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Characterizations for the fractional maximal operator and its commutators in generalized weighted Morrey spaces on Carnot groups

Profile image of Vagif S. GuliyevVagif S. Guliyev

Analysis and Mathematical Physics

https://doi.org/10.1007/S13324-020-00360-9
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20 pages

Abstract

In this paper, we shall give a characterization for the strong and weak type Spanne type boundedness of the fractional maximal operator M α , 0 ≤ α < Q on Carnot group G on generalized weighted Morrey spaces M p,ϕ (G, w), where Q is the homogeneous dimension of G. Also we give a characterization for the Spanne type boundedness of the fractional maximal commutator operator M b,α on generalized weighted Morrey spaces.

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In this paper we study the fractional maximal operator M α , 0 ≤ α < Q on the Heisenberg group H n in the generalized Morrey spaces M p,ϕ (H n), where Q = 2n + 2 is the homogeneous dimension of H n. We find the conditions on the pair (ϕ 1 , ϕ 2) which ensures the boundedness of the operator M α from one generalized Morrey space M p,ϕ 1 (H n) to another M q,ϕ 2 (H n), 1 < p < q < ∞, 1/p−1/q = α/Q, and from the space M 1,ϕ1 (H n) to the weak space W M q,ϕ2 (H n), 1 < q < ∞, 1 − 1/q = α/Q. We also find conditions on the ϕ which ensure the Adams type boundedness of M α from M p,ϕ 1 p (H n) to M q,ϕ

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Generalized fractional maximal operator on generalized local Morrey spaces
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Necessary and sufficient conditions for the boundedness of fractional maximal operators in local Morrey-type spaces
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The problem of the boundedness of the fractional maximal operator M , 0 < < n, in local and global Morrey-type spaces is reduced to the problem of the boundedness of the Hardy operator in weighted L p-spaces on the cone of non-negative non-increasing functions. This allows obtaining sharp sufficient conditions for the boundedness for all admissible values of the parameters. Moreover, in case of local Morrey-type spaces, for some values of the parameters, these sufficient conditions coincide with the necessary ones.

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In the present paper, we shall give necessary and sufficient conditions for the Spanne and Adams type boundedness of the commutators of fractional maximal operator on generalized Orlicz-Morrey spaces, respectively. The main advance in comparison with the existing results is that we manage to obtain conditions for the boundedness not in integral terms but in less restrictive terms of supremal operators.

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References (39)

  1. Bernardis, A., Hartzstein, S., Pradolini, G.: Weighted inequalities for commutators of fractional inte- grals on spaces of homogeneous type. J. Math. Anal. Appl. 322, 825-846 (2006)
  2. Bonfiglioli, A., Lanconelli, E., Uguzzoni, F.: Stratified Lie Groups and Potential Theory for Their Sub-Laplacians. Springer Monographs in Mathematics. Springer, Berlin (2007)
  3. Burenkov, V., Gogatishvili, A., Guliyev, V.S., Mustafayev, R.: Boundedness of the fractional maximal operator in local Morrey-type spaces. Complex Var. Elliptic Equ. 55(8-10), 739-758 (2010)
  4. Deringoz, F.: Parametric Marcinkiewicz integral operator and its higher order commutators on gener- alized weighted Morrey spaces. Trans. Natl. Acad. Sci. Azerb. Ser. Phys. Tech. Math. Sci. Math. 37(4), 24-32 (2017)
  5. Deringoz, F., Guliyev, V.S., Hasanov, S.G.: Commutators of fractional maximal operator on generalized Orlicz-Morrey spaces. Positivity 22(1), 141-158 (2018)
  6. Eroglu, A., Guliyev, V.S., Azizov, C.V.: Characterizations for the fractional integral operators in gen- eralized Morrey spaces on Carnot groups. Math. Notes. 102(5-6), 722-734 (2017)
  7. Eroglu, A., Omarova, M.N., Muradova, ShA: Elliptic equations with measurable coefficients in gen- eralized weighted Morrey spaces. Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb. 43(2), 197-213 (2017)
  8. Eroglu, A., Azizov, J.V., Guliyev, V.S.: Fractional maximal operator and its commutators in generalized Morrey spaces on Heisenberg group. Proc. Inst. Math. Mech. Natl. Acad. Sci. Azerb. 44(2), 304-317 (2018)
  9. Guliyev, V.S.: Integral operators on function spaces on the homogeneous groups and on domains in R n , [Russian Doctor's degree dissertation]. Mat. Inst. Steklov, Moscow, p. 329 (1994)
  10. Folland, G.B., Stein, E.M.: Hardy Spaces on Homogeneous Groups. Mathematical Notes, vol. 28. Princeton University Press, Princeton (1982)
  11. Guliyev, V.S.: Function Spaces, Integral Operators and Two Weighted Inequalities on Homogeneous Groups, Some applications (Russian), p. 332. Casioglu, Baku (1999)
  12. Guliyev, V.S.: Boundedness of the maximal, potential and singular operators in the generalized Morrey spaces. J Inequal Appl. 503948, 1-20 (2009)
  13. Guliyev, V.S., Akbulut, A., Mammadov, Y.Y.: Boundedness of fractional maximal operator and their higher order commutators in generalized Morrey spaces on Carnot groups. Acta Math Sci Ser B Engl Ed. 33(5), 1329-1346 (2013)
  14. Guliyev, V.S.: Generalized weighted Morrey spaces and higher order commutators of sublinear oper- ators. Eurasian Math J. 3(3), 33-61 (2012)
  15. Guliyev, V.S., Karaman, T., Mustafayev, RCh., Serbetci, A.: Commutators of sublinear operators gen- erated by Calderón-Zygmund operator on generalized weighted Morrey spaces. Czechoslovak Math. J. (139) 64(2), 365-386 (2014)
  16. Guliyev, V.S., Alizadeh, FCh.: Multilinear commutators of Calderon-Zygmund operator on generalized weighted Morrey spaces. J. Funct. Spaces 710542, 1-9 (2014)
  17. Guliyev, V.S., Omarova, M.N.: Multilinear singular and fractional integral operators on generalized weighted Morrey spaces. Azerb. J. Math. 5(1), 104-132 (2015)
  18. Guliyev, V.S., Hamzayev, V.H.: Rough singular integral operators and its commutators on generalized weighted Morrey spaces. Math. Ineq. Appl. 19(3), 863-881 (2016)
  19. Guliyev, V.S., Muradova, ShA, Omarova, M.N., Softova, L.: Gradient estimates for parabolic equations in generalized weighted Morrey spaces. Acta. Math. Sin. (Engl. Ser.) 32(8), 911-924 (2016)
  20. Guliyev, V.S., Guliyev, R.V., Omarova, M.N.: Riesz transforms associated with Schrödinger operator on vanishing generalized Morrey spaces. Appl. Comput. Math. 17(1), 56-71 (2018)
  21. Guliyev, V.S., Ekincioglu, I., Kaya, E., Safarov, Z.: Characterizations for the fractional maximal oper- ator and its commutators in generalized Morrey spaces on Carnot groups. Integr. Transforms Spec. Funct. 30(6), 453-470 (2019)
  22. Guliyev, V.S., Deringoz, F.: A characterization for fractional integral and its commutators in Orlicz and generalized Orlicz-Morrey spaces on spaces of homogeneous type. Anal. Math. Phys. 9(4), 1991-2019 (2019)
  23. Hamzayev, V.H.: Sublinear operators with rough kernel generated by Calderon-Zygmund operators and their commutators on generalized weighted Morrey spaces. Trans. Natl. Acad. Sci. Azerb. Ser. Phys. Tech. Math. Sci. Math. 38(1), 79-94 (2018)
  24. Ismayilova, A.F.: Fractional maximal operator and its commutators on generalized weighted Morrey spaces. Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. Math. 39(4), 84-95 (2019)
  25. Kaplan, A.: Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadrat- ics forms. Trans. Am. Math. Soc. 258, 147-153 (1980)
  26. Kokilashvili, V., Meskhi, A., Ragusa, M.A.: Weighted extrapolation in grand Morrey spaces and appli- cations to partial differential equations. Atti Accad. Naz. Lincei Rend. Lincei Mater. Appl. 30(1), 67-92 (2019)
  27. Komori, Y., Shirai, S.: Weighted Morrey spaces and a singular integral operator. Math. Nachr. 282(2), 219-231 (2009)
  28. Mizuhara, T.: Boundedness of some classical operators on generalized Morrey spaces. In: Igari, S. (ed.) Harmonic Analysis. ICM 90 Satellite Proceedings, pp. 183-189. Springer, Tokyo (1991)
  29. Morrey, C.B.: On the solutions of quasi-linear elliptic partial differential equations. Trans. Am. Math. Soc. 43, 126-166 (1938)
  30. Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Am. Math. Soc. 165, 207-226 (1972)
  31. Muckenhoupt, B., Wheeden, R.: Weighted norm inequalities for fractional integrals. Trans. Am. Math. Soc. 192, 261-274 (1974)
  32. Nakai, E.: Hardy-Littlewood maximal operator, singular integral operators and Riesz potentials on generalized Morrey spaces. Math. Nachr. 166, 95-103 (1994)
  33. Nakamura, S.: Generalized weighted Morrey spaces and classical operators. Math. Nachr. 289(17-18), 2235-2262 (2016)
  34. Perez, C., Wheeden, R.: Uncertainty principle estimates for vector fields. J. Funct. Anal. 181, 146-188 (1981)
  35. Ragusa, M.A.: Necessary and suffcient condition for a VMO function. Appl. Math. Comput. 218(24), 11952-11958 (2012)
  36. Ragusa, M.A., Scapellato, A.: Mixed Morrey spaces and their applications to partial differential equa- tions. Nonlinear Anal. 151, 51-65 (2017)
  37. Sawano, Y.: A thought on generalized Morrey spaces. J. Indones. Math. Soc. 25(3), 210-281 (2019)
  38. Stein, E.M.: Harmonic Analysis: Real-Variable Methods, Orthogonality and Oscillatory Integrals. Princeton University Press, Princeton (1993)
  39. Zhang, P., Wu, J.: Commutators of the fractional maximal function on variable exponent Lebesgue spaces. Czechoslov. Math. J. 64(1), 183-197 (2014)
About the author
Vagif S. Guliyev
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Related papers

Boundedness of Fractional Maximal Operator and Their Higher Order Commutators in Generalized Morrey Spaces on Carnot Groups
Vagif S. Guliyev

Acta Mathematica Scientia, 2013

We consider generalized Orlicz-Morrey spaces M Φ,ϕ (R n) including their weak versions W M Φ,ϕ (R n). We find the sufficient conditions on the pairs (ϕ 1 , ϕ 2) and (Φ, Ψ) which ensures the boundedness of the fractional maximal operator M α from M Φ,ϕ 1 (R n) to M Ψ,ϕ 2 (R n) and from M Φ,ϕ 1 (R n) to W M Ψ,ϕ 2 (R n). As applications of those results, the boundedness of the commutators of the fractional maximal operator M b,α with b ∈ BM O(R n) on the spaces M Φ,ϕ (R n) is also obtained. In all the cases the conditions for the boundedness are given in terms of supremal-type inequalities on weights ϕ(x, r), which do not assume any assumption on monotonicity of ϕ(x, r) on r.

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Characterizations for the fractional integral operator and its commutators in generalized weighted Morrey spaces on Carnot groups
Ismail Ekincioglu

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In this paper, we shall give a characterization for the strong and weak type Spanne type boundedness of the fractional integral operator I α , 0 < α < Q on Carnot group G on generalized weighted Morrey spaces M p,ϕ (G,w) , respectively, where Q is the homogeneous dimension of G. Also we give a characterization for the Spanne type boundedness of the commutator operator [b,I α ] on generalized weighted Morrey spaces. As applications of the properties of the fundamental solution of sub-Laplacian L on G , we prove two Sobolev-Stein embedding theorems on generalized weighted Morrey spaces in the Carnot group setting.

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Characterizations for the fractional integral operators in generalized Morrey spaces on Carnot groups
Vagif S. Guliyev

Mathematical Notes

In this paper, we study the boundedness of the fractional integral operator I α on Carnot group G in the generalized Morrey spaces M p,ϕ (G). We shall give a characterization for the strong and weak type boundedness of I α on the generalized Morrey spaces, respectively. As applications of the properties of the fundamental solution of sub-Laplacian L on G, we prove two Sobolev-Stein embedding theorems on generalized Morrey spaces in the Carnot group setting.

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Fractional Maximal Operator and Its Commutators in Generalized Morrey Spaces on Heisenberg Group
Vagif S. Guliyev

2018

In this paper we study the boundedness of the fractional maximal operator Mα on Heisenberg group H in the generalized Morrey spaces Mp,φ(H). We shall give a characterization for the strong and weak type Spanne and Adams type boundedness of Mα on the generalized Morrey spaces, respectively. Also we give a characterization for the Spanne and Adams type boundedness of fractional maximal commutator operator Mb,α on the generalized Morrey spaces.

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Necessary and sufficient conditions for boundedness of the fractional maximal operator in the local Morrey-type spaces
Victor Ivanovich Burenkov

Burenkov V. I., Guliyev H. V., Guliyev V. S. Necessary and su- cient conditions for boundedness of the fractional maximal operator in the local Morrey-type spaces. J. Comput. Appl. Math. 208, no. 1 (2007), 280-301., 2007

The problem of boundedness of the fractional maximal operator M α , 0 < α < n in local and global Morrey-type spaces is reduced to the problem of boundedness of the Hardy operator in weighted L p-spaces on the cone of non-negative non-increasing functions. This allows obtaining sharp sufficient conditions for boundedness for all admissible values of the parameters. Moreover, in case of local Morrey-type spaces, for some values of the parameters, these sufficient conditions coincide with the necessary ones. Key Words: maximal operator, fractional maximal operator, local and global Morrey-type spaces, weak Morrey-type spaces, Hardy operator on the cone of monotonic functions.

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