r)).Since 0 < κ < p/q, then κp1q<0. Thus r1+lntressinft<τ<,φ1(x,τ)(wp(B(x,τ)))1p(wq(B(x,t)))1qdtt=r1+lntr(wq(B(x,t)))κp1qdttj=0(1+j)2jr2j+1r(wq(B(x,t)))κp1qdttCj=0(1+j)(wq(B(x,2jr)))κp1qCj=0(1+j)2δj(κp1q)(wq(B(x,r)))κp1qC(wq(B(x,r)))κp1q=C,φ2(x,r).If wsAps,qs, then by Lemma 2.2 in Section 2 we know wqA1+q/pRn. Therefore, we have the following corollaries.

Corollary 1.8: Let 0 < α < n, let 1 ≤ s' < p < n/α, and let 1/q = 1/p — α/n. Let also, for 1 ≤ j ≤ k, |γj| = mj - 1, mj 2 and DγjAjBMORn. Suppose wsAps,qs,thenTΩ,αA1,A2,,Ak is bounded from Lp,k(wp,wq)(ℝn) to Lq,k q/p(wq,ℝn) and TΩ,αA1,A2,,AkfLq,κq/p(wq,Rn)Cj=1k|γj|=mj1DγjAjfLp,k(wp,wq)(Rn),where the constant C > 0 is independent of f and A1, A2 ,..., Ak.

Remark 1.9: Note that, in [2] the Nikolskii-Morrey type spaces were introduced and the authors studied some embedding theorems. In the next paper, we shall introduce the generalized weighted Nikolskii-Morrey spaces and will study some embedding theorems. We will also investigate the boundedness of fractional multilinear integral operators with rough kernels TΩ,αA1,A2,,Ak on the generalized weighted Nikolskii-Morrey spaces, see for example, [30]. These results may be applicable to some problems of partial differential equations; see for example [6, 7, 19, 20, 26, 28, 30, 43].

2 Some preliminaries

We begin with some properties of Ap weights which play a great role in the proofs of our main results. A weight w is a nonnegative, locally integrable function on ℝn. Let B = B(x0,rB) denote the ball with the center x0 and radius rB. For a given weight function w and a measurable set E, we also denote the Lebesgue measure of E by | E | and set weighted measure w(E)=Ew(x)dx. For any given weight function w on ℝn, Ω ⊆ ℝn and 0 < p < ∞, denote by Lp,w (Ω) the space of all function ƒ satisfying fLp,wΩ=Ωfxpwxdx1p<.

A weight w is said to belong to Ap for 1 < p < ∞, if there exists a constant 1|B|Bw(x)dx1|B|Bw(x)1pdxp1C,

where p' is the dual of ρ such that 1p+1p=1 The class α1 is defined by 1|B|Bw(y)dyCessinfxBw(x)foreveryballBRn.

A weight w is said to belong to A(ℝn) if there are positive numbers C and δ so that w(E)w(B)C|E||B|δ

for all balls B and all measurable E C B. It is well known that A=1p<Ap.

The classical Ap weight theory was first introduced by Muckenhoupt in the study of weighted Lp -boundedness of Hardy-Littlewood maximal function in [37].

Lemma 2.1: ([21, 37]). Suppose w ∈ Ap and the following statements hold.(i) For any 1 ≤ p < ∞, there is a positive number C such that w(Bk)w(Bj)C2np(kj)fork>j(ii) For any 1 ≤ p < ∞, there is a positive number C and S such that w(Bk)w(Bj)C2δ(kj)fork>j(2)(iii) For any 1 < p < ∞, one has w1pAp.We also need another weight class Ap,q introduced by Muckenhoupt and Wheeden in [38] to study weighted boundedness of fractional integral operators.Given 1 ≤ pq < ∞. We say that ωAp,q if there exists a constant C such that for every ball B ⊂ ℝn, the inequality 1|B|Bw(y)pdy1/p1|B|Bw(y)qdy1/qC(3) holds when 1 < p < ∞, and for every ball B ⊂ ℝn the inequality 1|B|Bw(y)qdy1/qCessinfxBw(x)holds when p = 1.By (3), we have Bw(y)pdy1/pBw(y)qdy1/qC|B|1/p+1/q.(4)We summarize some properties about weights Ap,q; see [21, 38].

Lemma 2.2: Given 1 ≤ pq < ∞.(i) w ∈ Ap,q if and only if wqA1+q/p;(ii) w ∈ Ap,q if and only if wpA1+p/q;(iii) w ∈ Ap,p if and only if wp Ap;(iv) If p1 < p2 and q2 > q1, then Ap1,q1Ap2,q2.In this paper, we need the following statement on the boundedness of the Hardy type operator (H1g)(t):=1t0tln(e+tr)g(r)dμ(r),0<t<,where μ is a non-negative Borei measure on (0, ).

Theorem 2.3: The inequality esssupt>0w(t)H1g(t)cesssupt>0v(t)g(t)holds for all non-negative and non-increasing g on (0, ) if and only if A1:=supt>0w(t)t0tln(e+tr)dμ(r)esssup0<s<rv(s)<,and c ≈ A1.Note that Theorem 2.3 is proved analogously to Theorem 4.3 in [24, 25].

Lemma 2.4: ([39, Theorem 5, p. 236]). Let wA. Then the norm of BMO(ℝn) is equivalent to the norm of BMO(w), where BMO(w)={b:b,w=supxRn,r>01w(B(x,r))B(x,r)|b(y)bB(x,r),w|w(y)dy<}and bB(x,r),w=1w(B(x,r))B(x,r)b(y)w(y)dy.

Remark 2.5: ([24]).(1) The John-Nirenberg inequality : there are constants C1,C2 > 0, such that for all bBMO(ℝn) and β > 0 |{xB:|b(x)bB|>β}|C1|B|e,C2β/||b||,BRn.(2) For 1 < p < ∞ the John-Nirenberg inequality implies that bsupB1|B|B|b(y)bB|pdy1pand for 1 < ρ < ∞ and w ∈ A bsupB1w(B)B|b(y)bB|pw(y)dy1p.The following lemma was proved by Guliyev in [24].

Lemma 2.6: ([24]).i) Let w ∈ A and b be a function in BMO(n). Let also 1 p < ∞, x ∈ n, and r1,r2 > 0. Then 1w(B(x,r1))B(x,r1)|b(y)bB(x,r2),w|pw(y)dy1pC1+lnr1r2b,where C > 0 is independent of f, x, r1 and r2.ii) Let w ∈ Ap and b be a function in BMO(n). Let also 1 < p < ∞, x ∈n, and r1,r2 > 0. Then 1w1p(B(x,r1))B(x,r1)|b(y)bB(x,r2),w|pw(y)1pdy1pC1+lnr1r2b,where C > 0 is independent of f, x, r1 and r2.Below we present some conclusions about Rm(A;x, y).

Lemma 2.7: ([22]). Suppose b is a function on ℝn with the m-th derivatives in Lq(ℝn), q > n. Then |Rm(b;x,y)|C|xy|m|γ|=m1B(x,5n|xy|)B(x,5n|xy|)|Dγb(z)|dz1/q.The following property is valid.

Lemma 2.8: Let xB(x0, r), yB(x0, 2j+1r)\B(x0, 2jr). Assume that A has derivatives of order m - 1 in BMO(ℝn). Then there exists a constant C, independent of A, such that RmA;x,y C|xy|m1|γ|=m1DγA+|γ|=m1|DγA(y)(DγA)B(x0,r)|.(5)

Proof: For fixed x ∈ ℝn, let A¯(x)=A(x)γ=m11γ!(DγA)B(x,5n|xy|)xγ.Then |Rm(A;x,y)|=|Rm(A¯;x,y)||Rm1(A¯;x,y)|+|γ|=m11γ!|(DγA¯(y))||xy|m1.(6)From Lemma 2.7 we have, |Rm1(A¯;x,y)|C|xy|m1|γ|=m1DγA.(7)When x B(x0,r), y ∈ B(x0, 2j+1r)\B(x0, 2jr), then 2j-1 r ≤ |x - y| ≤ 2j+2 r. Thus, we have B(x0,2j1r)B(x,5n|xy|)100nB(x0,2jr).Then |100nB(x0,2jr)||B(x,5n|xy|)||100nB(x0,2jr)||B(x0,2j1r)|C.Hence |(DγA)B(x,5n|xy|)(DγA)B(x0,2jr)|1|B(x,5n|xy|)|B(x,5n|xy|)|DγA(y)(DγA)B(x0,2jr)|dy1|100nB(x0,2jr)|100nB(x0,2jr)|DγA(y)(DγA)B(x0,2jr)|dyC||DγA||.Note that |(DγA)B(x0,2jr)(DγA)B(x0,r)|k=1j|(DγA)B(x0,2kr)(DγA)B(x0,2k1r)|2njDγA.Then |(DγA)B(x,5n|xy|)(DγA)B(x0,r)||(DγA)B(x,5n|xy|)(DγA)B(x0,2jr)|+|(DγA)B(x0,2jr)(DγA)B(x0,r)|Cj||DγA||.Thus |DγA¯(y)|=|DγA(y)(DγA)B(x,5n|xy|)||DγA(y)(DγA)B(x0;r)|+|(DγA)B(x,5n|xy|)(DγA)B(x0;r)||DγA(y)(DγA)B(x0;r)|+Cj||DγA||.(8)Combining with (6), (7) and (8), then (5) is proved.Finally, we present a relationship between essential supremum and essential infimum.

Lemma 2.9: ([10]). Let f be a real-valued nonnegative function and measurable on E. Then essinfxEfx1=esssupxE1fx.

3 A local weighted Guliyev type estimates

In the following theorem we get local weighted Guliyev type estimate (see, for example, [22, 23] in the case w = 1, m = 1 and [24] in the case w Ap, m = 1) for the operator TΩ,αA,m.

Theorem 3.1: Let 1 ≤ s' <p <n/α, and let 1/q = 1/p – α/n. Let also, for 1 ≤ jk, |γj | = mj 1, mj ≥ 2 and DγjAjBMO(Rn). AjBMO. (ℝn) Suppose that Ω is homogeneous of degree zero with Ω ∈ Ls(Sn-1), wsAps,qs, then for any r > 0, there is a constant C independent of f such that TΩ,αA1,A2,,AkfLq,wq(B(x0,r))Cj=1k|γj|=mj1DγjAj(wq(B(x0,r)))1q×2r(1+lntr)kfLp,wp(B(x0,t))(wq(B(x0,t)))1qdtt.(9)

Proof: We write f as f = f1 + f2, where f1(y)=f(y)χB(x0,2r)(y),χB(x0,2r) denotes the characteristic function of B(x0,2r) Then TΩ,αA1,A2,,AkfLq,wq(B(x0,r))<_TΩ,αA1,A2,,Akf1Lq,wq(B(x0,r))+TΩ,αA1,A2Since f1Lp,wp(Rn),, by the boundedness of TΩ,αA1,A2,,Ak from Lp,wp(Rn)toLq,wq(Rn) (Theorem 1.1) we get TΩ,αA1,A2,,Akf1Lq,wq(B(x0,r))TΩ,αA1,A2,Akf1Lq,wq(Rn)Cj=1k|γj|=mj1DγjAjf1Lp,wp(Rn)Cj=1k|γj|=mj1DγjAjfLp,wp(B(x0,2r)).Note that q > p > 1 and spp(ps)1, then by Hölder’s inequality, 1(1|B|Bw(y)pdy)1p(1|B|Bw(y)pdy)1p(1|B|Bw(y)qdy)1q(1|B|Bw(y)sppsdy)pssp.This means rnsα<_(wq(B(x0,r)))1qw1Lspps(B(x0,r))Then fLp(wp,B(x0,2r))CrnsαfLp,wp(B(x0,2r))2rtαns1dtC(wq(B(x0,r)))1qw1Lspps(B(x0,r))2rfLp,wp(B(x0,t))tαns1dt<_C(wq(B(x0,r)))1q2rfLp,wp(B(x0,t))w1Lspps(B(x0,t))tαns1dt.Since wsAps,qs by (4), for all r > 0 we get (wq(B(x0;r)))1qw1Lspps(B(x0;r))<_Crnsα.(10)Then TΩ;αA1;A2;;Akf1Lq,wq(B(x0;r))Cj=1k|γj|=mj1DγjAj(wq(B(x0;r)))1q2rfLp,wp(B(x0;t))(wq(B(x0;t)))1qdtt.To simplify process of Theorem 3.1, in the following discussion we consider only the case k = 2. The method can be used to deal with the case k >2 without any essential difficulty.Let N = m1 + m2 – 2, Δi = (B(x0, 2i+1 r)) \ (B(x0, 2i r)), and let xB(x0, r)) By Lemma 2.8, |TΩ,αA1,A2,,Akf2(x)|(B(x0,2r))c|Ω(xy)f(y)||xy|nα+m1Rm1(A1;x,y)Rm2(A2;x,y)f(y)dyi=1Δi|Ω(xy)f(y)||xy|nαj=12j+|γj|=mj1|DγjAj(y)(DγjAj)B(x0,r)|dyCj=i2j|γj|=mj1||DγjAj||i=1Δi|Ω(xy)f(y)||xy|nαdy+C|γ1|=m11||Dγ1A1|||γ2|=m21i=1Δi|Ω(xy)f(y)||xy|nα|Dγ2A2(y)(Dγ2A2)B(x0,r)|dy+C|γ2|=m21||Dγ2A2|||γ1|=m11i=1Δi|Ω(xy)f(y)||xy|nα|Dγ1A1(y)(Dγ1A1)B(x0,r)|dy+C|γ1|=m11|γ2|=m21i=1Δi|Ω(xy)f(y)||xy|nαj=12|DγjAj(y)(DγjAj)B(x0,r)|dyC(I1+I2+I3+I4).By Hölder’s inequalities, i|Ω(xy)f(y)||xy|nαdyi|Ω(xy)|sdy1si|f(y)|s|xy|(nα)sdy1s.When x ∈ B(x0, s) and y ∈ Δi, then by a direct calculation, we can see that 2i-1 r ≤|y–x| < 2i-1 r. Hence |Ω(xy)|sdy1sCΩLS(Sn1)|B(x0;2i+1r)|1s.(11)We also note that if xB(x0, r), yB(x0, 2r)c, then |yx| ≈ |yx0|. Consequently i|f(y)|s|xy|(nα)sdy1s1|B(x0;2i+1r)|1α/nB(x0;2i+1r)|f(y)|sdy1s.(12)Then I1Cj=12|γj|=mj1DγjAji=1j(2i+1r)αns(B(x0;2i+1r)|f(y)|sdy)1s.Since s' <p, it follows from Hölder’s inequality that B(x0;2i+1r)|f(y)|sdy1sCfLp,wp(B(x0;2i+1r))w1Lspps(B(x0;2i+1r)).Then I1Cj=12|γj|=mj1DγjAji=1j(2i+1r)αns(B(x0;2i+1s)|f(y)|qdy)1q.<_Cj=12|γj|=m+j1DγjAji=1(1+ln2i+1rr)(2i+1r)αnsfLp,wp(B(x0,2i+1r))w1Lspps(B(x0,2i+1r))<_Cj=12|γj|=mj1DγjAji=12iX2i+1r2i+2r(1+lntr)fLp,wp(B(x0,t))w1Lspps(B(x0,t))tαns1dt<_Cj=12|γj|=mj1DγjAj2r1(1+lntr)fLp,wp(B(x0,t))w1Lspps(B(x0,t))tαns1dt.From (10) we know w1Lspps(B(x0,r))<_Crnsα(wq(B(x0,r)))1q.(13)Then I1<_Cj=12|γj|=mj1DγjAj2r(1+lntr)fLp,wp(B(x0,t))(wq(B(x0,t)))1qdtt.On the other hand, by Hölder’s inequality and (11), (12), we have i|Ω(xy)f(y)||xy|nα|Dγ2A2(y)(Dγ2A2)B(x0,r)|dyi|Ω(xy)|sdy1si|Dγ2A2(y)(Dγ2A2)B(x0;r)f(y)|s|xy|(nα)sdy1sCi=1(2i+1r)αns(x0,2i+1r)|DγA(y)(DγA)B(x0,r)|s|f(y)|sdy1s.Applying Hölder’s inequality we get B(x0,2i+1r)|Dγ2A2(y)(Dγ2A2)B(x0,r)|s|f(y)|sdy1sCfLp,wp(B(x0,2i+1r))(Dγ2A2(y)(Dγ2A2)B(x0,r))w()1Lpsps(B(x0,2i+1r)).Consequently, I2C|γ1|=m11Dγ1A1|γ2|=m21i=1X2i+1r2i+2r(2i+1r)αns)αnsfLp,wp(B(x0t))×(Dγ2A2(y)(Dγ2A2)B(x0,r))ω()1Lpsps(B(x0,t))dtC||γ1=m11Dγ1A1|γ2|=m212rfLp,wp(B(x0,t))×(Dγ2A2(y)(Dγ2A2)B(x0,r))w()1Lpsps(B(x0,t))tαns1dt.By wsAps,qs and (ii) of Lemma 2.2 we know wsppsA1+ps(ps)q.. Then it follows from the Lemma 2.6 and the inequality (13) that (Dγ2A2(y)(Dγ2A2)B(x0,r))ω()1Lpsps(B(x0,t))(B(x0,t)|Dγ2A2(y)(Dγ2A2)B(x0,r)|pspswpsps(y)dy)psps<_CDγ2A2(1+lntr)(wpsps(B(x0,r)))psps=CDγ2A2(1+lntr)w1Lpsps(B(x0,r))<_CDγ2A2(1+lntr)rnsα(wq(B(x0,r)))1q.Then I2<_Cj=12|γj|=mj1DγjAj2r(1+lntr)fLp,wp(B(x0,t))(wq(B(x0,t)))1qdtt.Similarly to the estimates for I2, we have I3Cj=12|γj|=mj1DγjAj2r1(1+lntr)fLp,wp(B(x0;t))(wq(B(x0;t)))1qdtt.Finally, we come to estimate I4.By Hölder’s inequality and (11), (12), we have i|Ω(xy)f(y)||xy|nαj=12|DγjAj(y)(DγjAj)B(x0,r)|dy(|Ω(xy)|sdy)1s(ij=12|DγjAj(y)(DγjAj)B(x0,r)|s|xy|(nα)sdy)1sCi=1(2i+1r)αnsB(x0,2i+1r)j=12|DγjAj(y)(DγjAj)B(x0,r)|s|f(y)|sdy1s.Applying Hölder’s inequality we get (x0,2i+1r)j=12|DγjAj(y)(DγjAj)B(x0,r)|s|f(y)|sdy1sCfLp,wp(B(x0,2i+1r))j=12(DγjAj(y)(DγjAj)B(x0,r))w()1Lpsps(B(x0,2i+1r))CfLp,wp(B(x0,2i+1r))j=12(DγjAj(y)(DγjAj)B(x0,r))w()1/2L2psps(B(x0,2i+1r)).Then i=1i|Ω(xy)f(y)||xy|nαj=12|DγjAj(y)(DγjAj)B(x0,r)|dyC2rfLp,wp(B(x0,t))j=12(DγjAj(y)(DγjAj)B(x0,r))w()1/2L2psps(B(x0,2i+1r)). Since wsppsA1+ps(ps)q,, then from the Lemma2.6 and the inequality (13) we have (DγjAj(y)(Dγ2A2)B(x0,r))ω()1Lpsps(B(x0,t))(B(x0,t)|Dγ2A2(y)(Dγ2A2|pspswpsps(y)dy)pspsCDγ2A2(1+lntr)(wpsps(B(x0,r)))psps=CDγ2A2(1+lntr)w1Lpsps(B(x0,r))CDγ2A2(1+lntr)rnsα(wq(B(x0,r)))1q.(14)Then from (14) we have i=1i|Ω(xy)f(y)||xy|nαj=12|DγjAj(y)(DγjAj)B(x0,r)|dyCj=12DγjAj2r(1+lntr)2fLp,wp(B(x0,t))(wq(B(x0,r)))1qdrr.Combining with the estimates of I1, I2, I3 and I4, we have supxB(x0,r)|TΩ,αA1,A2,,Akf2(x)|Cj=12|γj|=mj1DγjAj2r1(1+lntr)2fLp,wp(B(x0,t))(wq(B(x0,t)))1qdtt.Then we get TΩ,αA1,A2,,Akf2Lq,wq(B(x0,r))<_Cj=12|γj|=mj1DγjAj(wq(B(x0,r)))1q×2r1(1+lntr)2fLp,wp(B(x0,t))(wq(B(x0,t)))1qdtt.This completes the proof of Theorem 3.1.

4 Proof of Theorem 1.5

First variant proof of Theorem 1.5

By Theorems 2.3 and 3.1 we have TΩ,αA1,A2,,AkfMq,φ(wq)=supx0Rn,r>0φ2(x0,r)1(wq(B(x0,r)))1qTΩ,αA1,A2,,AkfLq,wq(B(x0,r))2Csupx0Rn,rgt0φ2(x0,r)1r(1+lntr)kfLp,wp(B(x0,t))(wq(B(x0,t)))1qdttCsupx0Rn,r>0,φ1(x0,r)1wp(B(x,r)1pfLp,wp(B(x,r))=fMp,φ1(wp).

Second variant proof of Theorem 1.5

Since fMp,φ1wp, then by Lemma 2.9 and the fact fLp,wp(B(x0,t)) is a non-decreasing function of t, we get fLp,wp(B(x0,t))essinf0<τ<t<,φ1(x0,τ)(wp(B(x0,τ)))1pesssup0<τ<t<fLp,wp(B(x0,t))φ1(x0,τ)(wp(B(x0,τ)))1psupτ>0,x0RnfLp,wp(B(x0,τ))φ1(x0,τ)(wp(B(x0,τ)))1pfMp,φ(wp)1

Since (ϕ1, ϕ2) satisfies (1), we have r(1+lntr)kfLp,wp(B(x0,t))(wq(B(x0,t)))1qdttrf|Lp,wp(B(x0,t))essinf0<τ<t<,φ1(x0,τ)(wp(B(x0,τ)))1p(1+lntr)kessinf0<τ<t<,φ1(x0,τ)(wp(B(x0,τ)))1p(wq(B(x0,t)))1qdttCfMp,φ1(wp)1r(1+lntr)kessinf0<τ<t<,φ1(x0,τ)(wp(B(x0,τ)))1p(wq(B(x0,t)))1qdttCfMp,φ1(wp)φ2(x0,t).

Then by (9) we get TΩ,αA1,A2,,AkfMq,φ2(wq)Csupx0Rn,t>01φ2(x0,t)(1wq(B(x0,t))B(x0,t)|TΩ,αA1,A2,,Akf(y)|qwq(y)dy)1/qCj=1k|γj|=mj1DγjAjsupx0Rn,t>01φ2(x0,t)×r(1+lntr)kfLp,wp(B(x0,t))(wq(B(x0,t)))1qdttCj=1k|γj|=mj1DγjAjfMp,φ1(wp).

Acknowledgement

The authors would like to thank the referees for careful reading the paper and useful comments.

The research of A. Akbulut was partially supported by the grant of Ahi Evran University Scientific Research Project (FEF.A3.16.023).

References

  • [1]

    Akbulut A., Hamzayev V.H., Safarov Z.V., Rough fractional multilinear integral operators on generalized Morrey spaces, J. Inequal. Appl., 2015, 2015:234, 12 pp.

  • [2]

    Akbulut A., Eroglu A., Najafov A.M., Some embedding theorems on the Nikolskii-Morrey type spaces, Advances in Analysis, 2016, 1(1), 18-26.

  • [3]

    Bajsanski B., Coifman R., On singular integrals, Proc. Symp. Pure Math., Amer. Math. Soc. Providence, R. I., 1996, 10, 1-17.

  • [4]

    Belgacem F.B.M., Elliptic boundary value problems with indefinite weights: Variational formulations of the principal eigenvalue and applications, Pitman Research Notes Series, Vol. 368, New York, USA 1997.

  • [5]

    Belgacem, F.B.M., Theory and application with weight indefinite elliptic boundary value problems, International Journal on the Problems of Nonlinear Analysis in Engineering Systems, 1999, 5(2), 51-58.

  • [6]

    Byun, Sun-Sig, Cho, Y., Palagachev, D.K., Global weighted estimates for nonlinear elliptic obstacle problems over Reifenberg domains, Proc. Amer. Math. Soc., 2015, 143(6), 2527-2541.

  • [7]

    Byun, Sun-Sig, Softova, L.G., Gradient estimates in generalized Morrey spaces for parabolic operators, Math. Nachr., 2015, 288(14-15), 1602-1614.

  • [8]

    Chiarenza F., Frasca M., Morrey spaces and Hardy-Littlewood maximal function, Rend Math. Appl., 1987, 7, 273-279.

  • [9]

    Cohen J., A sharp estimate for a multilinear singular integral in Rn, Indiana Unit. Math. Jour., 1981, 30, 693-702.

  • [10]

    Cohen J., Gosselin J., A BMO estimate for multilinear singular integrals, Illinois J. Math., 1986, 30, 445-464.

  • [11]

    Ding Y., Lu S.Z., Higher order commutators for a class of rough operators, Ark. Mat., 1999, 37, 33-44.

  • [12]

    Ding Y., Lu S.Z., Weighted boundedness for a class of rough multilinear operators, Acta Math. Sin. (Engl. Ser.), 2001, 17, 517-526.

  • [13]

    Ding Y., A note on multilinear fractional integrals with rough kernel, Adv. Math. (China), 2001, 30, 238-246.

  • [14]

    Ding Y., Lu S., Higher order commutators for a class of rough operators, Ark. Mat., 1999, 37, 33-44.

  • [15]

    Chiarenza F., Frasca M., Longo P., Interior W2,p -estimates for nondivergence elliptic equations with discontinuous coefficients, Ric. Mat., 1991, 40(1), 149-168.

  • [16]

    Chiarenza F., Frasca M., Longo P., W2,p -solvability of Dirichlet problem for nondivergence elliptic equations with VMO coefficients, Trans. Amer. Math. Soc., 1993, 336(2), 841-853.

  • [17]

    Di Fazio G., Ragusa M.A., Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients, J. Funct. Anal., 1993, 112, 241-256.

  • [18]

    Fan D., Lu S., Yang D., Boundedness of operators in Morrey spaces on homogeneous spaces and its applications, Acta Math. Sinica (N. S.), 1998, 14, 625-634.

  • [19]

    Fua Z.J., Chena W., Yang H.T., Boundary particle method for Laplace transformed time fractional diffusion equations, J. Comput. Phy., 2013, 235(15), 52-66.

  • [20]

    Fu Z.J., Chen W., Ling L., Method of approximate particular solutions for constant- and variable-order fractional diffusion models. Eng. Anal. Bound. Elem., 2015, 57, 37-46.

  • [21]

    Garca-Cuerva J., Rubio de Francia J.L., Weighted Norm Inequalities and Related Topics, vol. 116 of North-Holland Mathematics Studies, North-Holland, Amsterdam, The Netherlands, 1985.

  • [22]

    Guliyev V.S., Integral operators on function spaces on the homogeneous groups and on domains in Rn, Doctoral dissertation, Moscow, Mat. Inst. Steklov, 1994, 329 pp. (in Russian)

  • [23]

    Guliyev V.S., Boundedness of the maximal, potential and singular operators in the generalized Morrey spaces, J. Inequal. Appl., Art. ID 503948, 2009, 20 pp.

  • [24]

    Guliyev V.S., Generalized weighted Morrey spaces and higher order commutators of sublinear operators, Eurasian Math. J., 2012, 3, 33-61.

  • [25]

    Guliyev V.S., Local generalized Morrey spaces and singular integrals with rough kernel, Azerb. J. Math., 2013, 3(2), 79-94.

  • [26]

    Guliyev V.S., Softova L., Global regularity in generalized Morrey spaces of solutions to nondivergence elliptic equations with VMO coefficients, Potential Anal., 2013, 38(4), 843-862.

  • [27]

    Guliyev V.S., Softova L., Generalized Morrey regularity for parabolic equations with discontinuity data, Proc. Edinb. Math. Soc., 2015, 58(1), 199-218.

  • [28]

    Guliyev V.S., Softova L., Generalized Morrey estimates for the gradient of divergence form parabolic operators with discontinuous coefficients, J. Differential Equations, 2015, 259 (6), 2368-2387.

  • [29]

    Guliyev V.S., Karaman T., Mustafayev R.Ch., Serbetci A., Commutators of sublinear operators generated by Calderón-Zygmund operator on generalized weighted Morrey spaces, Czechoslovak Math. J., 2014, 64(2), 365-386.

  • [30]

    Guliyev V.S., Muradova Sh., Omarova M.N., Softova L., Generalized weighted Morrey estimates for the gradient of divergence form parabolic operators with discontinuous coefficients, Acta Math. Sin. (Engl. Ser.), 2016, 32 (8), 911-924.

  • [31]

    Guliyev V.S., Omarova M.N., Multilinear singular and fractional integral operators on generalized weighted Morrey spaces, Azerb. J. Math., 2015, 5 (1), 104-132.

  • [32]

    Hofmann S., On some nonstandard Calderón-Zygmund operators, Studia Math., 1994, 109, 105-131.

  • [33]

    Karaman T., Guliyev V.S., Serbetci A., Boundedness of sublinear operators generated by Calderon-Zygmund operators on generalized weighted Morrey spaces, Scienti c Annals of “Al.I. Cuza” University of Iasi, 2014, 60(1), 227-244.

  • [34]

    Komori Y., Shirai S., Weighted Morrey spaces and a singular integral operator, Math. Nachr., 2009, 289, 219-231.

  • [35]

    Mizuhara T., Boundedness of some classical operators on generalized Morrey spaces, Harmonic Analysis (S. Igari, Editor), ICM 90 Satellite Proceedings, Springer - Verlag, Tokyo, 1991, 183-189.

  • [36]

    Morrey C.B., On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc., 1938, 43, 126-166.

  • [37]

    Muckenhoupt B., Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc., 1972, 165, 207-226.

  • [38]

    Muckenhoupt B., Wheeden R., Weighted norm inequalities for singular and fractional integrals, Trans. Amer. Math. Soc., 1971, 161, 249-258.

  • [39]

    Muckenhoupt B., Wheeden R., Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc., 1974, 192, 261-274.

  • [40]

    Mustafayev, R. Ch., On boundedness of sublinear operators in weighted Morrey spaces, Azerb. J. Math., 2012, 2 (1), 66-79.

  • [41]

    Palagachev D.K., Softova L.G., Singular integral operators, Morrey spaces and fine regularity of solutions to PDE’s, Potential Anal., 2004, 20, 237-263.

  • [42]

    Pérez C., Trujillo-González R., Sharp weighted estimates for multilinear commutators, J. London Math. Soc., 2002, 65, 672-692.

  • [43]

    Sun H., Chen W., Li C., Chen Y., Finite difference schemes for variable-order time fractional diffusion equation. Int. J. Bifurcation Chaos 22, 1250085, 2012, 16 pp.

  • [44]

    Takeshi I., Enji S., Yoshihiro S., Hitoshi T., Weighted norm inequalities for multilinear fractional operators on Morrey spaces, Studa Math., 2011, 205, 139-170.

  • [45]

    Ye X.F., Zhu X.S., Estimates of singular integrals and multilinear commutators in weighted Morrey spaces, J. Ineq. Appl., 2012, 2012:302, 13 pp.

About the article

Received: 2016-04-01

Accepted: 2016-11-02

Published Online: 2016-12-17

Published in Print: 2016-01-01



Citation Information: Open Mathematics, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2016-0090. Export Citation

© 2016 Akbulut and Hasanov. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. (CC BY-NC-ND 3.0)

Comments (0)

Please log in or register to comment.
Log in