# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Boundedness of a class of sublinear operators and their commutators on generalized Morrey spaces. (English) Zbl 1228.42017
Suppose that $T$ is a linear or a sublinear operator which satisfies for any $f\in L_1(\Bbb R^n)$ with compact support and $x\in (\operatorname{supp} f)^c$ $$|Tf(x)|\le c_0\int_{\Bbb R^n}\frac{|f(y)|}{|x-y|^n}\,dy,\tag1$$ where $c_0$ is independent of $f$ and $x$. For a function $a$, suppose that $T_a$ is a commutator generated by $T$ and $a$ satisfies for any $f\in L_1(\Bbb R^n)$ with compact support and $x\in (\operatorname{supp} f)^c$ $$|Tf(x)|\le c_0\int_{\Bbb R^n}\frac{|f(y)||a(x)-a(y)|}{|x-y|^n}\,dy, \tag2$$ where $c_0$ is independent of $f$ and $x$. Let $\varphi(x,r)$ be a positive measurable function on $\Bbb R^n\times (0,\infty)$ and $1\le p<\infty$. We denote by $M_{p,\varphi}$ the generalized Morrey space of all functions $f\in L^{\text{loc}}_p(\Bbb R^n)$ with finite quasinorm $$\|f\|_{M_{p,\varphi}}=\sup_{x\in\Bbb R^n,\,r>0} \varphi(x,r)^{-1}|B(x,r)|^{-\frac{1}{p}}\|f\|_{L_p(B(x,r))}.$$ Also, by $WM_{p,\varphi}$ we denote the weak generalized Morrey space of all functions $f\in WL^{\text{loc}}_p(\Bbb R^n)$ for which $$\|f\|_{WM_{p,\varphi}}=\sup_{x\in\Bbb R^n,\,r>0} \varphi(x,r)^{-1}|B(x,r)|^{-\frac{1}{p}}\|f\|_{WL_p(B(x,r))}<\infty.$$ The authors prove the boundedness of the sublinear operator $T$ satisfying condition (1) generated by the Calderón-Zygmund operator from one generalized Morrey space $M_{p,\varphi_1}$ to another space $M_{p,\varphi_2}$ for $1<p<\infty$ and from $M_{1,\varphi_1}$ to the weak space $WM_{1,\varphi_2}$. When $a\in \text{BMO}$, they find a sufficient condition on the pair $(\varphi_1,\varphi_2)$ which ensures the boundedness of the commutator $T_a$ from $M_{p,\varphi_1}$ to $M_{p,\varphi_2}$ for $1<p<\infty$. Finally, they apply their results to several particular operators such as pseudodifferential operators, the Littlewood-Paley operator, the Marcinkiewicz operator, and the Bochner-Riesz operator.
Reviewer: Yu Liu (Beijing)

##### MSC:
 42B20 Singular and oscillatory integrals, several variables 42B35 Function spaces arising in harmonic analysis
##### Keywords:
Calderón-Zygmund operator; Morrey spaces; boundedness
Full Text:
##### References:
 [1] R. R. Coifman and Y. Meyer, Au Delà des Opérateurs Pseudo-Différentiels, vol. 57 of Astérisque, Société Mathématique de France, Paris, France, 1978. · Zbl 0483.35082 [2] J. García-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, vol. 116 of North-Holland Mathematics Studies, North-Holland, Amsterdam, The Netherlands, 1985. · Zbl 0578.46046 [3] S. Lu, Y. Ding, and D. Yan, Singular Integrals and Related Topics, World Scientific Publishing, Hackensack, NJ, USA, 2007. · Zbl 1124.42011 · doi:10.1142/9789812770561 [4] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, NJ, USA, 1970. · Zbl 0207.13501 [5] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, vol. 43 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, USA, 1993. · Zbl 0821.42001 [6] A. Torchinsky, Real-Variable Methods in Harmonic Analysis, vol. 123 of Pure and Applied Mathematics, Academic Press, Orlando, Fla, USA, 1986. · Zbl 0621.42001 [7] F. Soria and G. Weiss, “A remark on singular integrals and power weights,” Indiana University Mathematics Journal, vol. 43, no. 1, pp. 187-204, 1994. · Zbl 0803.42004 · doi:10.1512/iumj.1994.43.43009 [8] G. Lu, S. Lu, and D. Yang, “Singular integrals and commutators on homogeneous groups,” Analysis Mathematica, vol. 28, no. 2, pp. 103-134, 2002. · Zbl 1026.43007 · doi:10.1023/A:1016568918973 [9] C. B. Morrey, Jr., “On the solutions of quasi-linear elliptic partial differential equations,” Transactions of the American Mathematical Society, vol. 43, no. 1, pp. 126-166, 1938. · Zbl 0018.40501 · doi:10.2307/1989904 [10] J. Peetre, “On the theory of Mp,\lambda ,” The Journal of Functional Analysis, vol. 4, pp. 71-87, 1969. · Zbl 0175.42602 · doi:10.1016/0022-1236(69)90022-6 [11] F. Chiarenza and M. Frasca, “Morrey spaces and Hardy-Littlewood maximal function,” Rendiconti di Matematica e delle sue Applicazioni. Serie VII, vol. 7, no. 3-4, pp. 273-279, 1987. · Zbl 0717.42023 [12] G. Di Fazio and M. A. Ragusa, “Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients,” Journal of Functional Analysis, vol. 112, no. 2, pp. 241-256, 1993. · Zbl 0822.35036 · doi:10.1006/jfan.1993.1032 [13] V. S. Guliyev, Integral operators on function spaces on the homogeneous groups and on domains in Rn, Doctor’s Degree Dissertation, Mathematical Institute, Moscow, Russia, 1994. [14] V. S. Guliyev, Function Spaces, Integral Operators and Two Weighted Inequalities on Homogeneous Groups, Some Applications, Baku, Azerbaijan, 1999. [15] V.S. Guliyev, “Boundedness of the maximal, potential and singular operators in the generalized Morrey spaces,” Journal of Inequalities and Applications, vol. 2009, Article ID 503948, 20 pages, 2009. · Zbl 1193.42082 · doi:10.1155/2009/503948 · eudml:229089 [16] V. S. Guliyev, J. J. Hasanov, and S. G. Samko, “Boundedness of the maximal, potential and singular operators in the generalized variable exponent Morrey spaces,” Mathematica Scandinavica, vol. 107, no. 2, pp. 285-304, 2010. · Zbl 1213.42077 [17] Y. Lin, “Strongly singular Calderón-Zygmund operator and commutator on Morrey type spaces,” Acta Mathematica Sinica (English Series), vol. 23, no. 11, pp. 2097-2110, 2007. · Zbl 1131.42014 · doi:10.1007/s10114-007-0974-0 [18] T. Mizuhara, “Boundedness of some classical operators on generalized Morrey spaces,” in Harmonic Analysis (Sendai, 1990), S. Igari, Ed., ICM 90 Satellite Conference Proceedings, pp. 183-189, Springer, Tokyo, Japan, 1991. · Zbl 0771.42007 [19] E. Nakai, “Hardy-Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces,” Mathematische Nachrichten, vol. 166, pp. 95-103, 1994. · Zbl 0837.42008 · doi:10.1002/mana.19941660108 [20] V. I. Burenkov, V. S. Guliev, and G. V. Guliev, “Necessary and sufficient conditions for the boundedness of the fractional maximal operator in local Morrey-type spaces,” Doklady Akademii Nauk, vol. 409, no. 4, pp. 443-447, 2006. · Zbl 1135.42011 · doi:10.1134/S1064562406040181 [21] V. I. Burenkov, H. V. Guliyev, and V. S. Guliyev, “Necessary and sufficient conditions for the boundedness of fractional maximal operators in local Morrey-type spaces,” Journal of Computational and Applied Mathematics, vol. 208, no. 1, pp. 280-301, 2007. · Zbl 1134.46014 · doi:10.1016/j.cam.2006.10.085 [22] V. I. Burenkov, V. S. Guliyev, A. Serbetci, and T. V. Tararykova, “Necessary and sufficient conditions for the boundedness of genuine singular integral operators in local Morrey-type spaces,” Eurasian Mathematical Journal, vol. 1, no. 1, pp. 32-53, 2010. · Zbl 1215.42019 [23] V. I. Burenkov, A. Gogatishvili, V. S. Guliyev, and R. Ch. Mustafayev, “Boundedness of the fractional maximal operator in local Morrey-type spaces,” Complex Variables and Elliptic Equations. An International Journal of Elliptic Equations and Complex Analysis, vol. 55, no. 8-10, pp. 739-758, 2010. · Zbl 1207.42015 · doi:10.1080/17476930903394697 [24] Y. Ding, D. Yang, and Z. Zhou, “Boundedness of sublinear operators and commutators on Lp,w(\Bbb Rn),” Yokohama Mathematical Journal, vol. 46, no. 1, pp. 15-27, 1998. · Zbl 0969.42009 [25] D. Fan, S. Lu, and D. Yang, “Boundedness of operators in Morrey spaces on homogeneous spaces and its applications,” Acta Mathematica Sinica. New Series, vol. 14, supplement, pp. 625-634, 1998. · Zbl 0916.43006 [26] S. Lu, D. Yang, and Z. Zhou, “Sublinear operators with rough kernel on generalized Morrey spaces,” Hokkaido Mathematical Journal, vol. 27, no. 1, pp. 219-232, 1998. · Zbl 0895.42005 [27] M. Carro, L. Pick, J. Soria, and V. D. Stepanov, “On embeddings between classical Lorentz spaces,” Mathematical Inequalities & Applications, vol. 4, no. 3, pp. 397-428, 2001. · Zbl 0996.46013 [28] A. Akbulut, V. S. Guliyev, and R. Mustafayev, “Boundedness of the maximal operator and singular integral operator in generalized Morrey spaces,” Preprint, Institute of Mathematics, AS CR, Prague, 2010. · Zbl 1250.42038 [29] R. R. Coifman, R. Rochberg, and G. Weiss, “Factorization theorems for Hardy spaces in several variables,” Annals of Mathematics. Second Series, vol. 103, no. 3, pp. 611-635, 1976. · Zbl 0326.32011 · doi:10.2307/1970954 [30] F. Chiarenza, M. Frasca, and P. Longo, “Interior W2,p-estimates for nondivergence elliptic equations with discontinuous coefficients,” Ricerche di Matematica, vol. 40, no. 1, pp. 149-168, 1991. · Zbl 0772.35017 [31] F. Chiarenza, M. Frasca, and P. Longo, “W2,p-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients,” Transactions of the American Mathematical Society, vol. 336, no. 2, pp. 841-853, 1993. · Zbl 0818.35023 · doi:10.2307/2154379 [32] L. Hörmander, “Pseudo-differential operators and hypo-elliptic operators,” in Proceedings of the Symposium in Pure Mathematics of the American Mathematical Society, pp. 138-183, 1967. [33] M. E. Taylor, Pseudo-Differential Operators and Nonlinear PDE, vol. 100 of Progress in Mathematics, Birkhäuser, Boston, Mass, USA, 1991. · Zbl 0746.35062 [34] N. Miller, “Weighted Sobolev spaces and pseudodifferential operators with smooth symbols,” Transactions of the American Mathematical Society, vol. 269, no. 1, pp. 91-109, 1982. · Zbl 0482.35082 · doi:10.2307/1998595 [35] E. M. Stein, “On the functions of Littlewood-Paley, Lusin, and Marcinkiewicz,” Transactions of the American Mathematical Society, vol. 88, pp. 430-466, 1958. · Zbl 0105.05104 · doi:10.2307/1993226 [36] L. Liu, “Weighted weak type estimates for commutators of Littlewood-Paley operator,” Japanese Journal of Mathematics. New Series, vol. 29, no. 1, pp. 1-13, 2003. · Zbl 1046.42013 [37] A. Torchinsky and S. L. Wang, “A note on the Marcinkiewicz integral,” Colloquium Mathematicum, vol. 60/61, no. 1, pp. 235-243, 1990. · Zbl 0731.42019 [38] Y. Liu and D. Chen, “The boundedness of maximal Bochner-Riesz operator and maximal commutator on Morrey type spaces,” Analysis in Theory and Applications, vol. 24, no. 4, pp. 321-329, 2008. · Zbl 1199.42105 · doi:10.1007/s10496-008-0321-z [39] L. Lanzhe and L. Shanzhen, “Weighted weak type inequalities for maximal commutators of Bochner-Riesz operator,” Hokkaido Mathematical Journal, vol. 32, no. 1, pp. 85-99, 2003. · Zbl 1034.42015