Commutators of sublinear operators generated by Calderón-Zygmund operator on generalized weighted Morrey spaces

Vagif Sabir Guliyev; Turhan Karaman; Rza Chingiz Mustafayev; Ayhan Şerbetçi

Czechoslovak Mathematical Journal (2014)

  • Volume: 64, Issue: 2, page 365-385
  • ISSN: 0011-4642

Abstract

top
In this paper, the boundedness of a large class of sublinear commutator operators T b generated by a Calderón-Zygmund type operator on a generalized weighted Morrey spaces M p , ϕ ( w ) with the weight function w belonging to Muckenhoupt’s class A p is studied. When 1 < p < ∞ and b ∈ BMO , sufficient conditions on the pair ( ϕ 1 , ϕ 2 ) which ensure the boundedness of the operator T b from M p , ϕ 1 ( w ) to M p , ϕ 2 ( w ) are found. In all cases the conditions for the boundedness of T b are given in terms of Zygmund-type integral inequalities on ( ϕ 1 , ϕ 2 ) , which do not require any assumption on monotonicity of ϕ 1 ( x , r ) , ϕ 2 ( x , r ) in r . Then these results are applied to several particular operators such as the pseudo-differential operators, Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.

How to cite

top

Guliyev, Vagif Sabir, et al. "Commutators of sublinear operators generated by Calderón-Zygmund operator on generalized weighted Morrey spaces." Czechoslovak Mathematical Journal 64.2 (2014): 365-385. <http://eudml.org/doc/261997>.

@article{Guliyev2014,
abstract = {In this paper, the boundedness of a large class of sublinear commutator operators $T_\{b\}$ generated by a Calderón-Zygmund type operator on a generalized weighted Morrey spaces $M_\{p,\varphi \}(w)$ with the weight function $w$ belonging to Muckenhoupt’s class $A_\{p\}$ is studied. When $1<p<\infty $ and $b \in \{\rm BMO\}$, sufficient conditions on the pair $(\varphi _1,\varphi _2)$ which ensure the boundedness of the operator $T_\{b\}$ from $M_\{p,\varphi _1\}(w)$ to $M_\{p,\varphi _2\}(w)$ are found. In all cases the conditions for the boundedness of $T_\{b\}$ are given in terms of Zygmund-type integral inequalities on $(\varphi _1,\varphi _2)$, which do not require any assumption on monotonicity of $\varphi _1(x,r)$, $\varphi _2(x,r)$ in $r$. Then these results are applied to several particular operators such as the pseudo-differential operators, Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.},
author = {Guliyev, Vagif Sabir, Karaman, Turhan, Mustafayev, Rza Chingiz, Şerbetçi, Ayhan},
journal = {Czechoslovak Mathematical Journal},
keywords = {generalized weighted Morrey space; sublinear operator; commutator; BMO space; maximal operator; Calderón-Zygmund operator; generalized weighted Morrey space; sublinear operator; commutator; BMO; maximal operator; Calderón-Zygmund operator},
language = {eng},
number = {2},
pages = {365-385},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Commutators of sublinear operators generated by Calderón-Zygmund operator on generalized weighted Morrey spaces},
url = {http://eudml.org/doc/261997},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Guliyev, Vagif Sabir
AU - Karaman, Turhan
AU - Mustafayev, Rza Chingiz
AU - Şerbetçi, Ayhan
TI - Commutators of sublinear operators generated by Calderón-Zygmund operator on generalized weighted Morrey spaces
JO - Czechoslovak Mathematical Journal
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 2
SP - 365
EP - 385
AB - In this paper, the boundedness of a large class of sublinear commutator operators $T_{b}$ generated by a Calderón-Zygmund type operator on a generalized weighted Morrey spaces $M_{p,\varphi }(w)$ with the weight function $w$ belonging to Muckenhoupt’s class $A_{p}$ is studied. When $1<p<\infty $ and $b \in {\rm BMO}$, sufficient conditions on the pair $(\varphi _1,\varphi _2)$ which ensure the boundedness of the operator $T_{b}$ from $M_{p,\varphi _1}(w)$ to $M_{p,\varphi _2}(w)$ are found. In all cases the conditions for the boundedness of $T_{b}$ are given in terms of Zygmund-type integral inequalities on $(\varphi _1,\varphi _2)$, which do not require any assumption on monotonicity of $\varphi _1(x,r)$, $\varphi _2(x,r)$ in $r$. Then these results are applied to several particular operators such as the pseudo-differential operators, Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.
LA - eng
KW - generalized weighted Morrey space; sublinear operator; commutator; BMO space; maximal operator; Calderón-Zygmund operator; generalized weighted Morrey space; sublinear operator; commutator; BMO; maximal operator; Calderón-Zygmund operator
UR - http://eudml.org/doc/261997
ER -

References

top
  1. Burenkov, V. I., Gogatishvili, A., Guliyev, V. S., Mustafayev, R. C., Boundedness of the fractional maximal operator in local Morrey-type spaces, Complex Var. Elliptic Equ. 55 739-758 (2010). (2010) Zbl1207.42015MR2674862
  2. Burenkov, V. I., Guliyev, H. V., Guliyev, V. S., 10.1016/j.cam.2006.10.085, J. Comput. Appl. Math. 208 280-301 (2007). (2007) Zbl1134.46014MR2347750DOI10.1016/j.cam.2006.10.085
  3. Burenkov, V. I., Guliyev, V. S., 10.1007/s11118-008-9113-5, Potential Anal. 30 211-249 (2009). (2009) Zbl1171.42003MR2480959DOI10.1007/s11118-008-9113-5
  4. Chiarenza, F., Frasca, M., Longo, P., Interior W 2 , p estimates for non-divergence elliptic equations with discontinuous coefficients, Ric. Mat. 40 149-168 (1991). (1991) MR1191890
  5. Chiarenza, F., Frasca, M., Longo, P., W 2 , p -solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients, Trans. Am. Math. Soc. 336 841-853 (1993). (1993) MR1088476
  6. Coifman, R. R., Meyer, Y., Beyond pseudodifferential operators, Asterisque 57 Société Mathématique de France, Paris (1978), French. (1978) MR0518170
  7. Coifman, R. R., Rochberg, R., Weiss, G., 10.2307/1970954, Ann. Math. 103 611-635 (1976). (1976) Zbl0326.32011MR0412721DOI10.2307/1970954
  8. Fazio, G. Di, Ragusa, M. A., 10.1006/jfan.1993.1032, J. Funct. Anal. 112 241-256 (1993). (1993) Zbl0822.35036MR1213138DOI10.1006/jfan.1993.1032
  9. Ding, Y., Yang, D., Zhou, Z., Boundedness of sublinear operators and commutators on L p , ω ( ℝ n ) , Yokohama Math. J. 46 15-27 (1998). (1998) MR1670757
  10. Garcí{a}-Cuerva, J., Francia, J. L. Rubio de, Weighted Norm Inequalities and Related Topics, North-Holland Mathematics Studies 116. Mathematical Notes 104 North-Holland, Amsterdam (1985). (1985) MR0807149
  11. Grafakos, L., Classical and Modern Fourier Analysis, Pearson/Prentice Hall Upper Saddle River (2004). (2004) Zbl1148.42001MR2449250
  12. Guliyev, V. S., Boundedness of the maximal, potential and singular operators in the generalized Morrey spaces, J. Inequal. Appl. 2009 Article ID 503948, 20 pages (2009). (2009) Zbl1193.42082MR2579556
  13. Guliyev, V. S., Function Spaces, Integral Operators and Two Weighted Inequalities on Homogeneous Groups, Some Applications Baku (1996). (1996) 
  14. Guliyev, V. S., Integral Operators on Function Spaces on the Homogeneous Groups and on Domains in ℝ n . Doctoral Degree Dissertation, Mat. Inst. Steklov Moskva (1994), Russian. (1994) 
  15. Guliyev, V. S., Aliyev, S. S., Karaman, T., Boundedness of a class of sublinear operators and their commutators on generalized Morrey spaces, Abstr. Appl. Anal. 2011 Article ID 356041, 18 pages (2011). (2011) Zbl1228.42017MR2819766
  16. Guliyev, V. S., Hasanov, J. J., Samko, S. G., 10.7146/math.scand.a-15156, Math. Scand. 107 285-304 (2010). (2010) Zbl1213.42077MR2735097DOI10.7146/math.scand.a-15156
  17. Hörmander, L., Pseudo-differential operators and hypoelliptic equations, Proc. Sympos. Pure Math. 10, Chicago, Ill., 1966 American Mathematical Society Providence 138-183 (1967). (1967) Zbl0167.09603MR0383152
  18. Karaman, T., Guliyev, V. S., Serbetci, A., Boundedness of sublinear operators generated by Calderón-Zygmund operators on generalized weighted Morrey spaces, An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) LX f.1, 18 pages (2014). (2014) MR3252469
  19. Komori, Y., Shirai, S., 10.1002/mana.200610733, Math. Nachr. 282 219-231 (2009). (2009) Zbl1160.42008MR2493512DOI10.1002/mana.200610733
  20. Lin, Y., 10.1007/s10114-007-0974-0, Acta Math. Sin., Engl. Ser. 23 2097-2110 (2007). (2007) Zbl1131.42014MR2359125DOI10.1007/s10114-007-0974-0
  21. Lin, Y., Lu, S., Strongly singular Calderón-Zygmund operators and their commutators, Jordan Journal of Mathematics and Statistics 1 31-49 (2008). (2008) Zbl1279.42018MR2261876
  22. Liu, L., 10.4099/math1924.29.1, Jap. J. Math., New Ser. 29 1-13 (2003). (2003) Zbl1046.42013MR1986863DOI10.4099/math1924.29.1
  23. Liu, L., Lu, S., 10.14492/hokmj/1350652427, Hokkaido Math. J. 32 85-99 (2003). (2003) MR1962028DOI10.14492/hokmj/1350652427
  24. Liu, Y., Chen, D., 10.1007/s10496-008-0321-z, Anal. Theory. Appl. 24 321-329 (2008). (2008) Zbl1199.42105MR2471861DOI10.1007/s10496-008-0321-z
  25. Lu, S., Ding, Y., Yan, D., Singular Integrals and Related Topics, World Scientific Publishing Hackensack (2007). (2007) Zbl1124.42011MR2354214
  26. Lu, G., Lu, S., Yang, D., 10.1023/A:1016568918973, Anal. Math. 28 103-134 (2002). (2002) Zbl1026.43007MR1918254DOI10.1023/A:1016568918973
  27. Miller, N., 10.1090/S0002-9947-1982-0637030-4, Trans. Am. Math. Soc. 269 91-109 (1982). (1982) Zbl0482.35082MR0637030DOI10.1090/S0002-9947-1982-0637030-4
  28. Mizuhara, T., Boundedness of some classical operators on generalized Morrey spaces, Harmonic Analysis. Proceedings of a conference in Sendai, Japan, 1990 S. Igari Springer Tokyo 183-189 (1991). (1991) Zbl0771.42007MR1261439
  29. Jr., C. B. Morrey, 10.1090/S0002-9947-1938-1501936-8, Trans. Am. Math. Soc. 43 126-166 (1938). (1938) Zbl0018.40501MR1501936DOI10.1090/S0002-9947-1938-1501936-8
  30. Muckenhoupt, B., Wheeden, R. L., 10.4064/sm-54-3-221-237, Stud. Math. 54 221-237 (1976). (1976) Zbl0318.26014MR0399741DOI10.4064/sm-54-3-221-237
  31. Nakai, E., 10.1002/mana.19941660108, Math. Nachr. 166 95-103 (1994). (1994) Zbl0837.42008MR1273325DOI10.1002/mana.19941660108
  32. Peetre, J., 10.1016/0022-1236(69)90022-6, J. Funct. Anal. 4 71-87 (1969). (1969) MR0241965DOI10.1016/0022-1236(69)90022-6
  33. Polidoro, S., Ragusa, M. A., 10.1023/A:1011261019736, Potential Anal. 14 341-350 (2001). (2001) Zbl0980.35081MR1825690DOI10.1023/A:1011261019736
  34. Sawano, Y., 10.1007/s00030-008-6032-5, NoDEA, Nonlinear Differ. Equ. Appl. 15 413-425 (2008). (2008) Zbl1173.42317MR2465971DOI10.1007/s00030-008-6032-5
  35. Shi, X., Sun, Q., 10.1090/S0002-9939-1992-1136237-1, Proc. Am. Math. Soc. 116 665-673 (1992). (1992) Zbl0786.42006MR1136237DOI10.1090/S0002-9939-1992-1136237-1
  36. Sjölin, P., 10.1007/BF02383638, Ark. Mat. 9 (1971), 65-90. (1971) Zbl0212.41703MR0336222DOI10.1007/BF02383638
  37. Soria, F., Weiss, G., 10.1512/iumj.1994.43.43009, Indiana Univ. Math. J. 43 187-204 (1994). (1994) Zbl0803.42004MR1275458DOI10.1512/iumj.1994.43.43009
  38. Stein, E. M., Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals. With the assistance of Timothy S. Murphy, Princeton Mathematical Series 43. Monographs in Harmonic Analysis III Princeton University Press, Princeton (1993). (1993) Zbl0821.42001MR1232192
  39. Stein, E. M., 10.1090/S0002-9947-1958-0112932-2, Trans. Am. Math. Soc. 88 430-466 (1958). (1958) MR0112932DOI10.1090/S0002-9947-1958-0112932-2
  40. Stein, E. M., Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series 30 Princeton University Press, Princeton (1970). (1970) Zbl0207.13501MR0290095
  41. Taylor, M. E., Pseudodifferential Operators and Nonlinear PDE, Progress in Mathematics 100 Birkhäuser, Boston (1991). (1991) Zbl0746.35062MR1121019
  42. Torchinsky, A., Real-Variable Methods in Harmonic Analysis, Pure and Applied Mathematics 123 Academic Press, Orlando (1986). (1986) Zbl0621.42001MR0869816
  43. Torchinsky, A., Wang, S., 10.4064/cm-60-61-1-235-243, Colloq. Math. 60/61 235-243 (1990). (1990) Zbl0731.42019MR1096373DOI10.4064/cm-60-61-1-235-243
  44. Vargas, A. M., 10.1112/jlms/54.2.297, J. Lond. Math. Soc., II. Ser. 54 297-310 (1996). (1996) Zbl0884.42011MR1405057DOI10.1112/jlms/54.2.297

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.