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Boundedness of multilinear oscillatory singular integral on weighted Morrey spaces

  • 1.

    School of Mathematical Science, Huaibei Normal University, Huaibei 235000, China

  • 2.

    School of Mathematics, Hefei University of Technology, Hefei 230009, China

Funds:  Supported by the National Natural Science Foundation of China (11501233), China Postdoctoral Science Foundation (2015M572327), Humanities and Social Sciences Program of the Ministry of Education (15YJC630053), Natural Science Foundation of Anhui Province (1408085MA08; 1508085SMA204), Natural Science Foundation of the Education Department of Anhui Province (KJ2015A335; KJ2015A270).
Cite this:
https://doi.org/10.3969/j.issn.0253-2778.2015.12.004
More Information
  • Author Bio:

    PAN Yali, female, born in 1981, master. Research field： harmonic analysis. E-mail: panliiyang@163.com

  • Corresponding author: LI Changwen
  • Received Date: 11 December 2013
  • Accepted Date: 10 May 2014
  • Rev Recd Date: 10 May 2014
  • Publish Date: 30 December 2015
    Abstract

  • Abstract

    Using the boundedness result of the multilinear oscillatory integral operators on weighted Lebesgue spaces, based on the extrapolation theory, some inequality techniques and decomposition of space, some boundedness of two classes of multilinear oscillatory singular integrals on the weighted Morrey spaces Lp,κ(ω) were obtained.

    Abstract

    Using the boundedness result of the multilinear oscillatory integral operators on weighted Lebesgue spaces, based on the extrapolation theory, some inequality techniques and decomposition of space, some boundedness of two classes of multilinear oscillatory singular integrals on the weighted Morrey spaces Lp,κ(ω) were obtained.
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    • References(24)
  • [1]
    Morrey C B. On the solutions of quasi-linear elliptic partial differential equations[J]. Trans Amer Math Soc, 1938, 43: 126-166.
    [2]
    Muckenhoupt B. Weighted norm inqualities for the Hardy maximal function[J]. Trans Amer Math Soc, 1972, 165: 207-226.
    [3]
    Komori Y,Shirai S. Weighted Morrey spaces and a singular integral operator[J]. Math Nachr, 2009, 282: 219-231.
    [4]
    Chiarenza F, Frasca M. Morrey spaces and Hardy-Littlewood maximal function[J]. Rend MathAppl, 1987, 7: 273-279.
    [5]
    Peetre J. On the theory of Lp,λ spaces[J]. J Funct Anal, 1969, 4: 71-87.
    [6]
    Lu S Z, Ding Y, Yan D Y. Singular integrals and related topics[M]. NJ: World Scientific Publishing, 2007.
    [7]
    Nakai E. Hardy-Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces[J]. Math Nachr, 1994, 166: 95-103.
    [8]
    Sawano Y, Tanaka H. Morrey spaces for non-doubling measures [J]. Acta Math Sini, 2005,21: 1 535-1 544.
    [9]
    Wang H, Liu H P. Some estimates for Bochner-Riesz operators on the weighted Morrey spaces[J].Acta Math Sinica (Chin Ser),2012, 55: 551-560.
    [10]
    Wang H, Liu H P. Weak type estimates of intrinsic square functions on the weighted Hardy spaces [J]. Arch Math, 2011, 97: 49-59.
    [11]
    Mustafayev R Ch. On boundedness of sublinear operators in weighted Morrey spaces[J]. Azerbaijan J Math, 2012, 2(1): 66-79.
    [12]
    Ye X F, Zhu X S. Estimates of singular integrals and multilinear commutators in weighted Morrey spaces [J]. J Inequ and Appl, 2012, 2012: 302.
    [13]
    Wang H. The boundedness of some operators with rough kernel on the weighted Morrey spaces [EB/OL]. http://arxiv.org/abs/1011.5763.
    [14]
    Wang H. The boundedness of fractional integral operators with rough kernels on the weighted Morrey spaces [EB/OL]. http://arxiv.org/abs/1203.1441.
    [15]
    He S. The boundedness of some multilinear operator with rough kernel on the weighted Morrey spaces [EB/OL]. http://arxiv.org/abs/1111. 5463v3.
    [16]
    Wang H. Intrinsic square functions on the weighted Morrey spaces[J]. J Math Anal Appl, 2012, 396: 302-314.
    [17]
    Shi S G, Fu Z W, Lu S Z. Boundedness of oscillatory integral operators and their commutators on weighted Morrey spaces[J]. Sci Sin Math, 2013, 43: 147-158. (in Chinese)
    [18]
    Pan Y L, Li C W, Wang X X. Boundedness of oscillatory integrals with variable Calderón-Zygmund kernel on weighted Morrey spaces[J]. Journal of Function Spaces and Applications, 2013, 2013: Article ID 946435.
    [19]
    Hu G, Yang D C. Multilinear oscillatory singular integral operators on Hardy spaces [J]. Chinese J Contemporary Math, 1997, 18: 403-413.
    [20]
    Hu G E, Wu Q, Yang D C. Boundedness of multilinear oscillatory singular integrals on Hardy type spaces[J]. Hiroshima Math J, 2000, 30: 301-315.
    [21]
    Wu C M, Yang D C. Uniform estimate for multilinear oscillatory singular integrals on weighted Hardy-type spaces[J].Acta in Math, 2002, 31(6): 527-536.
    [22]
    Garcia-Cuerva J, Rubio de Francia J. Weighted norm inequalities and related topics[M]. Amsterdam: North-Holland, 1985.
    [23]
    Gundy R F,Wheeden R L. Weighted integral inequalities for nontangential maximal function, Lusin area integral, and Walsh-Paley series[J]. Studia Math, 1974, 49: 107-124.
    [24]
    Cohen J,Gosselin J. A BMO estimate for multilinear singular integrals [J]. Illinois J Math, 1986, 30: 445-464.
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    [1]
    Morrey C B. On the solutions of quasi-linear elliptic partial differential equations[J]. Trans Amer Math Soc, 1938, 43: 126-166.
    [2]
    Muckenhoupt B. Weighted norm inqualities for the Hardy maximal function[J]. Trans Amer Math Soc, 1972, 165: 207-226.
    [3]
    Komori Y,Shirai S. Weighted Morrey spaces and a singular integral operator[J]. Math Nachr, 2009, 282: 219-231.
    [4]
    Chiarenza F, Frasca M. Morrey spaces and Hardy-Littlewood maximal function[J]. Rend MathAppl, 1987, 7: 273-279.
    [5]
    Peetre J. On the theory of Lp,λ spaces[J]. J Funct Anal, 1969, 4: 71-87.
    [6]
    Lu S Z, Ding Y, Yan D Y. Singular integrals and related topics[M]. NJ: World Scientific Publishing, 2007.
    [7]
    Nakai E. Hardy-Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces[J]. Math Nachr, 1994, 166: 95-103.
    [8]
    Sawano Y, Tanaka H. Morrey spaces for non-doubling measures [J]. Acta Math Sini, 2005,21: 1 535-1 544.
    [9]
    Wang H, Liu H P. Some estimates for Bochner-Riesz operators on the weighted Morrey spaces[J].Acta Math Sinica (Chin Ser),2012, 55: 551-560.
    [10]
    Wang H, Liu H P. Weak type estimates of intrinsic square functions on the weighted Hardy spaces [J]. Arch Math, 2011, 97: 49-59.
    [11]
    Mustafayev R Ch. On boundedness of sublinear operators in weighted Morrey spaces[J]. Azerbaijan J Math, 2012, 2(1): 66-79.
    [12]
    Ye X F, Zhu X S. Estimates of singular integrals and multilinear commutators in weighted Morrey spaces [J]. J Inequ and Appl, 2012, 2012: 302.
    [13]
    Wang H. The boundedness of some operators with rough kernel on the weighted Morrey spaces [EB/OL]. http://arxiv.org/abs/1011.5763.
    [14]
    Wang H. The boundedness of fractional integral operators with rough kernels on the weighted Morrey spaces [EB/OL]. http://arxiv.org/abs/1203.1441.
    [15]
    He S. The boundedness of some multilinear operator with rough kernel on the weighted Morrey spaces [EB/OL]. http://arxiv.org/abs/1111. 5463v3.
    [16]
    Wang H. Intrinsic square functions on the weighted Morrey spaces[J]. J Math Anal Appl, 2012, 396: 302-314.
    [17]
    Shi S G, Fu Z W, Lu S Z. Boundedness of oscillatory integral operators and their commutators on weighted Morrey spaces[J]. Sci Sin Math, 2013, 43: 147-158. (in Chinese)
    [18]
    Pan Y L, Li C W, Wang X X. Boundedness of oscillatory integrals with variable Calderón-Zygmund kernel on weighted Morrey spaces[J]. Journal of Function Spaces and Applications, 2013, 2013: Article ID 946435.
    [19]
    Hu G, Yang D C. Multilinear oscillatory singular integral operators on Hardy spaces [J]. Chinese J Contemporary Math, 1997, 18: 403-413.
    [20]
    Hu G E, Wu Q, Yang D C. Boundedness of multilinear oscillatory singular integrals on Hardy type spaces[J]. Hiroshima Math J, 2000, 30: 301-315.
    [21]
    Wu C M, Yang D C. Uniform estimate for multilinear oscillatory singular integrals on weighted Hardy-type spaces[J].Acta in Math, 2002, 31(6): 527-536.
    [22]
    Garcia-Cuerva J, Rubio de Francia J. Weighted norm inequalities and related topics[M]. Amsterdam: North-Holland, 1985.
    [23]
    Gundy R F,Wheeden R L. Weighted integral inequalities for nontangential maximal function, Lusin area integral, and Walsh-Paley series[J]. Studia Math, 1974, 49: 107-124.
    [24]
    Cohen J,Gosselin J. A BMO estimate for multilinear singular integrals [J]. Illinois J Math, 1986, 30: 445-464.
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