Skip to content
Accessible Unlicensed Requires Authentication Published by De Gruyter March 17, 2016

On the boundedness of a B-Riesz potential in the generalized weighted B-Morrey spaces

Rabil Ayazoglu (Mashiyev) and Javanshir J. Hasanov


We consider the generalized shift operator associated with the Laplace–Bessel differential operator


The maximal operator Mγ (B-maximal operator) and the Riesz potential Iα,γ (B-Riesz potential), associated with the generalized shift operator are investigated. We prove that the B-maximal operator Mγ and the B-singular integral operator are bounded from the generalized weighted B-Morrey space p,ω1,φ,γ(k,+n) to p,ω2,φ,γ(k,+n) for all 1<p<, φAp,γ(k,+n). Furthermore, we prove that the B-Riesz potential Iα,γ, 0<α<n+|γ|, is bounded from the generalized weighted B-Morrey space p,ω1,φ,γ(k,+n) to q,ω2,φ,γ(k,+n), where α/(n+|γ|)=1/p-1/q, 1<p<(n+|γ|)/α, φA1+q/p',γ(k,+n) and 1/p+1/p'=1.

The authors would like to express their gratitude to the referee for very valuable comments and suggestions.


1 D. R. Adams, A note on Riesz potentials, Duke Math. J. 42 (1975), 4, 765–778. Search in Google Scholar

2 I. A. Aliev and A. D. Gadjiev, Weighted estimates for multidimensional singular integrals generated by a generalized shift operator (in Russian), Mat. Sb. 183 (1992), 9, 45–66; translation in Russian Acad. Sci. Sb. Math. 77 (1994), no. 1, 37–55. Search in Google Scholar

3 V. I. Burenkov and H. V. Guliyev, Necessary and sufficient conditions for boundedness of the maximal operator in local Morrey-type spaces, Studia Math. 163 (2004), 2, 157–176. Search in Google Scholar

4 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), 3, 211–249. Search in Google Scholar

5 V. I. Burenkov, V. S. Guliyev, T. V. Tararykova and A. Sherbetchi, Necessary and sufficient conditions for the boundedness of genuine singular integral operators in Morrey-type local spaces (in Russian), Dokl. Akad. Nauk 422 (2008), 1, 11–14; translation in Dokl. Math. 78 (2008), no. 2, 651–654. Search in Google Scholar

6 F. Chiarenza and M. Frasca, Morrey spaces and Hardy–Littlewood maximal function, Rend. Mat. Appl. (7) 7 (1987), 3–4, 273–279. Search in Google Scholar

7 R. R. Coifman and G. Weiss, Analyse harmonique non-commutative sur certains espaces homogènes. Étude de certaines intégrales singulières, Lecture Notes in Math. 242, Springer, Berlin, 1971. Search in Google Scholar

8 A. D. Gadjiev and I. A. Aliev, On classes of operators of potential types, generated by a generalized shift (in Russian), Rep. Enlarged Sess. Semin. I. Vekua Appl. Math. 3 (1987), 2, 21–24. Search in Google Scholar

9 E. V. Guliyev, Weighted inequality for fractional maximal functions and fractional integrals, associated with the Laplace–Bessel differential operator, Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. 26 (2006), 1, 71–80. Search in Google Scholar

10 V. S. Guliyev, Integral Operators on Function Spaces on the Homogeneous Groups and on Domains in ℝn (in Russian), Doctor's degree dissertation, Steklov Institute of Mathematics, Moscow, 1994. Search in Google Scholar

11 V. S. Guliyev, Sobolev's theorem for Riesz B-potentials (in Russian), Dokl. Akad. Nauk 358 (1998), 4, 450–451. Search in Google Scholar

12 V. S. Guliyev, Function Spaces, Integral Operators and Two Weighted Inequalities on Homogeneous Groups. Some Applications (in Russian), Casioglu, Baku,1999. Search in Google Scholar

13 V. S. Guliyev, Sobolev's theorem for the anisotropic Riesz–Bessel potential in Morrey–Bessel spaces (in Russian), Dokl. Akad. Nauk 367 (1999), 2, 155–156. Search in Google Scholar

14 V. S. Guliyev, On maximal function and fractional integral, associated with the Bessel differential operator, Math. Inequal. Appl. 6 (2003), 2, 317–330. Search in Google Scholar

15 V. S. Guliyev and J. J. Hasanov, Sobolev–Morrey type inequality for Riesz potentials, associated with the Laplace–Bessel differential operator, Fract. Calc. Appl. Anal. 9 (2006), 1, 17–32. Search in Google Scholar

16 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), 1, 113–122. Search in Google Scholar

17 V. S. Guliyev and R. C. Mustafaev, Integral operators of potential type in spaces of homogeneous type (in Russian), Dokl. Akad. Nauk 354 (1997), 6, 730–732. Search in Google Scholar

18 V. S. Guliyev and R. C. Mustafayev, Fractional integrals in spaces of functions defined on spaces of homogeneous type (in Russian), Anal. Math. 24 (1998), 3, 181–200. Search in Google Scholar

19 J. J. Hasanov, A note on anisotropic potentials associated with the Laplace–Bessel differential operator, Oper. Matrices 2 (2008), 4, 465–481. Search in Google Scholar

20 I. A. Kiprijanov, Fourier–Bessel transforms and imbedding theorems for weight classes (in Russian), Tr. Mat. Inst. Steklova 89 (1967), 130–213; translated as Proc. Steklov Inst. Math. 89 (1967), 149–246. Search in Google Scholar

21 V. M. Kokilashvili and A. Kufner, Fractional integrals on spaces of homogeneous type, Comment. Math. Univ. Carolin. 30 (1989), 3, 511–523. Search in Google Scholar

22 Y. Komori and S. Shirai, Weighted Morrey spaces and a singular integral operator, Math. Nachr. 282 (2009), 2, 219–231. Search in Google Scholar

23 A. Kufner, O. John and S. Fuçik, Function Spaces, Monogr. Textb. Mech. Solids Fluids Mech. Anal., Noordhoff, Leyden, 1977. Search in Google Scholar

24 B. M. Levitan, Expansion in Fourier series and integrals with Bessel functions (in Russian), Uspekhi Mat. Nauk (N.S.) 6 (1951), 2(42), 102–143. Search in Google Scholar

25 L. N. Lyakhov, Multipliers of the mixed Fourier–Bessel transform (in Russian), Tr. Mat. Inst. Steklova 214 (1997), 234–249; translation in Proc. Steklov Inst. Math. 214 (1996), no. 3 227–242. Search in Google Scholar

26 R. A. Macias and C. Segovia, A Well-Behaved Quasi-Distance for Spaces of Homogeneous Type, CONICET, Buenos Aires, 1981. Search in Google Scholar

27 T. Mizuhara, Boundedness of some classical operators on generalized Morrey spaces, Harmonic Analysis. ICM-90 (Sendai 1990), Satell. Conf. Proc., Springer, Tokyo (1991), 183–189. Search in Google Scholar

28 C. B. Morrey Jr., On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc. 43 (1938), 1, 126–166. Search in Google Scholar

29 B. Muckenhoupt and E. M. Stein, Classical expansions and their relation to conjugate harmonic functions, Trans. Amer. Math. Soc. 118 (1965), 17–92. Search in Google Scholar

30 E. Nakai, Hardy-Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces, Math. Nachr. 166 (1994), 95–103. Search in Google Scholar

31 E. Nakai, Generalized fractional integrals on generalized Morrey spaces, Math. Nachr. 287 (2014), 2–3, 339–351. Search in Google Scholar

32 J. Peetre, On the theory of p,λ spaces, J. Funct. Anal. 4 (1969), 71–87. Search in Google Scholar

33 S. G. Samko, A. A. Kilbas and O. I. Marichev, Fractional Integrals and Derivatives. Theory and Applications, Gordon and Breach Science Publishers, Yverdon, 1993. Search in Google Scholar

34 Y. Sawano, Generalized Morrey spaces for non-doubling measures, NoDEA Nonlinear Differential Equations Appl. 15 (2008), 4–5, 413–425. Search in Google Scholar

35 Y. Sawano, S. Sugano and H. Tanaka, Generalized fractional integral operators and fractional maximal operators in the framework of Morrey spaces, Trans. Amer. Math. Soc. 363 (2011), 12, 6481–6503. Search in Google Scholar

36 A. Ṣerbetçi and I. Ekincioǧlu, Boundedness of Riesz potential generated by generalized shift operator on Ba spaces, Czechoslovak Math. J. 54(129) (2004), 3, 579–589. Search in Google Scholar

37 E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Math. Ser. 30, Princeton University Press, Princeton, 1970. Search in Google Scholar

38 E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Math. Ser. 32, Princeton University Press, Princeton, 1971. Search in Google Scholar

39 K. Stempak, Almost everywhere summability of Laguerre series, Studia Math. 100 (1991), 2, 129–147. Search in Google Scholar

40 K. Trimèche, Inversion of the Lions transmutation operators using generalized wavelets, Appl. Comput. Harmon. Anal. 4 (1997), 1, 97–112. Search in Google Scholar

41 G. V. Welland, Weighted norm inequalities for fractional integrals, Proc. Amer. Math. Soc. 51 (1975), 143–148. Search in Google Scholar

Received: 2014-5-7
Revised: 2014-6-28
Accepted: 2014-11-19
Published Online: 2016-3-17
Published in Print: 2016-6-1

© 2016 by De Gruyter