Necessary and sufficient condition for the boundedness of the Gegenbauer–Riesz potential on Morrey spaces
-
Abstract
In this paper, we study the Riesz potential (G-Riesz potential) generated by the Gegenbauer differential operator
We prove that the G-Riesz potential
Also, we prove that the G-Riesz potential
Dedicated to the 80th birthday of Professor V. Kokilashvili
Acknowledgements
The authors thank the anonymous referees for careful reading of the paper and very useful comments.
References
[1] V. I. Burenkov, A. Gogatishvili, V. S. Guliyev and R. Ch. Mustafayev, Boundedness of the Riesz potential in local Morrey-type spaces, Potential Anal. 35 (2011), no. 1, 67–87. 10.1007/s11118-010-9205-xSuche in Google Scholar
[2] V. I. Burenkov and V. S. Guliyev, Necessary and sufficient conditions for the boundedness of the Riesz potential in local Morrey-type spaces, Potential Anal. 30 (2009), no. 3, 211–249. 10.1007/s11118-008-9113-5Suche in Google Scholar
[3] L. Durand, P. M. Fisbane and L. M. Simmons, Exspansion formulas and addition theorems for Gegenbauer functions, J. Math. Phys. 17 (1976), no. 11, 1993–1948. 10.1063/1.522831Suche in Google Scholar
[4] C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107–115. 10.2307/2373450Suche in Google Scholar
[5] I. S. Gradshteyn and I. M. Ryzhik, The Tables of Integrals, Sums, Series and Derivativies (in Russian), Nauka, Moscow, 1971. Suche in Google Scholar
[6] V. S. Guliev, Sobolev’s theorem for the anisotropic Riesz–Bessel potential in Morrey–Bessel spaces (in Russian), Dokl. Akad. Nauk 367 (1999), no. 2, 155–156. Suche in Google Scholar
[7] V. S. Guliyev and J. J. Hasanov, Necessary and sufficient conditions for the boundedness of B-Riesz potential in the B-Morrey spaces, J. Math. Anal. Appl. 347 (2008), no. 1, 113–122. 10.1016/j.jmaa.2008.03.077Suche in Google Scholar
[8]
V. S. Guliyev and E. J. Ibrahimov,
Generalized Gegenbauer shift and some problems of the theory of approximation of functions on the metric of
[9] V. S. Guliyev, M. N. Omarova, M. A. Ragusa and A. Scapellato, Commutators and generalized local Morrey spaces, J. Math. Anal. Appl. 457 (2018), no. 2, 1388–1402. 10.1016/j.jmaa.2016.09.070Suche in Google Scholar
[10] E. I. Ibrahimov, On Gegenbauer transformation on the half-line, Georgian Math. J. 18 (2011), no. 3, 497–515. 10.1515/gmj.2011.0024Suche in Google Scholar
[11] E. J. Ibrahimov and A. Akbulut, The Hardy–Littlewood–Sobolev theorem for Riesz potential generated by Gegenbauer operator, Trans. A. Razmadze Math. Inst. 170 (2016), no. 2, 166–199. 10.1016/j.trmi.2016.05.004Suche in Google Scholar
[12] A. Meskhi, H. Rafeiro and M. A. Zaighum, Interpolation on variable Morrey spaces defined on quasi-metric measure spaces, J. Funct. Anal. 270 (2016), no. 10, 3946–3961. 10.1016/j.jfa.2015.11.013Suche in Google Scholar
[13] A. Scapellato, On some qualitative results for the solution to a Dirichlet problem in local generalized Morrey spaces, AIP Conf. Proc. 1798 (2017), 10.1063/1.4972730. 10.1063/1.4972730Suche in Google Scholar
[14] A. Scapellato, Some properties of integral operators on generalized Morrey spaces, AIP Conf. Proc. 1863 (2017), 10.1063/1.4992662. 10.1063/1.4992662Suche in Google Scholar
© 2018 Walter de Gruyter GmbH, Berlin/Boston
Artikel in diesem Heft
- Frontmatter
- Approximation in generalized Morrey spaces
- On the Cauchy problem for a generalized nonlinear heat equation
- A new principle for arbitrary meromorphic functions in a given domain
- On some generalized Painlevé and Hayman type equations with meromorphic solutions in a bounded domain
- On measurability of real-valued functions in infinite-dimensional topological vector spaces
- A modular variable Orlicz inequality for the local maximal operator
- On the Rellich inequality in Lp(·)(a,b)
- On a generalization of Smirnov’s theorem with some applications
- Space quasiconformal mappings and Neumann eigenvalues in fractal type domains
- Necessary and sufficient condition for the boundedness of the Gegenbauer–Riesz potential on Morrey spaces
- Summability on non-rectifiable Jordan curves
- On generalized fractional cosine and sine transforms
- On mixed norm Bergman–Orlicz–Morrey spaces
- Sharp estimates for the gradient of the generalized Poisson integral for a half-space
- Grand Lebesgue sequence spaces
- Generalized fractional integral operators on generalized Orlicz–Morrey spaces of the second kind over non-doubling metric measure spaces
- A note on N. Bary’s one conjecture
Artikel in diesem Heft
- Frontmatter
- Approximation in generalized Morrey spaces
- On the Cauchy problem for a generalized nonlinear heat equation
- A new principle for arbitrary meromorphic functions in a given domain
- On some generalized Painlevé and Hayman type equations with meromorphic solutions in a bounded domain
- On measurability of real-valued functions in infinite-dimensional topological vector spaces
- A modular variable Orlicz inequality for the local maximal operator
- On the Rellich inequality in Lp(·)(a,b)
- On a generalization of Smirnov’s theorem with some applications
- Space quasiconformal mappings and Neumann eigenvalues in fractal type domains
- Necessary and sufficient condition for the boundedness of the Gegenbauer–Riesz potential on Morrey spaces
- Summability on non-rectifiable Jordan curves
- On generalized fractional cosine and sine transforms
- On mixed norm Bergman–Orlicz–Morrey spaces
- Sharp estimates for the gradient of the generalized Poisson integral for a half-space
- Grand Lebesgue sequence spaces
- Generalized fractional integral operators on generalized Orlicz–Morrey spaces of the second kind over non-doubling metric measure spaces
- A note on N. Bary’s one conjecture
