Tbilisi Mathematical Journal

Marcinkiewicz integrals with rough kernel associated with Schrödinger operators and commutators on generalized vanishing local Morrey spaces

Ferit Gürbüz

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Abstract

Let $L=-\Delta+V\left( x\right) $ be a Schrödinger operator, where $\Delta$ is the Laplacian on ${\mathbb{R}^{n}}$, while nonnegative potential $V\left(x\right)$ belonging to the reverse Hölder class. In this paper, using the some conditions on $\varphi\left(x.r\right) $, we dwell on the boundedness of Marcinkiewicz integrals with rough kernel associated with schrödinger operators and commutators generated by these operators and local Campanato functions both on generalized local Morrey spaces and on generalized vanishing local Morrey spaces, respectively. As an application of the above results, the boundedness of parametric Marcinkiewicz integral and its commutator both on generalized local Morrey spaces and on generalized vanishing local Morrey spaces is also obtained.

Article information

Source
Tbilisi Math. J., Volume 11, Issue 3 (2018), 133-156.

Dates
Received: 11 April 2017
Accepted: 15 May 2018
First available in Project Euclid: 3 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.tbilisi/1538532032

Digital Object Identifier
doi:10.32513/tbilisi/1538532032

Mathematical Reviews number (MathSciNet)
MR3954200

Subjects
Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Secondary: 42B25: Maximal functions, Littlewood-Paley theory 42B35: Function spaces arising in harmonic analysis

Keywords
Marcinkiewicz operator rough kernel Schrödinger operator generalized local Morrey space generalized vanishing local Morrey space commutator local Campanato space parametric Marcinkiewicz integral

Citation

Gürbüz, Ferit. Marcinkiewicz integrals with rough kernel associated with Schrödinger operators and commutators on generalized vanishing local Morrey spaces. Tbilisi Math. J. 11 (2018), no. 3, 133--156. doi:10.32513/tbilisi/1538532032. https://projecteuclid.org/euclid.tbilisi/1538532032


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References

  • D. R. Adams, Morrey spaces, Lecture Notes in Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, Cham, 2015.
  • A. Akbulut and O. Kuzu, Marcinkiewicz integrals with rough kernel associated with Schrödinger operator on vanishing generalized Morrey Spaces. Azerb. J. Math., 4 (2014), 40-54.
  • A. S. Balakishiyev, V. S. Guliyev, F. Gürbüz and A. Serbetci, Sublinear operators with rough kernel generated by Calderon-Zygmund operators and their commutators on generalized local Morrey spaces, J. Inequal. Appl., 2015:61, (2015), 1-18.
  • X. N. Cao and D. X. Chen, The boundedness of Toeplitz-type operators on vanishing Morrey spaces, Anal. Theory Appl., 27 (2011), 309-319.
  • S. Chanillo, A note on commutators, Indiana Univ. Math. J., 31 (1982), 7-16.
  • Y. Chen, Y. Ding and X. Wang, Compactness of commutators for singular integrals on Morrey spaces, Canad. J. Math., 64 (2012), 257-281.
  • Y. Chen, Y. Ding and G. Hong, Commutators with fractional differentiation and new characterizations of BMO-Sobolev spaces, Analysis & PDE, 9 (2016), 1497-1522.
  • F. Chiarenza, M. Frasca and P. Longo, Interior $W^{2,p}$-estimates for nondivergence elliptic equations with discontinuous coefficients, Ricerche Mat., 40 (1991), 149-168.
  • F. Chiarenza, M. Frasca and P. Longo, $W^{2,p} $-solvability of Dirichlet problem for nondivergence elliptic equations with VMO coefficients, Trans. Amer. Math. Soc., 336 (1993), 841-853.
  • W. Gao and L. Tang, Boundedness for Marcinkiewicz integrals associated with Schrödinger operators, Proc. Indian Acad. Sci., 124 (2014), 193-203.
  • V. S. Guliyev and L. G. Softova, Global regularity in generalized Morrey spaces of solutions to nondivergence elliptic equations with VMO coefficients, Potential Anal., 38 (2013), 843-862.
  • F. Gürbüz, Boundedness of some potential type sublinear operators and their commutators with rough kernels on generalized local Morrey spaces, (Turkish) $\left[ \text{\textit{Ph.D. thesis}}\right] $, Ankara University, Ankara, Turkey, 2015.
  • F. Gürbüz, Sublinear operators with rough kernel generated by fractional integrals and commutators on generalized vanishing local Morrey spaces, arXiv:1602.07853 [math.AP].
  • F. Gürbüz, Multi-sublinear operators generated by multilinear fractional integral operators and commutators on the product generalized local Morrey spaces, Adv. Math. (China) (to appear), 47 (2018), doi: 10.11845/sxjz.2017093b.
  • F. Gürbüz, Parabolic sublinear operators with rough kernel generated by parabolic Calderón-Zygmund operators and parabolic local Campanato space estimates for their commutators on the parabolic generalized local Morrey spaces, Open Math., 14 (2016), 300-323.
  • F. Gürbüz, Parabolic generalized local Morrey space estimates of rough parabolic sublinear operators and commutators, Adv. Math. (China), 46 (2017), 765-792.
  • F. Gürbüz, Some estimates for generalized commutators of rough fractional maximal and integral operators on generalized weighted Morrey spaces, Canad. Math. Bull., 60 (2017), 131-145.
  • F. Gürbüz, Sublinear operators with rough kernel generated by Calderón-Zygmund operators and their commutators on generalized Morrey spaces, Math. Notes, 101 (2017), 429-442.
  • L. Hörmander, Translation invariant operators, Acta Math., 104 (1960), 93-139.
  • S. Janson, Mean oscillation and commutators of singular integral operators, Ark. Mat., 16 (1978), 263-270.
  • S. Z. Lu, Y. Ding and D. Y. Yan, Singular integrals and related topics, World Scientific Publishing, Singapore, (2006).
  • C. Miranda, Sulle equazioni ellittiche del secondo ordine di tipo non variazionale, a coefficienti discontinui, Ann. Math. Pura E Appl., 63 (1963), 353-386.
  • C. B. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc., 43 (1938), 126-166.
  • D. K. Palagachev and L. G. Softova, Singular integral operators, Morrey spaces and fine regularity of solutions to PDE's, Potential Anal., 20 (2004), 237-263.
  • M. Paluszynski, Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss, Indiana Univ. Math. J., 44 (1995), 1-17.
  • M. A. Ragusa, Commutators of fractional integral operators on vanishing-Morrey spaces, J. Global Optim., 40 (2008), 361–368.
  • N. Samko, Maximal, Potential and Singular Operators in vanishing generalized Morrey Spaces, J. Global Optim., 57 (2013), 1385-1399.
  • Z. Shen, $L_{p}$ estimates for Schrödinger operators with certain potentials, Ann. Inst. Fourier (Grenoble), 45 (1995), 513-546.
  • L. G. Softova, Singular integrals and commutators in generalized Morrey spaces, Acta Math. Sin. (Engl. Ser.), 22 (2006), 757-766.
  • E. M. Stein, On the functions of Littlewood-Paley, Lusin and Marcinkiewicz, Trans. Amer. Math. Soc., 88 (1958), 430-466.
  • E. M. Stein, Singular integrals and differentiability of functions, Princeton University Press, Princeton, NJ, (1970).
  • A. Torchinsky, Real Variable Methods in Harmonic Analysis, Pure and Applied Math. 123, Academic Press, New York, (1986).
  • C. Vitanza, Functions with vanishing Morrey norm and elliptic partial differential equations. In: Proceedings of Methods of Real Analysis and Partial Differential Equations,Capri, pp. 147-150. Springer (1990).
  • C. Vitanza, Regularity results for a class of elliptic equations with coefficients in Morrey spaces, Ricerche Mat., 42 (1993), 265-281.
  • R. L. Wheeden and A. Zygmund, Measure and Integral: An Introduction to Real Analysis, vol. 43 of Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, (1977).