Tbilisi Mathematical Journal

Oscillatory integrals with variable Calderón-Zygmund kernel on vanishing generalized Morrey spaces

V. S. Guliyev, A. Ahmadli, and S. E. Ekincioglu

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In this paper, the authors investigate the boundedness of the oscillatory singular integrals with variable Calderón-Zygmund kernel on generalized Morrey spaces $M^{p,\varphi}(\mathbb R^n)$ and the vanishing generalized Morrey spaces $VM^{p,\varphi}(\mathbb R^n)$. When $1<p<\infty$ and $(\varphi_1,\varphi_2)$ satisfies some conditions, we show that the oscillatory singular integral operators $T_{\lambda}$ and $T_{\lambda}^{*}$ are bounded from $M^{p,\varphi_1}(\mathbb R^n)$ to $M^{p,\varphi_2}(\mathbb R^n)$ and from $VM^{p,\varphi_1}(\mathbb R^n)$ to $VM^{p,\varphi_2}(\mathbb R^n)$. Meanwhile, the corresponding result for the oscillatory singular integrals with standard Calderón-Zygmund kernel are established.

Note

The research of V. Guliyev was partially supported by the Grant of 1st Azerbaijan-Russia Joint Grant Competition (Agreement Number No. EIF-BGM-4-RFTF-1/2017-21/01/1).

Note

We thank the referee(s) for careful reading the paper and useful comments.

Article information

Source
Tbilisi Math. J., Volume 13, Issue 1 (2020), 69-82.

Dates
Received: 4 September 2019
Accepted: 4 December 2019
First available in Project Euclid: 24 March 2020

Permanent link to this document
https://projecteuclid.org/euclid.tbilisi/1585015221

Digital Object Identifier
doi:10.32513/tbilisi/1585015221

Mathematical Reviews number (MathSciNet)
MR4079451

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
vanishing generalized Morrey space oscillatory integral variable Calderón-Zygmund kernels

Citation

Guliyev, V. S.; Ahmadli, A.; Ekincioglu, S. E. Oscillatory integrals with variable Calderón-Zygmund kernel on vanishing generalized Morrey spaces. Tbilisi Math. J. 13 (2020), no. 1, 69--82. doi:10.32513/tbilisi/1585015221. https://projecteuclid.org/euclid.tbilisi/1585015221


Export citation

References

  • A. Akbulut, V.S. Guliyev and R. Mustafayev, On the boundedness of the maximal operator and singular integral operators in generalized Morrey spaces. Math. Bohem. 137 (1) 2012, 27-43.
  • A. Akbulut, V.S. Guliyev, M.N Omarova, Marcinkiewicz integrals associated with Schrödinger operators and their commutators on vanishing generalized Morrey spaces, Bound. Value Probl. 2017, Paper No. 121, 16 pp.
  • V. Burenkov, V.S. Guliyev, Necessary and sufficient conditions for the boundedness of the Riesz potential in local Morrey-type spaces, Pot. Anal. 30 (3) 2009, 211-249.
  • M. Carro, L. Pick, J. Soria, V.D. Stepanov, On embeddings between classical Lorentz spaces, Math. Inequal. Appl. 4 (3) (2001), 397-428.
  • X. Cao, D. Chen, The boundedness of Toeplitz-type operators on vanishing-Morrey spaces, Anal. Theory Appl. 27 (4) (2011), 309-319.
  • G. Di Fazio and M.A. Ragusa, Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients, J. Funct. Anal. 112 (1993), 241-256.
  • D. Fan, S. Lu and D. Yang, Boundedness of operators in Morrey spaces on homogeneous spaces and its applications, Acta Math. Sinica (N. S.) 14 (1998), suppl., 625-634.
  • Y. Ding, D. Yang, Z. Zhou, Boundedness of sublinear operators and commutators on $L^{p,w}(\Rn)$, Yokohama Math. J. 46 (1) (1998), 15-27.
  • A. Eroglu, V.S. Guliyev, C.V. Azizov, Characterizations for the fractional integral operators in generalized Morrey spaces on Carnot groups, Math. Notes 102 (5) (2017), 127-139.
  • V.S. Guliyev, Integral operators on function spaces on the homogeneous groups and on domains in $\Rn$. Doctor of Sciences, Mat. Inst. Steklova, Moscow, 1994, 329 pp. (in Russian)
  • V.S. Guliyev, Function spaces, integral operators and two weighted inequalities on homogeneous groups. Some applications, Baku. 1999, 1-332. (Russian)
  • V.S. Guliyev, Boundedness of the maximal, potential and singular operators in the generalized Morrey spaces, J. Inequal. Appl. 2009, Art. ID 503948.
  • V.S. Guliyev, S.S. Aliyev, T. Karaman, P. S. Shukurov, Boundedness of sublinear operators and commutators on generalized Morrey Space, Integral Equations Operator Theory 71 (2011), 327-355.
  • V.S. Guliyev, R.V. Guliyev, M.N. Omarova, Riesz transforms associated with Schrödinger operator on vanishing generalized Morrey spaces, Appl. Comput. Math. 17 (1) (2018), 56-71.
  • A. Eroglu, Boundedness of fractional oscillatory integral operators and their commutators on generalized Morrey spaces, Bound. Value Probl. 2013, 2013:70, 12 pp.
  • S. Lu, D. Yang, and Z. Zhou, On local oscillatory integrals with variable Calder´on-Zygmund kernels, Integral Equations Operator Theory 33 (4) (1999), 456-470.
  • T. Mizuhara, Boundedness of some classical operators on generalized Morrey spaces, Harmonic Analysis (S. Igari, Editor), ICM 90 Satellite Proceedings, Springer - Verlag, Tokyo (1991), 183-189.
  • C.B. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (1938), 126-166.
  • E. Nakai, Hardy–Littlewood maximal operator, singular integral operators and Riesz potentials on generalized Morrey spaces, Math. Nachr. 166 (1994), 95-103.
  • 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.
  • Y. Pan, Uniform estimate for oscillatory integral operators, J. Funct. Anal. 130 (1991), 207-220.
  • M.A. Ragusa, Commutators of fractional integral operators on vanishing-Morrey spaces, J. Global Optim. 40 (1-3) (2008), 361-368.
  • M.A. Ragusa, Embeddings for Morrey-Lorentz Spaces, J. Optim. Theory Appl. 154 (2) (2012), 491-499.
  • M.A. Ragusa, Necessary and sufficient condition for a VMO function, Appl. Math. Comput. 218 (24) (2012), 11952-11958.
  • N. Samko, Maximal, potential and singular operators in vanishing generalized Morrey spaces, J. Global Optim. 57 (4) (2013), 1385-1399.
  • Y. Sawano, H. Gunawan, V. Guliyev, H. Tanaka, Morrey spaces and related function spaces [Editorial]. J. Funct. Spaces 2014, Art. ID 867192, 2 pp.
  • Y. Sawano, A thought on generalized Morrey spaces, arXiv:1812.08394v1, 2018, 78 pp.
  • Y. Sawano, S. Sugano, H. Tanaka, A note on generalized fractional integral operators on generalized Morrey spaces, Bound. Value Probl. 2009, Art. ID 835865, 18 pp.
  • L. Softova, Singular integrals and commutators in generalized Morrey spaces, Acta Math. Sin. (Engl. Ser.) 22 (2006), no. 3, 757-766.
  • E.M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, Princeton, NJ, USA, 1971.
  • C. Vitanza, Functions with vanishing Morrey norm and elliptic partial differential equations, In: Proceedings of Methods of Real Analysis and Partial Differential Equations, Capri, 1990, pp. 147-150, Springer.