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# 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 ${T}_{\mathrm{\Omega },\alpha }^{{A}_{1},{A}_{2},\dots ,{A}_{k}},$ 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 ${T}_{\mathrm{\Omega },\alpha }^{{A}_{1},{A}_{2},\dots ,{A}_{k}},$ from ${M}_{p,{\phi }_{1}}\left({w}^{p}\right)\phantom{\rule{thinmathspace}{0ex}}\mathrm{t}\mathrm{o}\phantom{\rule{thinmathspace}{0ex}}{M}_{p,{\phi }_{2}}\left({w}^{q}\right)$ for 1 < p < q < ∞. In all cases the conditions for the boundedness of the operator ${T}_{\mathrm{\Omega },\alpha }^{{A}_{1},{A}_{2},\dots ,{A}_{k}},$ 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.

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 $TAf(x)=p.v.∫RnΩ(x−y)|x−y|n+m−1Rm(A;x,y)f(y)dy,$ 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|

Using the method of good — λ inequality, in 1986, Cohen and Gosselin [10] proved that if Ω ∈ Lip1 (Sn-1) and DγABMO(ℝn), then $∥TAf∥Lp≤C∑|γ|=m−1∥DγA∥BMO∥f∥Lp,1 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 ${T}_{\mathrm{\Omega }}^{A}.$ The purpose of [12] is to study this problem. Let us give the definition of the multilinear fractional integral operator as follows: $TΩ,αA1,A2,…,Akf(x)=∫RnΩ(x−y)|x−y|n−α+N∏j=1kRmj(Aj;x,y)f(y)dy,$

where $0<\alpha Ω is homogeneous of degree zero and $\mathrm{\Omega }\in {L}_{s}\left({S}^{n-1}\right),\phantom{\rule{thinmathspace}{0ex}}s>1,\phantom{\rule{thinmathspace}{0ex}}{R}_{{m}_{j}}\left({A}_{j};x,\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}y\right)$ is as above.

When k = 1 and m = 1, then ${T}_{\mathrm{\Omega },\alpha }^{A}$ is just the commutator of the fractional integral TΩ,αf with function A, $TΩ,αAf(x)=∫RnΩ(x−y)|x−y|n−α(A(x)−A(y))f(y)dy.$

When mj = 1 and Aj = A for j = 1,..., k, then ${T}_{\mathrm{\Omega },\alpha }^{{A}_{1},{A}_{2},\dots ,{A}_{k}}$ is just the higher-order commutator ${T}_{\mathrm{\Omega },\alpha }^{A,k}f$ given in [11], $TΩ,αA,kf(x)=∫RnΩ(x−y)|x−y|n−α(A(x)−A(y))kf(y)dy.$

When mj ≥ 2, ${T}_{\mathrm{\Omega },\alpha }^{{A}_{1},{A}_{2},\dots ,{A}_{k}}$ 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<\alpha and let Ω be homogeneous of degree zero with Ω ∈ Ls(Sn-1). Moreover,for 1 ≤ jk, |γj| = mj - 1, mj 2 and ${D}^{{\gamma }_{j}}{A}_{j}\in BMO\left({\mathbb{R}}^{n}\right).$ Then there exists a constant C, independent of Aj,1 ≤ jk and f, such that $∥TΩ,αA1,A2,…,Akf∥Lq,wq(Rn)≤C∏j=1k∑|γj|=mj−1∥DγjAj∥∗∥f∥Lp,wp(Rn).$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 $f\in {L}_{p,w}^{loc}\left({\mathbb{R}}^{n}\right)$ with finite norm $∥f∥Mp,φ(w)=supx∈Rn,r>0φ(x,r)−1w(B(x,r))−1p∥f∥Lp,w(B(x,r)),$where Lp,w(B(x, r)) denotes the weighted Lp-space of measurable functions f for which $∥f∥Lp,w(B(x,r))=∫B(x,r)|f(y)|pw(y)dy1p.$Furthermore, by WMp,ϕ(w) we denote the weak generalized weighted Morrey space of all functions $f\in W{L}_{p,w}^{loc}\left({\mathbb{R}}^{n}\right)$ for which $∥f∥WMp,φ,(w)=supx∈Rn,r>0φ(x,r)−1w(B(x,r))−1p∥f∥WLp,w(B(x,r))<∞,$where WLp,w(B(x, r)) denotes the weak Lp,w-space of measurable functions f for which $∥f∥WLp,w(B(x,r))=supt>0t∫{y∈B(x,r):|f(y)|>t}w(y)dy1p.$

Remark 1.3: (1) If w ≡ 1, then Mp,ϕ(1) = Mp,ϕ is the generalized Morrey space.(2) If $\phi \left(x,r\right)\equiv w\left(B\left(x,r\right){\right)}^{\frac{\kappa -1}{p}},\phantom{\rule{thinmathspace}{0ex}}then\phantom{\rule{thinmathspace}{0ex}}{M}_{p,\phi }\left(w\right)={L}_{p,\kappa }\left(w\right)$ is the weighted Morrey space.(3) If $\phi \left(x,r\right)\equiv v\left(B\left(x,r\right){\right)}^{\frac{\kappa }{p}}w\left(B\left(x,r\right){\right)}^{-\frac{1}{p}},\phantom{\rule{thinmathspace}{0ex}}then\phantom{\rule{thinmathspace}{0ex}}{M}_{p,\phi }\left(w\right)={L}_{p,\kappa }\left(v,w\right)$ is the two weighted Morrey space.(4) If $w\equiv 1\phantom{\rule{0.056em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}and\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{0.056em}{0ex}}\phi \left(x,r\right)={r}^{\frac{\lambda -n}{p}}with\phantom{\rule{thinmathspace}{0ex}}0<\phantom{\rule{thinmathspace}{0ex}}\lambda is the classical Morrey space and WMρ,ϕ(w) = WLp(ℝn) is the weak Morrey space.(5) If $\phi \left(x,r\right)\equiv w\left(B\left(x,r\right){\right)}^{-\frac{1}{p}},\phantom{\rule{thinmathspace}{0ex}}then\phantom{\rule{thinmathspace}{0ex}}{M}_{p,\phi },\left(w\right)={L}_{p,w}\left({\mathbb{R}}^{n}\right)$ 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 $\stackrel{\to }{b}$ 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 $∫r∞lnke+tresssupt(1)where C does not depend on χ and r. Then the operator ${T}_{\mathrm{\Omega },\alpha }^{A,k}$ is bounded from ${M}_{p,{\phi }_{1}}\left({w}^{p}\right)\phantom{\rule{thinmathspace}{0ex}}to\phantom{\rule{thinmathspace}{0ex}}{M}_{q,{\phi }_{2}}\left({w}^{q}\right).$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 ${T}_{\mathrm{\Omega },\alpha }^{{A}_{1},{A}_{2},\dots ,{A}_{k}}$ 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 ${D}^{{\gamma }_{j}}{A}_{j}\in \phantom{\rule{thinmathspace}{0ex}}BMO\left({\mathbb{R}}^{n}\right).$ Suppose ${w}^{{s}^{\prime }}\in {A}_{\frac{p}{{s}^{\prime }},\frac{q}{{s}^{\prime }}},$ then the operator ${T}_{\mathrm{\Omega },\alpha }^{{A}_{1},{A}_{2},\dots ,{A}_{k}}$ is bounded from ${M}_{p,{\phi }_{1}}\left({w}^{p}\right)\phantom{\rule{thinmathspace}{0ex}}to\phantom{\rule{thinmathspace}{0ex}}{M}_{q,{\phi }_{2}}\left({w}^{q}\right).$ Moreover, then there is a constant C > 0 independent of f and A1, A2, ... , Ak such that $∥TΩ,αA1,A2,…,Akf∥Mq,φ2(wq)≤C∏j=1k∑|γj|=mj−1∥DγjAj∥∗∥f∥Mp,φ1(wp).$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 $∫r∞lnke+tressinft<τ<∞φ1(x,τ)τnptnqdtt≤C0φ2(x,r),$where C0 does not depend on x and r. Let also, for 1 ≤ jk, |γj| = mj - 1, mj ≥ 2 and ${D}^{{\gamma }_{j}}{A}_{j}\in \phantom{\rule{thinmathspace}{0ex}}BMO\left({\mathbb{R}}^{n}\right).$ Then the operator ${T}_{\mathrm{\Omega },\alpha }^{{A}_{1},{A}_{2},\dots ,{A}_{k}}$ is bounded from ${M}_{p,{\phi }_{1}}\left({\mathbb{R}}^{n}\right)\phantom{\rule{thinmathspace}{0ex}}to\phantom{\rule{thinmathspace}{0ex}}{M}_{q,{\phi }_{2}}\left({\mathbb{R}}^{n}\right).$ Moreover, there is a constant C > 0 independent of f such that $∥TΩ,αA1,A2,…,Akf∥Mq,φ2≤C∏j=1k∑|γj|=mj−1∥DγjAj∥∗∥f∥Mp,φ1.$

Example 1.7: Let ${\phi }_{1}\left(x,t\right)=\left({w}^{q}\left(B\left(x,t\right)\right){\right)}^{\frac{k}{p}}\left({w}^{p}\left(B\left(x,t\right)\right){\right)}^{-\frac{1}{p}},{\phi }_{2}\left(x,t\right)=\left({w}^{q}\left(B\left(x,t\right)\right){\right)}^{\frac{k}{p}-\frac{1}{q}},0 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 $\frac{\kappa }{p}-\frac{1}{q}<0.$ Thus $∫r∞1+ln⁡tressinft<τ<∞,φ1(x,τ)(wp(B(x,τ)))1p(wq(B(x,t)))1qdtt=∫r∞1+ln⁡tr(wq(B(x,t)))κp−1qdtt≤∑j=0∞(1+j)∫2jr2j+1r(wq(B(x,t)))κp−1qdtt≤C∑j=0∞(1+j)(wq(B(x,2jr)))κp−1q≤C∑j=0∞(1+j)2δj(κp−1q)(wq(B(x,r)))κp−1q≤C(wq(B(x,r)))κp−1q=C,φ2(x,r).$If ${w}^{{s}^{\prime }}\in {A}_{\frac{p}{{s}^{\prime }},\frac{q}{{s}^{\prime }}},$ then by Lemma 2.2 in Section 2 we know ${w}^{q}\in {A}_{1+q/{p}^{\prime }}\left({\mathbb{R}}^{n}\right).$ 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 ${D}^{{\gamma }_{j}}{A}_{j}\in \phantom{\rule{thinmathspace}{0ex}}BMO\left({\mathbb{R}}^{n}\right).$ Suppose ${w}^{{s}^{\prime }}\in {A}_{\frac{p}{{s}^{\prime }},\frac{q}{{s}^{\prime }}},\phantom{\rule{thinmathspace}{0ex}}then\phantom{\rule{thinmathspace}{0ex}}{T}_{\mathrm{\Omega },\alpha }^{{A}_{1},{A}_{2},\dots ,{A}_{k}}$ is bounded from Lp,k(wp,wq)(ℝn) to Lq,k q/p(wq,ℝn) and $∥TΩ,αA1,A2,…,Akf∥Lq,κq/p(wq,Rn)≤C∏j=1k∑|γj|=mj−1∥DγjAj∥∗∥f∥Lp,k(wp,wq)(Rn),$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 ${T}_{\mathrm{\Omega },\alpha }^{{A}_{1},{A}_{2},\dots ,{A}_{k}}$ 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 $w\left(E\right)={\int }_{E}w\left(x\right)dx.$ 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 $1|B|∫Bw(x)dx1|B|∫Bw(x)1−p′dxp−1≤C,$

where p' is the dual of ρ such that $\frac{1}{p}+\frac{1}{{p}^{\prime }}=1$ The class α1 is defined by $1|B|∫Bw(y)dy≤C⋅essinfx∈B⁡w(x)foreveryballB⊂Rn.$

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 $A∞=⋃1≤p<∞Ap.$

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 $w(Bk)w(Bj)≤C2np(k−j)fork>j$(ii) For any 1 ≤ p < ∞, there is a positive number C and S such that $w(Bk)w(Bj)≥C2δ(k−j)fork>j$(2)(iii) For any 1 < p < ∞, one has ${w}^{1-{p}^{\prime }}\in {A}_{{p}^{\prime }}.$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 $1|B|∫Bw(y)−p′dy1/p′1|B|∫Bw(y)qdy1/q≤C$(3) holds when 1 < p < ∞, and for every ball B ⊂ ℝn the inequality $1|B|∫Bw(y)qdy1/q≤C⋅essinfx∈B⁡w(x)$holds when p = 1.By (3), we have $∫Bw(y)−p′dy1/p′∫Bw(y)qdy1/q≤C|B|1/p′+1/q.$(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 ${w}^{q}\in {A}_{1+q/{p}^{\prime }};$(ii) w ∈ Ap,q if and only if ${w}^{-{p}^{\prime }}\in {A}_{1+{p}^{\prime }/q};$(iii) w ∈ Ap,p if and only if wp Ap;(iv) If p1 < p2 and q2 > q1, then ${A}_{{p}_{1},{q}_{1}}\subset {A}_{{p}_{2},{q}_{2}}.$In this paper, we need the following statement on the boundedness of the Hardy type operator $(H1g)(t):=1t∫0tln⁡(e+tr)g(r)dμ(r),0where μ is a non-negative Borei measure on (0, ).

Theorem 2.3: The inequality $esssupt>0⁡w(t)H1g(t)≤cesssupt>0⁡v(t)g(t)$holds for all non-negative and non-increasing g on (0, ) if and only if $A1:=supt>0w(t)t∫0tln⁡(e+tr)dμ(r)esssup0and 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 $BMO(w)={b:∥b∥∗,w=supx∈Rn,r>01w(B(x,r))∫B(x,r)|b(y)−bB(x,r),w|w(y)dy<∞}$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 $|{x∈B:|b(x)−bB|>β}|≤C1|B|e,−C2β/||b||∗,∀B⊂Rn.$(2) For 1 < p < ∞ the John-Nirenberg inequality implies that $∥b∥∗≈supB1|B|∫B|b(y)−bB|pdy1p$and for 1 < ρ < ∞ and w ∈ A $∥b∥∗≈supB1w(B)∫B|b(y)−bB|pw(y)dy1p.$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 $1w(B(x,r1))∫B(x,r1)|b(y)−bB(x,r2),w|pw(y)dy1p≤C1+ln⁡r1r2∥b∥∗,$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 $1w1−p′(B(x,r1))∫B(x,r1)|b(y)−bB(x,r2),w|p′w(y)1−p′dy1p≤C1+ln⁡r1r2∥b∥∗,$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 $|Rm(b;x,y)|≤C|x−y|m∑|γ|=m1B(x,5n|x−y|)∫B(x,5n|x−y|)|Dγb(z)|dz1/q.$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 $RmA;x,y$ $≤C|x−y|m−1∑|γ|=m−1∥DγA∥∗+∑|γ|=m−1|DγA(y)−(DγA)B(x0,r)|.$(5)

Proof: For fixed x ∈ ℝn, let $A¯(x)=A(x)−∑γ=m−11γ!(DγA)B(x,5n|x−y|)xγ.$Then $|Rm(A;x,y)|=|Rm(A¯;x,y)|≤|Rm−1(A¯;x,y)|+∑|γ|=m−11γ!|(DγA¯(y))||x−y|m−1.$(6)From Lemma 2.7 we have, $|Rm−1(A¯;x,y)|≤C|x−y|m−1∑|γ|=m−1∥DγA∥∗.$(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 $B(x0,2j−1r)⊂B(x,5n|x−y|)⊂100nB(x0,2jr).$Then $|100nB(x0,2jr)||B(x,5n|x−y|)|≤|100nB(x0,2jr)||B(x0,2j−1r)|≤C.$Hence $|(DγA)B(x,5n|x−y|)−(DγA)B(x0,2jr)|≤1|B(x,5n|x−y|)|∫B(x,5n|x−y|)|DγA(y)−(DγA)B(x0,2jr)|dy≤1|100nB(x0,2jr)|∫100nB(x0,2jr)|DγA(y)−(DγA)B(x0,2jr)|dy≤C||DγA||∗.$Note that $|(DγA)B(x0,2jr)−(DγA)B(x0,r)|≤∑k=1j|(DγA)B(x0,2kr)−(DγA)B(x0,2k−1r)|≤2nj∥DγA∥∗.$Then $|(DγA)B(x,5n|x−y|)−(DγA)B(x0,r)|≤|(DγA)B(x,5n|x−y|)−(DγA)B(x0,2jr)|+|(DγA)B(x0,2jr)−(DγA)B(x0,r)|≤Cj||DγA||∗.$Thus $|DγA¯(y)|=|DγA(y)−(DγA)B(x,5n|x−y|)|≤|DγA(y)−(DγA)B(x0;r)|+|(DγA)B(x,5n|x−y|)−(DγA)B(x0;r)|≤|DγA(y)−(DγA)B(x0;r)|+Cj||DγA||∗.$(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 $essinfx∈E⁡fx−1=esssupx∈E⁡1fx.$

## 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 ${T}_{\mathrm{\Omega },\alpha }^{A,m}.$

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 ${D}^{{\gamma }_{j}}{A}_{j}\in BMO\left({\mathbb{R}}^{n}\right).$ AjBMO. (ℝn) Suppose that Ω is homogeneous of degree zero with Ω ∈ Ls(Sn-1), ${w}^{{s}^{\prime }}\in {A}_{\frac{p}{{s}^{\prime }},\frac{q}{{s}^{\prime }}},$ then for any r > 0, there is a constant C independent of f such that $TΩ,αA1,A2,…,AkfLq,wq(B(x0,r))C∏j=1k∑|γj|=mj−1DγjAj∗(wq(B(x0,r)))1q×∫2r(1+ln⁡tr)kfLp,wp(B(x0,t))(wq(B(x0,t)))−1qdtt.$(9)

Proof: We write f as f = f1 + f2, where ${f}_{1}\left(y\right)=f\left(y{\right)}_{{\chi }_{B\left({x}_{0},2r\right)}\left(y\right)},{\chi }_{B\left({x}_{0},2r\right)}$ denotes the characteristic function of B(x0,2r) Then $∥TΩ,αA1,A2,…,Akf∥Lq,wq(B(x0,r))<_∥TΩ,αA1,A2,…,Akf1∥Lq,wq(B(x0,r))+∥TΩ,αA1,A2$Since ${f}_{1}\in {L}_{p,{w}^{p}}\left({\mathbb{R}}^{n}\right),$, by the boundedness of ${T}_{\mathrm{\Omega },\alpha }^{{A}_{1},{A}_{2},\dots ,{A}_{k}}$ from ${L}_{p,{w}^{p}}\left({\mathbb{R}}^{n}\right)\phantom{\rule{thinmathspace}{0ex}}\text{to}\phantom{\rule{thinmathspace}{0ex}}{L}_{q,{w}^{q}}\left({\mathbb{R}}^{n}\right)$ (Theorem 1.1) we get $TΩ,αA1,A2,…,Akf1∥Lq,wq(B(x0,r))≤TΩ,αA1,A2,…Akf1∥Lq,wq(Rn)C∏j=1k⁡∑|γj|=mj−1∥DγjAj∥∗∥f1∥Lp,wp(Rn)C∏j=1k⁡∑|γj|=mj−1∥DγjAj∥∗∥f∥Lp,wp(B(x0,2r)).$Note that q > p > 1 and $\frac{{s}^{\prime }p}{{p}^{\prime }\left(p-{s}^{\prime }\right)}\ge 1$, then by Hölder’s inequality, $1≤(1|B|∫Bw(y)pdy)1p(1|B|∫Bw(y)−p′dy)1p′≤(1|B|∫Bw(y)qdy)1q(1|B|∫Bw(y)−s′pp−s′dy)p−s′s′p.$This means $rns′−α<_(wq(B(x0,r)))1q∥w−1∥Ls′pp−s′(B(x0,r))$Then $∥f∥Lp(wp,B(x0,2r))≤Crns′−α∥f∥Lp,wp(B(x0,2r))∫2r∞tα−ns′−1dt≤C(wq(B(x0,r)))1q∥w−1∥Ls′pp−s′(B(x0,r))∫2r∞∥f∥Lp,wp(B(x0,t))tα−ns′−1dt<_C(wq(B(x0,r)))1q∫2r∞∥f∥Lp,wp(B(x0,t))∥w−1∥Ls′pp−s′(B(x0,t))tα−ns′−1dt.$Since ${w}^{{s}^{\prime }}\in {A}_{\frac{p}{{s}^{\prime }},\frac{q}{{s}^{\prime }}}$ by (4), for all r > 0 we get $(wq(B(x0;r)))1q∥w−1∥Ls′pp−s′(B(x0;r))<_Crns′−α.$(10)Then $∥TΩ;αA1;A2;⋯;Akf1∥Lq,wq(B(x0;r))≤C∏j=1k∑|γj|=mj−1∥DγjAj∥∗(wq(B(x0;r)))1q∫2r∞∥f∥Lp,wp(B(x0;t))(wq(B(x0;t)))−1qdtt.$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, $|TΩ,αA1,A2,⋯,Akf2(x)|≤∫(B(x0,2r))c|Ω(x−y)f(y)||x−y|n−α+m−1Rm1(A1;x,y)Rm2(A2;x,y)f(y)dy≤∑i=1∞∫Δi|Ω(x−y)f(y)||x−y|n−α∏j=12j+∑|γj|=mj−1|DγjAj(y)−(DγjAj)B(x0,r)|dy≤C∏j=i2j∑|γj|=mj−1||DγjAj||∗∑i=1∞∫Δi|Ω(x−y)f(y)||x−y|n−αdy+C∑|γ1|=m1−1||Dγ1A1||∗∑|γ2|=m2−1∑i=1∞∫Δi|Ω(x−y)f(y)||x−y|n−α|Dγ2A2(y)−(Dγ2A2)B(x0,r)|dy+C∑|γ2|=m2−1||Dγ2A2||∗∑|γ1|=m1−1∑i=1∞∫Δi|Ω(x−y)f(y)||x−y|n−α|Dγ1A1(y)−(Dγ1A1)B(x0,r)|dy+C∑|γ1|=m1−1∑|γ2|=m2−1∑i=1∞∫Δi|Ω(x−y)f(y)||x−y|n−α∏j=12|DγjAj(y)−(DγjAj)B(x0,r)|dy≤C(I1+I2+I3+I4).$By Hölder’s inequalities, $∫△i|Ω(x−y)f(y)||x−y|n−αdy≤∫△i|Ω(x−y)|sdy1s∫△i|f(y)|s′|x−y|(n−α)s′dy1s′.$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 $∫|Ω(x−y)|sdy1s≤C∥Ω∥LS(Sn−1)|B(x0;2i+1r)|1s.$(11)We also note that if xB(x0, r), yB(x0, 2r)c, then |yx| ≈ |yx0|. Consequently $∫△i|f(y)|s′|x−y|(n−α)s′dy1s′≤1|B(x0;2i+1r)|1−α/n∫B(x0;2i+1r)|f(y)|s′dy1s.$(12)Then $I1≤C∏j=12∑|γj|=mj−1∥DγjAj∥∗∑i=1∞j(2i+1r)α−ns′(∫B(x0;2i+1r)|f(y)|s′dy)1s′.$Since s' <p, it follows from Hölder’s inequality that $∫B(x0;2i+1r)|f(y)|s′dy1s′≤C∥f∥Lp,wp(B(x0;2i+1r))∥w−1∥Ls′pp−s′(B(x0;2i+1r)).$Then $I1≤C∏j=12∑|γj|=mj−1∥DγjAj∥∗∑i=1∞j(2i+1r)α−ns′(∫B(x0;2i+1s)|f(y)|q′dy)1q.<_C∏j=12∑|γj|=m+j−1∥DγjAj∥∗∑i=1∞(1+ln⁡2i+1rr)(2i+1r)α−ns′∥f∥Lp,wp(B(x0,2i+1r))∥w−1∥Ls′pp−s′(B(x0,2i+1r))<_C∏j=12∑|γj|=mj−1∥DγjAj∥∗∑i=12iX∫2i+1r2i+2r(1+ln⁡tr)∥f∥Lp,wp(B(x0,t))∥w−1∥Ls′pp−s′(B(x0,t))tα−ns′−1dt<_C∏j=12∑|γj|=mj−1∥DγjAj∥∗∫2r1(1+ln⁡tr)∥f∥Lp,wp(B(x0,t))∥w−1∥Ls′pp−s′(B(x0,t))tα−ns′−1dt.$From (10) we know $∥w−1∥Ls′pp−s′(B(x0,r))<_Crns′−α(wq(B(x0,r)))−1q.$(13)Then $I1<_C∏j=12∑|γj|=mj−1∥DγjAj∥∗∫2r∞(1+ln⁡tr)∥f∥Lp,wp(B(x0,t))(wq(B(x0,t)))−1qdtt.$On the other hand, by Hölder’s inequality and (11), (12), we have $∫△i|Ω(x−y)f(y)||x−y|n−α|Dγ2A2(y)−(Dγ2A2)B(x0,r)|dy≤∫△i|Ω(x−y)|sdy1s∫△i|Dγ2A2(y)−(Dγ2A2)B(x0;r)f(y)|s′|x−y|(n−α)sdy1s′≤C∑i=1∞(2i+1r)α−ns′∫(x0,2i+1r)|DγA(y)−(DγA)B(x0,r)|s′|f(y)|s′dy1s′.$Applying Hölder’s inequality we get $∫B(x0,2i+1r)|Dγ2A2(y)−(Dγ2A2)B(x0,r)|s′|f(y)|s′dy1s′≤C∥f∥Lp,wp(B(x0,2i+1r))∥(Dγ2A2(y)−(Dγ2A2)B(x0,r))w(⋅)−1∥Lps′p−s′(B(x0,2i+1r)).$Consequently, $I2C∑|γ1|=m1−1∥Dγ1A1∥∗∑|γ2|=m2−1∑i=1X∫2i+1r2i+2r(2i+1r)α−ns′)α−ns′∥f∥Lp,wp(B(x0t))$$×∥(Dγ2A2(y)−(Dγ2A2)B(x0,r))ω(⋅)−1∥Lps′p−s′(B(x0,t))dt≤C∑||γ1=m1−1∥Dγ1A1∥∗∑|γ2|=m2−1∫2r∞∥f∥Lp,wp(B(x0,t))×∥(Dγ2A2(y)−(Dγ2A2)B(x0,r))w(⋅)−1∥Lps′p−s′(B(x0,t))tα−ns′−1dt.$By ${w}^{{s}^{\prime }}\in {A}_{\frac{p}{{s}^{\prime }},\frac{q}{{s}^{\prime }}}$ and (ii) of Lemma 2.2 we know ${w}^{-\frac{{s}^{\prime }p}{p-{s}^{\prime }}}\in {A}_{1+\frac{p{s}^{\prime }}{\left(p-{s}^{\prime }\right)q}}.$. Then it follows from the Lemma 2.6 and the inequality (13) that $∥(Dγ2A2(y)−(Dγ2A2)B(x0,r))ω(⋅)−1∥Lps′p−s′(B(x0,t))≤(∫B(x0,t)|Dγ2A2(y)−(Dγ2A2)B(x0,r)|ps′p−s′w−ps′p−s′(y)dy)p−s′ps′<_C∥Dγ2A2∥∗(1+ln⁡tr)(w−ps′p−s′(B(x0,r)))p−s′ps′=C∥Dγ2A2∥∗(1+ln⁡tr)∥w−1∥Lps′p−s′(B(x0,r))<_C∥Dγ2A2∥∗(1+ln⁡tr)rns′−α(wq(B(x0,r)))−1q.$Then $I2<_C∏j=12∑|γj|=mj−1∥DγjAj∥∗∫2r∞(1+ln⁡tr)∥f∥Lp,wp(B(x0,t))(wq(B(x0,t)))−1qdtt.$Similarly to the estimates for I2, we have $I3≤C∏j=12∑|γj|=mj−1∥DγjAj∥∗∫2r1(1+ln⁡tr)∥f∥Lp,wp(B(x0;t))(wq(B(x0;t)))−1qdtt.$Finally, we come to estimate I4.By Hölder’s inequality and (11), (12), we have $∫△i|Ω(x−y)f(y)||x−y|n−α∏j=12|DγjAj(y)−(DγjAj)B(x0,r)|dy≤(∫|Ω(x−y)|sdy)1s(∫△i∏j=12|DγjAj(y)−(DγjAj)B(x0,r)|s′|x−y|(n−α)s′dy)1s′≤C∑i=1∞(2i+1r)α−ns′∫B(x0,2i+1r)∏j=12|DγjAj(y)−(DγjAj)B(x0,r)|s′|f(y)|s′dy1s′.$Applying Hölder’s inequality we get $∫(x0,2i+1r)∏j=12|DγjAj(y)−(DγjAj)B(x0,r)|s′|f(y)|s′dy1s′$$≤C∥f∥Lp,wp(B(x0,2i+1r))∥∏j=12(DγjAj(y)−(DγjAj)B(x0,r))w(⋅)−1∥Lps′p−s′(B(x0,2i+1r))≤C∥f∥Lp,wp(B(x0,2i+1r))∏j=12∥(DγjAj(y)−(DγjAj)B(x0,r))w(⋅)−1/2∥L2ps′p−s′(B(x0,2i+1r)).$Then $∑i=1∞∫△i|Ω(x−y)f(y)||x−y|n−α∏j=12|DγjAj(y)−(DγjAj)B(x0,r)|dy≤C∫2r∞∥f∥Lp,wp(B(x0,t))∏j=12∥(DγjAj(y)−(DγjAj)B(x0,r))w(⋅)−1/2∥L2ps′p−s(B(x0,2i+1r)).$ Since ${w}^{-\frac{{s}^{\prime }p}{p-{s}^{\prime }}}\in {A}_{1+\frac{p{s}^{\prime }}{\left(p-{s}^{\prime }\right)q}},$, then from the Lemma2.6 and the inequality (13) we have $∥(DγjAj(y)−(Dγ2A2)B(x0,r))ω(⋅)−1∥Lps′p−s′(B(x0,t))≤(∫B(x0,t)|Dγ2A2(y)−(Dγ2A2|ps′p−s′w−ps′p−s′(y)dy)p−s′ps′≤C∥Dγ2A2∥∗(1+ln⁡tr)(w−ps′p−s′(B(x0,r)))p−s′ps′=C∥Dγ2A2∥∗(1+ln⁡tr)∥w−1∥Lps′p−s(B(x0,r))≤C∥Dγ2A2∥∗(1+ln⁡tr)rns′−α(wq(B(x0,r)))−1q.$(14)Then from (14) we have $∑i=1∞∫△i|Ω(x−y)f(y)||x−y|n−α∏j=12|DγjAj(y)−(DγjAj)B(x0,r)|dy≤C∏j=12∥DγjAj∥∗∫2r∞(1+ln⁡tr)2∥f∥Lp,wp(B(x0,t))(wq(B(x0,r)))−1qdrr.$Combining with the estimates of I1, I2, I3 and I4, we have $supx∈B(x0,r)|TΩ,αA1,A2,⋯,Akf2(x)|≤C∏j=12∑|γj|=mj−1∥DγjAj∥∗∫2r1(1+ln⁡tr)2∥f∥Lp,wp(B(x0,t))(wq(B(x0,t)))−1qdtt.$Then we get $∥TΩ,αA1,A2,⋯,Akf2∥Lq,wq(B(x0,r))<_C∏j=12∑|γj|=mj−1∥DγjAj∥∗(wq(B(x0,r)))1q×∫2r1(1+ln⁡tr)2∥f∥Lp,wp(B(x0,t))(wq(B(x0,t)))−1qdtt.$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 $∥TΩ,αA1,A2,⋯,Akf∥Mq,φ(wq)=supx0∈Rn,r>0φ2(x0,r)−1(wq(B(x0,r)))−1q∥TΩ,αA1,A2,⋯,Akf∥Lq,wq(B(x0,r))2≤Csupx0∈Rn,rgt0φ2(x0,r)−1∫r∞(1+ln⁡tr)k∥f∥Lp,wp(B(x0,t))(wq(B(x0,t)))−1qdtt≤Csupx0∈Rn,r>0,φ1(x0,r)−1wp(B(x,r)−1p∥f∥Lp,wp(B(x,r))=∥f∥Mp,φ1(wp).$

Second variant proof of Theorem 1.5

Since $f\in {M}_{p,{\phi }_{1}}\left({w}^{p}\right),$ then by Lemma 2.9 and the fact $\parallel f{\parallel }_{{L}_{p,{w}^{p}}\left(B\left({x}_{0},t\right)\right)}$ is a non-decreasing function of t, we get $∥f∥Lp,wp(B(x0,t))essinf0<τ0,x0∈Rn∥f∥Lp,wp(B(x0,τ))φ1(x0,τ)(wp(B(x0,τ)))1p≤∥f∥Mp,φ(wp)1⋅$

Since (ϕ1, ϕ2) satisfies (1), we have $∫r∞(1+ln⁡tr)k∥f∥Lp,wp(B(x0,t))(wq(B(x0,t)))−1qdtt≤∫r∞∥f|Lp,wp(B(x0,t))essinf0<τ

Then by (9) we get $∥TΩ,αA1,A2,⋯,Akf∥Mq,φ2(wq)≤Csupx0∈Rn,t>01φ2(x0,t)(1wq(B(x0,t))∫B(x0,t)|TΩ,αA1,A2,⋯,Akf(y)|qwq(y)dy)1/q≤C∏j=1k∑|γj|=mj−1∥DγjAj∥∗supx0∈Rn,t>01φ2(x0,t)×∫r∞(1+ln⁡tr)k∥f∥Lp,wp(B(x0,t))(wq(B(x0,t)))−1qdtt≤C∏j=1k∑|γj|=mj−1∥DγjAj∥∗∥f∥Mp,φ1(wp).$

## 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).

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Accepted: 2016-11-02

Published Online: 2016-12-17

Published in Print: 2016-01-01

Citation Information: Open Mathematics, ISSN (Online) 2391-5455, Export Citation