Abstract
In this paper, we study the boundedness of fractional multilinear integral operators with rough kernels
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
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,
Using the method of good — λ inequality, in 1986, Cohen and Gosselin [10] proved that if Ω ∈ Lip1 (Sn-1) and DγA ∈ BMO(ℝn), then
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 w ∈ Ap.
It is natural to ask whether the multilinear fractional integral operator with a rough kernel has the mapping properties similar to those of
where
When k = 1 and m = 1, then
When mj = 1 and Aj = A for j = 1,..., k, then
When mj ≥ 2,
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].
([12]). Let
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.
Let 1 ≤ ρ < ∞, ϕ be a positive measurable function on ℝ × (0, ∞) and w be non-negative measurable function on ℝn. We denote by Mp,ϕ(w) the generalized weighted Morrey space, the space of all functions
where Lp,w(B(x, r)) denotes the weighted Lp-space of measurable functions f for which
Furthermore, by WMp,ϕ(w) we denote the weak generalized weighted Morrey space of all functions
where WLp,w(B(x, r)) denotes the weak Lp,w-space of measurable functions f for which
(1) If w ≡ 1, then Mp,ϕ(1) = Mp,ϕ is the generalized Morrey space.
(2) If
(3) If
(4) If
(5) If
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
([24]). Let 0 < α < n, 1 < p < n/α and 1/q = 1/p — α/n, Ω ∈ L∞(𝕊n-1), w ∈ Ap,q, A ∈ BMO(ℝn), and (ϕ1, ϕ2) satisfies the condition
where C does not depend on χ and r. Then the operator
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
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 (ϕ1,ϕ2) satisfy the condition (1). Let also, for 1 ≤ j ≤ k, |γj| = mj - 1, mj ≥ 2 and
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].
([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
where C0 does not depend on x and r. Let also, for 1 ≤ j ≤ k, |γj| = mj - 1, mj ≥ 2 and
Let
In fact, from (2.8) in Section 2 we have constant δ > 0 such that
Since 0 < κ < p/q, then
If
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
where the constant C > 0 is independent of f and A1, A2,..., Ak.
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
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
A weight w is said to belong to Ap for 1 < p < ∞, if there exists a constant
where p' is the dual of ρ such that
A weight w is said to belong to A∞(ℝn) if there are positive numbers C and δ so that
for all balls B and all measurable E C B. It is well known that
The classical Ap weight theory was first introduced by Muckenhoupt in the study of weighted Lp -boundedness of Hardy-Littlewood maximal function in [37].
([21, 37]). Suppose w ∈Ap and the following statements hold.
(i) For any 1 ≤ p < ∞, there is a positive number C such that
(ii) For any 1 ≤ p < ∞, there is a positive number C and S such that
(iii) For any 1 < p < ∞, one has
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 ≤ p ≤ q < ∞. We say that ω ∈ Ap,q if there exists a constant C such that for every ball B ⊂ ℝn, the inequality
holds when 1 < p < ∞, and for every ball B ⊂ ℝn the inequality
holds when p = 1.
By (3), we have
We summarize some properties about weights Ap,q; see [21, 38].
Given 1 ≤ p ≤ q < ∞.
(i) w ∈Ap,q if and only if
(ii) w ∈ Ap,q if and only if
(iii) w ∈Ap,p if and only if wp ∈Ap;
(iv) If p1 < p2 and q2 > q1, then
In this paper, we need the following statement on the boundedness of the Hardy type operator
where μ is a non-negative Borei measure on (0, ∞).
The inequality
holds for all non-negative and non-increasing g on (0, ∞) if and only if
and c ≈ A1.
Note that Theorem 2.3 is proved analogously to Theorem 4.3 in [24, 25].
([39, Theorem 5, p. 236]). Let w ∈ A∞. Then the norm of BMO(ℝn) is equivalent to the norm of BMO(w), where
and
([24]).
(1) The John-Nirenberg inequality : there are constants C1,C2 > 0, such that for all b ∈ BMO(ℝn) and β > 0
(2) For 1 < p < ∞ the John-Nirenberg inequality implies that
and for 1 < ρ < ∞ and w ∈ A∞
The following lemma was proved by Guliyev in [24].
([24]).
i) Let w ∈ A∞and b be a function in BMO(ℝn). Let also 1 ≤p < ∞, x ∈ℝn, and r1,r2 > 0. Then
where C > 0 is independent of f, x, r1 and r2.
ii) Let w ∈Ap and b be a function in BMO(ℝn). Let also 1 < p < ∞, x ∈ ℝn, and r1,r2 > 0. Then
where C > 0 is independent of f, x, r1 and r2.
Below we present some conclusions about Rm(A;x, y).
([22]). Suppose b is a function on ℝnwith the m-th derivatives in Lq(ℝn), q > n. Then
The following property is valid.
Let x ∈ B(x0, r), y ∈ B(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
For fixed x ∈ ℝn, let
Then
From Lemma 2.7 we have,
When x ∈ B(x0,r), y ∈ B(x0, 2j+1r)\B(x0, 2jr), then 2j-1r ≤ |x - y| ≤2j+2 r. Thus, we have
Then
Hence
Note that
Then
Thus
Combining with (6), (7) and (8), then (5) is proved.
Finally, we present a relationship between essential supremum and essential infimum.
([10]). Let f be a real-valued nonnegative function and measurable on E. Then
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
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
We write f as f = f1 + f2, where
Since
Note that q > p > 1 and
This means
Then
Since
Then
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+1r)) \ (B(x0, 2ir)), and let x ∈ B(x0, r)) By Lemma 2.8,
By Hölder’s inequalities,
When x ∈B(x0, s) and y ∈ Δi, then by a direct calculation, we can see that 2i-1 r ≤|y–x| < 2i-1r. Hence
We also note that if x ∈ B(x0, r), y ∈ B(x0, 2r)c, then |y – x| ≈ |y – x0|. Consequently
Then
Since s' <p, it follows from Hölder’s inequality that
Then
From (10) we know
Then
On the other hand, by Hölder’s inequality and (11), (12), we have
Applying Hölder’s inequality we get
Consequently,
By
Then
Similarly to the estimates for I2, we have
Finally, we come to estimate I4.
By Hölder’s inequality and (11), (12), we have
Applying Hölder’s inequality we get
Then
Since
Then from (14) we have
Combining with the estimates of I1, I2, I3 and I4, we have
Then we get
This completes the proof of Theorem 3.1.
4 Proof of Theorem 1.5
First variant proof of Theorem 1.5
By Theorems 2.3and 3.1 we have
Second variant proof of Theorem 1.5
Since
Since (ϕ1, ϕ2) satisfies (1), we have
Then by (9) we get
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.10.1186/s13660-015-0751-zSearch in Google Scholar
[2] Akbulut A., Eroglu A., Najafov A.M., Some embedding theorems on the Nikolskii-Morrey type spaces, Advances in Analysis, 2016, 1(1), 18-26.10.22606/aan.2016.11003Search in Google Scholar
[3] Bajsanski B., Coifman R., On singular integrals, Proc. Symp. Pure Math., Amer. Math. Soc. Providence, R. I., 1996, 10, 1-17.10.1090/pspum/010/0238129Search in Google Scholar
[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.Search in Google Scholar
[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.Search in Google Scholar
[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), 2527-2541.10.1090/S0002-9939-2015-12458-6Search in Google Scholar
[7] Byun, Sun-Sig, Softova, L.G., Gradient estimates in generalized Morrey spaces for parabolic operators, Math. Nachr., 2015, 288(14-15), 1602-1614.10.1002/mana.201400113Search in Google Scholar
[8] Chiarenza F., Frasca M., Morrey spaces and Hardy-Littlewood maximal function, Rend Math. Appl., 1987, 7, 273-279.Search in Google Scholar
[9] Cohen J., A sharp estimate for a multilinear singular integral in Rn, Indiana Unit. Math. Jour., 1981, 30, 693-702.10.1512/iumj.1981.30.30053Search in Google Scholar
[10] Cohen J., Gosselin J., A BMO estimate for multilinear singular integrals, Illinois J. Math., 1986, 30, 445-464.10.1215/ijm/1256044539Search in Google Scholar
[11] Ding Y., Lu S.Z., Higher order commutators for a class of rough operators, Ark. Mat., 1999, 37, 33-44.10.1007/BF02384827Search in Google Scholar
[12] Ding Y., Lu S.Z., Weighted boundedness for a class of rough multilinear operators, Acta Math. Sin. (Engl. Ser.), 2001, 17, 517-526.10.1007/s101140100113Search in Google Scholar
[13] Ding Y., A note on multilinear fractional integrals with rough kernel, Adv. Math. (China), 2001, 30, 238-246.Search in Google Scholar
[14] Ding Y., Lu S., Higher order commutators for a class of rough operators, Ark. Mat., 1999, 37, 33-44.10.1007/BF02384827Search in Google Scholar
[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.Search in Google Scholar
[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), 841-853.10.1090/S0002-9947-1993-1088476-1Search in Google Scholar
[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, 241-256.10.1006/jfan.1993.1032Search in Google Scholar
[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.Search in Google Scholar
[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), 52-66.10.1016/j.jcp.2012.10.018Search in Google Scholar
[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, 37-46.10.1016/j.enganabound.2014.09.003Search in Google Scholar
[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.Search in Google Scholar
[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)Search in Google Scholar
[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.10.1155/2009/503948Search in Google Scholar
[24] Guliyev V.S., Generalized weighted Morrey spaces and higher order commutators of sublinear operators, Eurasian Math. J., 2012, 3, 33-61.10.1016/S0252-9602(13)60085-5Search in Google Scholar
[25] Guliyev V.S., Local generalized Morrey spaces and singular integrals with rough kernel, Azerb. J. Math., 2013, 3(2), 79-94.10.1007/s10958-013-1448-9Search in Google Scholar
[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), 843-862.10.1007/s11118-012-9299-4Search in Google Scholar
[27] Guliyev V.S., Softova L., Generalized Morrey regularity for parabolic equations with discontinuity data, Proc. Edinb. Math. Soc., 2015, 58(1), 199-218.10.1017/S0013091513000758Search in Google Scholar
[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), 2368-2387.10.1016/j.jde.2015.03.032Search in Google Scholar
[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), 365-386.10.1007/s10587-014-0107-8Search in Google Scholar
[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), 911-924.10.1007/s10114-016-5530-3Search in Google Scholar
[31] Guliyev V.S., Omarova M.N., Multilinear singular and fractional integral operators on generalized weighted Morrey spaces, Azerb. J. Math., 2015, 5 (1), 104-132.10.1155/2015/594323Search in Google Scholar
[32] Hofmann S., On some nonstandard Calderón-Zygmund operators, Studia Math., 1994, 109, 105-131.10.4064/sm-109-2-105-131Search in Google Scholar
[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), 227-244.10.2478/aicu-2013-0009Search in Google Scholar
[34] Komori Y., Shirai S., Weighted Morrey spaces and a singular integral operator, Math. Nachr., 2009, 289, 219-231.10.1002/mana.200610733Search in Google Scholar
[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, 183-189.10.1007/978-4-431-68168-7_16Search in Google Scholar
[36] Morrey C.B., On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc., 1938, 43, 126-166.10.1090/S0002-9947-1938-1501936-8Search in Google Scholar
[37] Muckenhoupt B., Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc., 1972, 165, 207-226.10.1090/S0002-9947-1972-0293384-6Search in Google Scholar
[38] Muckenhoupt B., Wheeden R., Weighted norm inequalities for singular and fractional integrals, Trans. Amer. Math. Soc., 1971, 161, 249-258.10.1090/S0002-9947-1971-0285938-7Search in Google Scholar
[39] Muckenhoupt B., Wheeden R., Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc., 1974, 192, 261-274.10.1090/S0002-9947-1974-0340523-6Search in Google Scholar
[40] Mustafayev, R. Ch., On boundedness of sublinear operators in weighted Morrey spaces, Azerb. J. Math., 2012, 2 (1), 66-79.Search in Google Scholar
[41] Palagachev D.K., Softova L.G., Singular integral operators, Morrey spaces and fine regularity of solutions to PDE’s, Potential Anal., 2004, 20, 237-263.10.1023/B:POTA.0000010664.71807.f6Search in Google Scholar
[42] Pérez C., Trujillo-González R., Sharp weighted estimates for multilinear commutators, J. London Math. Soc., 2002, 65, 672-692.10.1112/S0024610702003174Search in Google Scholar
[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.10.1142/S021812741250085XSearch in Google Scholar
[44] Takeshi I., Enji S., Yoshihiro S., Hitoshi T., Weighted norm inequalities for multilinear fractional operators on Morrey spaces, Studa Math., 2011, 205, 139-170.10.4064/sm205-2-2Search in Google Scholar
[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.10.1186/1029-242X-2012-302Search in Google Scholar
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