## Tbilisi Mathematical Journal

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

#### 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

#### 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

#### 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.