Φ-Admissible singular operators and their commutators on vanishing generalized Orlicz-Morrey spaces
© Guliyev et al.; licensee Springer. 2014
Received: 5 November 2013
Accepted: 17 March 2014
Published: 8 April 2014
We study the boundedness of Φ-admissible singular operators and their commutators on vanishing generalized Orlicz-Morrey spaces including their weak versions. These conditions are satisfied by most of the operators in harmonic analysis, such as the Hardy-Littlewood maximal operator, the Calderón-Zygmund singular integral operator and so on. In all the cases the conditions for the boundedness are given in terms of Zygmund-type integral inequalities on weights without assuming any monotonicity property of on r.
MSC:42B20, 42B25, 42B35, 46E30.
where denotes the weak -space.
Morrey found that many properties of solutions to PDE can be attributed to the boundedness of some operators on Morrey spaces. Maximal functions and singular integrals play a key role in harmonic analysis, since maximal functions could control crucial quantitative information concerning the given functions, despite their larger size, while singular integrals, with the Hilbert transform as their prototype, nowadays are intimately connected with PDE, operator theory, and other fields.
Orlicz spaces, introduced in [2, 3], are generalizations of Lebesgue spaces . They are useful tools in harmonic analysis and its applications. For example, the Hardy-Littlewood maximal operator is bounded on for , but not on . Using Orlicz spaces, we can investigate the boundedness of the maximal operator near more precisely (see [4, 5] and ).
for some ball B which contains x, proving the absolute convergence of the integral in the second term and the independence of the choice of the ball B (see [9, 10] for example). Also, is dense in the Orlicz spaces if and only if Φ satisfies the condition.
The main purpose of this paper is to find sufficient conditions on general Young function Φ and functions , which ensure the boundedness of the sublinear operators generated by singular integral operators from vanishing generalized Orlicz-Morrey spaces to another , from to vanishing weak generalized Orlicz-Morrey spaces and the boundedness of the commutator of the sublinear operators from to .
There are several kinds of Orlicz-Morrey spaces in the literature. The first kind is due to Nakai  and the second kind is due to Sawano et al. . Our definition (see ) should be called ‘generalized Orlicz-Morrey space of the third kind’. For the boundedness of the operators of harmonic analysis on Orlicz-Morrey spaces, see also [10–19]. For details see Remark 9 in  and references therein.
By we mean that with some positive constant C independent of appropriate quantities. If and , we write and say that A and B are equivalent.
We recall the definition of Young functions.
Definition 2.1 A function is called a Young function if Φ is convex, left-continuous, , and .
From the convexity and it follows that any Young function is increasing. If there exists such that , then for .
If , then Φ is absolutely continuous on every closed interval in and bijective from to itself.
Definition 2.2 (Orlicz space)
is called Orlicz space. If , , then . If () and (), then . The space endowed with the natural topology is defined as the set of all functions f such that for all balls . We refer to the books [20–22] for the theory of Orlicz spaces.
In the case , we shortly denote it by .
for some . The function satisfies the -condition but does not satisfy the -condition. If , then satisfies both conditions. The function satisfies the -condition but does not satisfy the -condition.
The following analog of the Hölder inequality is well known; see .
Theorem 2.4 
In the next sections where we prove our main estimates, we use the following lemma, which follows from Theorem 2.4, Lemma 2.5, and (2.2).
Let T be a sublinear operator, that is, .
Definition 2.7 (Φ-admissible singular operator)
- (1)T satisfies the size condition of the form(2.3)
for and ;
T is bounded in .
In the case , , the Φ-admissible singular operator will be called the p-admissible singular operator.
Definition 2.8 (Weak Φ-admissible singular operator)
T satisfies the size condition (2.3).
T is bounded from to the weak .
In the case , . the weak Φ-admissible singular operator will be called weak p-admissible singular operator.
Remark 2.9 Note that in , Φ-admissible singular operators and weak Φ-admissible singular operators were introduced and their boundedness on generalized Orlicz-Morrey spaces was studied. Also in , p-admissible singular operators were introduced and their boundedness on vanishing generalized Morrey spaces was studied.
Definition 2.10 (Generalized Orlicz-Morrey space)
The vanishing generalized Morrey space which was introduced and studied by Samko  is defined as follows.
Definition 2.11 (Vanishing generalized Morrey space)
Extending the definition of vanishing generalized Morrey spaces to the case of Orlicz-Morrey spaces, we introduce the following definitions.
Definition 2.12 (Vanishing generalized Orlicz-Morrey space)
Definition 2.13 (Vanishing weak generalized Orlicz-Morrey space)
If we choose , at Definition 2.12, we get Definition 2.11. The vanishing Morrey space of the classical Morrey space was introduced by Vitanza in  and applied there to obtain a regularity result for elliptic partial differential equations. Later in  Vitanza proved an existence theorem for a Dirichlet problem, under weaker assumptions than those introduced by Miranda in , and a regularity result assuming that the partial derivatives of the coefficients of the highest and lower order terms belong to a vanishing Morrey space depending on the dimension. Also Ragusa  proved a sufficient condition for commutators of fractional integral operators to belong to vanishing Morrey spaces . About commutator operators in vanishing Morrey spaces see the papers [30, 31].
which makes the spaces and non-trivial, because bounded functions with compact support belong then to this space.
respectively. The spaces and are closed subspaces of the Banach spaces and , respectively, which may be shown by standard means.
3 Φ-Admissible singular operators in the spaces
In this section, sufficient conditions on φ for the boundedness of the Φ-admissible singular operator T in vanishing generalized Orlicz-Morrey spaces are obtained.
The known boundedness statement for the Hardy-Littlewood maximal operator M and the Calderón-Zygmund singular integral operators K in Orlicz spaces runs as follows. For details of these results see .
Let Φ any Young function. Then the maximal operator M is bounded from to and for bounded in .
Let Φ be a Young function. If , then the operator K is bounded on and if , then the operator K is bounded from to .
By using Lemma 3.3 the following statement was proved in .
where C does not depend on x and r. Then a Φ-admissible singular operator T is bounded from to and a weak Φ-admissible singular operator T is bounded from to .
where does not depend on and . Then a Φ-admissible singular operator T is bounded from to and a weak Φ-admissible singular operator T is bounded from to .
Proof The statement is derived from Theorem 3.4.
In this estimation we follow some ideas of  in such a passage to the limit in the case , but we base ourselves on Lemma 3.3.
which completes the proof of (3.4).
The proof of (3.5) is similar to the proof of (3.4). □
Remark 3.6 The condition (3.2) is not needed in the case where does not depend on x, since (3.2) follows from (3.3) in these cases.
Remark 3.7 Note that from Theorems 3.1 and 3.2 that it is found that the maximal operator M and the singular integral operator K are the weak Φ-admissible singular operators for any Young function Φ and , respectively. Also the maximal operator M and the singular integral operator K are the Φ-admissible singular operators for the Young functions and , respectively.
From Remark 3.7 we get the following corollaries which were proved in .
Corollary 3.8 Let Φ be a Young function, , , and Φ satisfy the conditions (2.4)-(2.5) and (3.2)-(3.3). Then the maximal operator M is bounded from to and for , the operator M is bounded from to .
Corollary 3.9 Let Φ be a Young function, K be a Calderon-Zygmund singular operator with standard kernel and , , Φ satisfy the conditions (2.4)-(2.5) and (3.2)-(3.3). If , then the operator K is bounded from to and if , then the operator K is bounded from to .
4 Commutators of the Φ-admissible singular operators in the spaces and
It is well known that the commutator is an important integral operator and plays a key role in harmonic analysis. In 1965, Calderon [37, 38] studied a kind of commutators appearing in Cauchy integral problems of Lip-line. Let K be a Calderón-Zygmund singular integral operator and . A well-known result of Coifman et al.  states that the commutator operator is bounded on for . The commutator of Calderón-Zygmund operators plays an important role in studying the regularity of solutions of elliptic partial differential equations of second order. The boundedness result was generalized to other contexts and important applications to some non-linear PDEs were given by Coifman et al. .
We recall the definition of the space of .
Modulo constants, the space is a Banach space with respect to the norm .
- (1)The John-Nirenberg inequality states that there are constants , such that for all and
- (2)The John-Nirenberg inequality implies that(4.1)
- (3)Let . Then there is a constant such that(4.2)
where C is independent of b, x, r, and t.
Remark 4.4 We know that if Φ is lower type and upper type with , then . Conversely if , then Φ is lower type and upper type with (see ).
In the following lemma, which was proved in , we provide a generalization of the property (4.1) from -norms to Orlicz norms.
Remark 4.6 Note that the Lemma 4.5 for the variable exponent Lebesgue space case was proved in .
Remark 4.8 It is well known that if and only if (see, for example, ).
Remark 4.9 Remark 4.8 and Remark 4.4 show us that a Young function Φ is lower type and upper type with if and only if .
Definition 4.10 (Φ-admissible commutator singular operator)
- (1)satisfies the size condition of the form
for and ;
is bounded in .
In the case , , the Φ-admissible commutator singular operator will be called a p-admissible commutator singular operator.
where w is a weight.
The following theorem was proved in .
Moreover, the value is the best constant for (4.3).
Remark 4.12 In (4.3) and (4.4) it is assumed that and .
holds for any ball and for all .
and the statement of Lemma 4.13 follows by (4.9). □
Proof The statement of Theorem 4.14 follows by Lemma 4.13 and Theorem 4.11 in the same manner as in the proof of Theorem 3.4. □
If we take , at Theorem 4.14 we get the following result, which was proved at .
where C does not depend on x and r. Then the operator is bounded from to .
The known boundedness statement for the commutator operators and on Orlicz spaces runs as follows.
Theorem 4.16 
Let Φ be a Young function with , . Then the operators and are bounded on .
For the commutator operators and from Theorem 4.14 we get the following corollaries, which were proved in .
Corollary 4.17 Let Φ be a Young function with , and , , and Φ satisfy the condition (4.10). Then the operators and are bounded from to .
for every . Then the operator is bounded from to .
Proof The proof follows more or less the same lines as for Theorem 3.5, but now the arguments are different due to the necessity to introduce the logarithmic factor into the assumptions.
where C and are constants from (4.15) and (4.11), which yields the estimate of the first term uniform in , .
where is the constant from (4.13) with and is a similar constant with omitted logarithmic factor in the integrand. Then by (4.12) we can choose a small r such that , which completes the proof. □
Corollary 4.19 
Let Φ be a Young function with , , and , , and Φ satisfy the conditions (4.11), (4.12), and (4.13). Then the operators and are bounded from to .
The research of V Guliyev and F Deringoz were partially supported by the grant of Ahi Evran University Scientific Research Projects (PYO.FEN.4003.13.003), (PYO.FEN.4003-2.13.007) and (PYO.FEN.4009.14.001). We thank both referees for some good suggestions, which helped to improve the final version of this paper.
- Morrey CB: On the solutions of quasi-linear elliptic partial differential equations. Trans. Am. Math. Soc. 1938, 43: 126-166. 10.1090/S0002-9947-1938-1501936-8MathSciNetView ArticleGoogle Scholar
- Orlicz, W: Über eine gewisse Klasse von Räumen vom Typus B. Bull. Acad. Polon. A, 207-220 (1932). Reprinted in: Collected Papers, PWN, Warszawa, 217-230 (1988)Google Scholar
- Orlicz, W: Über Räume ( L M ) . Bull. Acad. Polon. A, 93-107 (1936). Reprinted in: Collected Papers, PWN, Warszawa, 345-359 (1988)Google Scholar
- Kita H: On maximal functions in Orlicz spaces. Proc. Am. Math. Soc. 1996, 124: 3019-3025. 10.1090/S0002-9939-96-03807-5MATHMathSciNetView ArticleGoogle Scholar
- Kita H: On Hardy-Littlewood maximal functions in Orlicz spaces. Math. Nachr. 1997, 183: 135-155. 10.1002/mana.19971830109MATHMathSciNetView ArticleGoogle Scholar
- Cianchi A: Strong and weak type inequalities for some classical operators in Orlicz spaces. J. Lond. Math. Soc. (2) 1999,60(1):187-202. 10.1112/S0024610799007711MathSciNetView ArticleGoogle Scholar
- Stein EM: Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals. Princeton University Press, Princeton; 1993.MATHGoogle Scholar
- Torchinsky A Pure and Applied Math. 123. In Real Variable Methods in Harmonic Analysis. Academic Press, New York; 1986.Google Scholar
- Nakai E: Generalized fractional integrals on Orlicz-Morrey spaces. In Banach and Function Spaces. Yokohama Publishers, Yokohama; 2004:323-333.Google Scholar
- Nakai E: Calderón-Zygmund operators on Orlicz-Morrey spaces and modular inequalities. In Banach and Function Spaces II. Yokohama Publishers, Yokohama; 2008:393-410.Google Scholar
- Sawano Y, Sugano S, Tanaka H: Orlicz-Morrey spaces and fractional operators. Potential Anal. 2012,36(4):517-556. 10.1007/s11118-011-9239-8MATHMathSciNetView ArticleGoogle Scholar
- Deringoz F, Guliyev VS, Samko S: Boundedness of maximal and singular operators on generalized Orlicz-Morrey spaces. Operator Theory: Advances and Applications 235. Operator Theory, Operator Algebras and Applications 2014, 139-158.View ArticleGoogle Scholar
- Guliyev VS, Deringoz F: On the Riesz potential and its commutators on generalized Orlicz-Morrey spaces. J. Funct. Spaces Appl. 2014., 2014: Article ID 617414Google Scholar
- Hasanov JJ: Φ-admissible sublinear singular operators and generalized Orlicz-Morrey spaces. J. Funct. Spaces Appl. 2014., 2014: Article ID 505237Google Scholar
- Liang Y, Nakai E, Yang D, Zhang J: Boundedness of intrinsic Littlewood-Paley functions on Musielak-Orlicz Morrey and Campanato spaces. Banach J. Math. Anal. 2014,8(1):221-268.MATHMathSciNetView ArticleGoogle Scholar
- Lu Q, Tao X: Characterization of maximal operators in Orlicz-Morrey spaces of homogeneous type. Appl. Math. J. Chin. Univ. Ser. B 2006,21(1):52-58. 10.1007/s11766-996-0022-3MATHMathSciNetView ArticleGoogle Scholar
- Maeda FY, Mizuta Y, Ohno T, Shimomura T: Boundedness of maximal operators and Sobolev’s inequality on Musielak-Orlicz-Morrey spaces. Bull. Sci. Math. 2013,137(1):76-96. 10.1016/j.bulsci.2012.03.008MATHMathSciNetView ArticleGoogle Scholar
- Mizuta Y, Nakai E, Ohno T, Shimomura T:Maximal functions, Riesz potentials and Sobolev embeddings on Musielak-Orlicz-Morrey spaces of variable exponent in . Rev. Mat. Complut. 2012,25(2):413-434. 10.1007/s13163-011-0074-7MATHMathSciNetView ArticleGoogle Scholar
- Nakai E: Orlicz-Morrey spaces and the Hardy-Littlewood maximal function. Stud. Math. 2008,188(3):193-221. 10.4064/sm188-3-1MATHMathSciNetView ArticleGoogle Scholar
- Kokilashvili V, Krbec MM: Weighted Inequalities in Lorentz and Orlicz Spaces. World Scientific, Singapore; 1991.MATHView ArticleGoogle Scholar
- Krasnoselskii MA, Rutickii YB: Convex Functions and Orlicz Spaces. Noordhoff, Groningen; 1961. (English translation)Google Scholar
- Rao MM, Ren ZD: Theory of Orlicz Spaces. Dekker, New York; 1991.MATHGoogle Scholar
- Weiss G: A note on Orlicz spaces. Port. Math. 1956, 15: 35-47.MATHGoogle Scholar
- Bennett C, Sharpley R: Interpolation of Operators. Academic Press, Boston; 1988.MATHGoogle Scholar
- Liu PD, Wang MF: Weak Orlicz spaces: some basic properties and their applications to harmonic analysis. Sci. China Math. 2013,56(4):789-802. 10.1007/s11425-012-4452-5MATHMathSciNetView ArticleGoogle Scholar
- Samko N: Maximal, potential and singular operators in vanishing generalized Morrey spaces. J. Global Optim. 2014. 10.1007/s10898-012-9997-xGoogle Scholar
- Vitanza C: Functions with vanishing Morrey norm and elliptic partial differential equations. In Proceedings of Methods of Real Analysis and Partial Differential Equations. Springer, Berlin; 1990:147-150.Google Scholar
- Vitanza C: Regularity results for a class of elliptic equations with coefficients in Morrey spaces. Ric. Mat. 1993,42(2):265-281.MATHMathSciNetGoogle Scholar
- Miranda C: Sulle equazioni ellittiche del secondo ordine di tipo non variazionale, a coefficienti discontinui. Ann. Mat. Pura Appl. (4) 1963, 63: 353-386. 10.1007/BF02412185MATHMathSciNetView ArticleGoogle Scholar
- Ragusa MA: Commutators of fractional integral operators on vanishing-Morrey spaces. J. Glob. Optim. 2008,40(1-3):361-368. 10.1007/s10898-007-9176-7MATHMathSciNetView ArticleGoogle Scholar
- Persson LE, Ragusa MA, Samko N, Wall P: Commutators of Hardy operators in vanishing Morrey spaces. 1493. AIP Conf. Proc. 2012, 859.Google Scholar
- Genebashvili I, Gogatishvili A, Kokilashvili V, Krbec M: Weight Theory for Integral Transforms on Spaces of Homogeneous Type. Longman, Harlow; 1998.MATHGoogle Scholar
- Guliyev, VS: Integral operators on function spaces on the homogeneous groups and on domains in R n . Doctor’s degree dissertation. Mat. Inst. Steklov, Moscow, 329 pp. (1994) (in Russian)Google Scholar
- Guliyev, VS: Function spaces, Integral Operators and Two Weighted Inequalities on Homogeneous Groups. Some Applications. Casioglu, Baku, 332 pp. (1999) (in Russian)Google Scholar
- Guliyev VS: Boundedness of the maximal, potential and singular operators in the generalized Morrey spaces. J. Inequal. Appl. 2009., 2009: Article ID 503948Google Scholar
- Deringoz, F, Guliyev, VS, Samko, S: Boundedness of maximal and singular operators and their commutators on vanishing generalized Orlicz-Morrey spaces (submitted)Google Scholar
- Calderon AP: Commutators of singular integral operators. Proc. Natl. Acad. Sci. USA 1965, 53: 1092-1099. 10.1073/pnas.53.5.1092MATHMathSciNetView ArticleGoogle Scholar
- Calderon AP: Cauchy integrals on Lipschitz curves and related operators. Proc. Natl. Acad. Sci. USA 1977,74(4):1324-1327. 10.1073/pnas.74.4.1324MATHMathSciNetView ArticleGoogle Scholar
- Coifman RR, Rochberg R, Weiss G: Factorization theorems for Hardy spaces in several variables. Ann. Math. (2) 1976,103(3):611-635. 10.2307/1970954MATHMathSciNetView ArticleGoogle Scholar
- Coifman R, Lions P, Meyer Y, Semmes S: Compensated compactness and Hardy spaces. J. Math. Pures Appl. 1993, 72: 247-286.MATHMathSciNetGoogle Scholar
- Izuki M, Sawano Y: Variable Lebesgue norm estimates for BMO functions. Czechoslov. Math. J. 2012,62(137)(3):717-727.MathSciNetView ArticleGoogle Scholar
- Guliyev VS: Generalized weighted Morrey spaces and higher order commutators of sublinear operators. Eurasian Math. J. 2012,3(3):33-61.MATHMathSciNetGoogle Scholar
- Guliyev VS, Aliyev SS, Karaman T, Shukurov PS: Boundedness of sublinear operators and commutators on generalized Morrey space. Integral Equ. Oper. Theory 2011,71(3):327-355. 10.1007/s00020-011-1904-1MATHMathSciNetView ArticleGoogle Scholar
- Fu X, Yang D, Yuan W: Boundedness on Orlicz spaces for multilinear commutators of Calderón-Zygmund operators on non-homogeneous spaces. Taiwan. J. Math. 2012, 16: 2203-2238.MATHMathSciNetGoogle Scholar
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