Research Article
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Year 2020, Volume: 12 Issue: 1, 56 - 65, 29.06.2020

Abstract

Supporting Institution

TÜBİTAK

Project Number

1059B191600675

References

  • Adams, D.R., \textit{A note on Riesz potentials}, Duke Math., \textbf{42(4)}(1975), 765-778.
  • Burenkov, V., Guliyev, V.S., \textit{Necessary and sufficient conditions for the boundedness of the Riesz operator in local Morrey-type spaces}, Potential Anal., \textbf{30(3)}(2009), 211-249.
  • Burenkov, V., Guliyev, H.V., Guliyev, V.S., \textit{Necessary and sufficient conditions for boundedness of the fractional maximal operator in the local Morrey-type spaces}, J. Comput. Appl. Math., \textbf{208(1)}(2007), 280-301.
  • Cafarelli, L., \textit{Elliptic second order equations}, Rend. Sem. Mat. Fis. Milano, \textbf{58}(1998), 253-284 (1990), DOI 10.1007/BF02925245.
  • Carro, M., Pick, L., Soria, J., Stepanov, V.D., \textit{On embeddings between classical Lorentz spaces}, Math. Inequal. Appl., \textbf{4(3)}(2001), 397-428.
  • Coifman, R.R., Fefferman, C., \textit{Weighted norm inequalities for maximal functions and singular integrals}, Tamkang J. Math., Studia Math., \textbf{51}(1974), 241-250.
  • Eridani, A., \textit{On the boundedness of a generalized fractional integral on generalized Morrey spaces}, Tamkang J. Math., \textbf{33(4)}(2002), 335-340.
  • Eridani, A., Gunawan, H., Nakai, E., Sawano, Y., \textit{Characterizations for the generalized fractional integral operators on Morrey spaces}, Math. Inequal. Appl., \textbf{17(2)}(2014), 761-777.
  • Eridani, A., Gunawan, H., Nakai, E., \textit{On generalized fractional integral operators}, Sci. Math. Jpn., \textbf{60(3)}(2004), 539-550.
  • Gadjiev, A.D., \textit{On generalized potential-type integral operators}, Dedicated to Roman Taberski on the occasion of his 70th birthday. Funct. Approx. Comment. Math., \textbf{25}(1997), 37-44.
  • Garcia-Cuerva, J., Rubio de Francia, J.L., Weighted Norm Inequalities and Related Topics, North-Holland Math., 16, Amsterdam, 1985.
  • Grafakos, L., Classical and Modern Fourier Analysis, Pearson Education, Inc. Upper Saddle River, New Jersey, 2004.
  • Guliyev, V.S., Integral Operators on Function Spaces on The Homogeneous Groups and on Domains in $\Rn$ [in Russian], Diss. Steklov Math. Inst., Moscow, 1994.
  • Guliyev, V.S, \textit{Boundedness of the maximal, potential and singular operators in the generalized Morrey spaces,} J. Inequal. Appl., 2009, Art. ID 503948, 20 pp.
  • Guliyev, V.S., Shukurov, P.S., \textit{On the boundedness of the fractional maximal operator, Riesz potential and their commutators in generalized Morrey spaces}, Advances in Harmonic Analysis and Operator Theory, Series: Operator Theory: Advances and Applications, \textbf{229}(2013), 175-199.
  • Guliyev, V.S., \textit{Generalized weighted Morrey spaces and higher order commutators of sublinear operators}, Eurasian Math. J., \textbf{3(3)}(2012), 33-61.
  • Guliyev, V.S., Ismayilova, A.F., Kucukaslan, A., Serbetci, A., \textit{Generalized fractional integral operators on generalized local Morrey spaces}, Journal of Function Spaces, Volume 2015, Article ID 594323, 8 pages.
  • Gunawan, H., \textit{A note on the generalized fractional integral operators}, J. Indones. Math. Soc., \textbf{9(1)}(2003), 39-43.
  • Komori, T.Y., Shirai, S., \textit{Weighted Morrey spaces and a singular integral operator}, Math. Nachr., \textbf{282(2)}, (2009), 219-231.
  • Kucukaslan, A., Hasanov, S.G, Aykol, C., \textit{Generalized fractional integral operators on vanishing generalized local Morrey spaces}, Int. J. of Math. Anal., \textbf{11(6)}(2017), 277--291.
  • Kucukaslan, A., Guliyev, V.S., Serbetci, A., \textit{Generalized fractional maximal operators on generalized local Morrey spaces}, Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat., \textbf{69(1)}(2020), 73-87, DOI: 10.31801/cfsuasmas.
  • Kucukaslan, A., \textit{Equivalence of norms of the generalized fractional integral operator and the generalized fractional maximal operator on generalized weighted Morrey spaces}, Annals of Funct. Analysis, 2020, DOI: 10.1007/s43034-020-00066-w
  • Mazzucato, A.L., \textit{Besov-Morrey spaces: function space theory and applications to non-linear PDE}, Trans. Amer. Math. Soc., \textbf{355(4)}(2003), 1297-1364, DOI 10.1090/S0002-9947-02-03214-2.
  • Morrey, C.B., \textit{On the solutions of quasi-linear elliptic partial differential equations}, Trans. Amer. Math. Soc., \textbf{43}(1938), 126-166.
  • Muckenhoupt, B., Wheeden, R., \textit{Weighted norm inequalities for the Hardy maximal function}, Trans. Amer. Math. Soc., \textbf{165}(1972), 261-274.
  • Muckenhoupt, B., \textit{Weighted norm inequalities for fractional integrals}, Trans. Amer. Math. Soc., \textbf{192}(1974), 207-226.
  • Mustafayev, R., Kucukaslan, A., \textit{An extension of Muckenhoupt-Wheeden theorem to generalized weighted Morrey spaces}, Georgian Math. Journal, DOI: https://doi.org/10.1515/gmj-2020-2056.
  • Nakai, E., \textit{Hardy-Littlewood maximal operator, singular integral operators and Riesz potentials on generalized Morrey spaces}, Math. Nachr., \textbf{166}(1994), 95-103.
  • Nakai, E., On Generalized Fractional Integrals on The Weak Orlicz Spaces, $BMO^\varphi$, The Morrey Spaces and The Campanato Spaces, Function Spaces, Interpolation Theory and Related Topics (Lund, 2000), 389-401, de Gruyter, Berlin, 2002.
  • Nakai, E., \textit{Generalized fractional integrals on generalized Morrey spaces}, Math. Nachr., \textbf{287(2-3)}(2014), 339--351.
  • Nakamura, S., \textit{Generalized weighted Morrey spaces and classical operators}, Math. Nachr., \textbf{289}, No. 1718, (2016). DOI:10.1002/mana.201500260.
  • Peetre, J., \textit{On the theory of $M_{p,\lambda}$}, J. Funct. Anal., \textbf{4}(1969) 71-87.
  • Ruiz, A., Vega, L., \textit{Unique continuation for Schr\"{o}dinger operators with potentials in the Morrey class}, Publ. Math., \textbf{35(2)}(1991), 291-298, Conference of Mathematical Analysis (El Escorial, 1989).
  • Sawano, Y., Sugano, S., Tanaka, H., \textit{Generalized fractional integral operators and fractional maximal operators in the framework of Morrey spaces}, Trans. Amer. Math. Soc., \textbf{363(12)}(2011), 6481-6503.
  • Sawano, Y., \textit{Boundedness of the Generalized Fractional Integral Operators on Generalized Morrey Spaces over Metric Measure Spaces}, Journal of Analysis and its Applications, \textbf{36(2)}(2017) 159-190 DOI: 10.4171/ZAA/1584
  • Sugano, S., \textit{Some inequalities for generalized fractional integral operators on generalized Morrey spaces}, Math. Inequal. Appl., \textbf{14(4)}(2011), 849--865.

The Two-Type Estimates for The Boundedness of Generalized Fractional Maximal Operator on The Generalized Weighted Local Morrey Spaces

Year 2020, Volume: 12 Issue: 1, 56 - 65, 29.06.2020

Abstract

In this paper, we study two-type estimates which are the Spanne and Adams type estimates for the continuity properties of the generalized fractional maximal operator $M_{\rho}$ on the generalized weighted local Morrey spaces $M^{\{x_0\}}_{p,\varphi}(w^{p})$ and generalized weighted Morrey spaces $M_{p,\varphi^{\frac{1}{p}}}(w)$, including weak estimates. We prove the Spanne type boundedness of the generalized fractional maximal operator $M_{\rho}$ from generalized weighted local Morrey spaces $M^{\{x_0\}}_{p,\varphi_{1}}(w^{p})$ to the weighted weak space $WM^{\{x_0\}}_{q,\varphi_2}(w^{q})$ for $1\leq p< q<\infty$ and from $M^{\{x_0\}}_{p,\varphi_1}(w^{p})$ to another space $M^{\{x_0\}}_{q,\varphi_{2}}(w^{q})$ for $1< p< q<\infty$ with $w^{q} \in A_{1+\frac{q}{p'}}$. We also prove the Adams type boundedness of $M_{\rho}$ from $M_{p,\varphi^{\frac{1}{p}}}(w)$ to the weighted weak space $WM_{q,\varphi^{\frac{1}{q}}}(w)$ for $1\leq pIn all cases the conditions for the boundedness of the operator $M_{\rho}$ are given in terms of supremal-type integral inequalities on the all $\varphi$ functions and $r$ which do not assume any assumption on monotonicity of $\varphi_1(x,r)$, $\varphi_2(x,r)$ and $\varphi(x,r)$ in $r$.

Project Number

1059B191600675

References

  • Adams, D.R., \textit{A note on Riesz potentials}, Duke Math., \textbf{42(4)}(1975), 765-778.
  • Burenkov, V., Guliyev, V.S., \textit{Necessary and sufficient conditions for the boundedness of the Riesz operator in local Morrey-type spaces}, Potential Anal., \textbf{30(3)}(2009), 211-249.
  • Burenkov, V., Guliyev, H.V., Guliyev, V.S., \textit{Necessary and sufficient conditions for boundedness of the fractional maximal operator in the local Morrey-type spaces}, J. Comput. Appl. Math., \textbf{208(1)}(2007), 280-301.
  • Cafarelli, L., \textit{Elliptic second order equations}, Rend. Sem. Mat. Fis. Milano, \textbf{58}(1998), 253-284 (1990), DOI 10.1007/BF02925245.
  • Carro, M., Pick, L., Soria, J., Stepanov, V.D., \textit{On embeddings between classical Lorentz spaces}, Math. Inequal. Appl., \textbf{4(3)}(2001), 397-428.
  • Coifman, R.R., Fefferman, C., \textit{Weighted norm inequalities for maximal functions and singular integrals}, Tamkang J. Math., Studia Math., \textbf{51}(1974), 241-250.
  • Eridani, A., \textit{On the boundedness of a generalized fractional integral on generalized Morrey spaces}, Tamkang J. Math., \textbf{33(4)}(2002), 335-340.
  • Eridani, A., Gunawan, H., Nakai, E., Sawano, Y., \textit{Characterizations for the generalized fractional integral operators on Morrey spaces}, Math. Inequal. Appl., \textbf{17(2)}(2014), 761-777.
  • Eridani, A., Gunawan, H., Nakai, E., \textit{On generalized fractional integral operators}, Sci. Math. Jpn., \textbf{60(3)}(2004), 539-550.
  • Gadjiev, A.D., \textit{On generalized potential-type integral operators}, Dedicated to Roman Taberski on the occasion of his 70th birthday. Funct. Approx. Comment. Math., \textbf{25}(1997), 37-44.
  • Garcia-Cuerva, J., Rubio de Francia, J.L., Weighted Norm Inequalities and Related Topics, North-Holland Math., 16, Amsterdam, 1985.
  • Grafakos, L., Classical and Modern Fourier Analysis, Pearson Education, Inc. Upper Saddle River, New Jersey, 2004.
  • Guliyev, V.S., Integral Operators on Function Spaces on The Homogeneous Groups and on Domains in $\Rn$ [in Russian], Diss. Steklov Math. Inst., Moscow, 1994.
  • Guliyev, V.S, \textit{Boundedness of the maximal, potential and singular operators in the generalized Morrey spaces,} J. Inequal. Appl., 2009, Art. ID 503948, 20 pp.
  • Guliyev, V.S., Shukurov, P.S., \textit{On the boundedness of the fractional maximal operator, Riesz potential and their commutators in generalized Morrey spaces}, Advances in Harmonic Analysis and Operator Theory, Series: Operator Theory: Advances and Applications, \textbf{229}(2013), 175-199.
  • Guliyev, V.S., \textit{Generalized weighted Morrey spaces and higher order commutators of sublinear operators}, Eurasian Math. J., \textbf{3(3)}(2012), 33-61.
  • Guliyev, V.S., Ismayilova, A.F., Kucukaslan, A., Serbetci, A., \textit{Generalized fractional integral operators on generalized local Morrey spaces}, Journal of Function Spaces, Volume 2015, Article ID 594323, 8 pages.
  • Gunawan, H., \textit{A note on the generalized fractional integral operators}, J. Indones. Math. Soc., \textbf{9(1)}(2003), 39-43.
  • Komori, T.Y., Shirai, S., \textit{Weighted Morrey spaces and a singular integral operator}, Math. Nachr., \textbf{282(2)}, (2009), 219-231.
  • Kucukaslan, A., Hasanov, S.G, Aykol, C., \textit{Generalized fractional integral operators on vanishing generalized local Morrey spaces}, Int. J. of Math. Anal., \textbf{11(6)}(2017), 277--291.
  • Kucukaslan, A., Guliyev, V.S., Serbetci, A., \textit{Generalized fractional maximal operators on generalized local Morrey spaces}, Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat., \textbf{69(1)}(2020), 73-87, DOI: 10.31801/cfsuasmas.
  • Kucukaslan, A., \textit{Equivalence of norms of the generalized fractional integral operator and the generalized fractional maximal operator on generalized weighted Morrey spaces}, Annals of Funct. Analysis, 2020, DOI: 10.1007/s43034-020-00066-w
  • Mazzucato, A.L., \textit{Besov-Morrey spaces: function space theory and applications to non-linear PDE}, Trans. Amer. Math. Soc., \textbf{355(4)}(2003), 1297-1364, DOI 10.1090/S0002-9947-02-03214-2.
  • Morrey, C.B., \textit{On the solutions of quasi-linear elliptic partial differential equations}, Trans. Amer. Math. Soc., \textbf{43}(1938), 126-166.
  • Muckenhoupt, B., Wheeden, R., \textit{Weighted norm inequalities for the Hardy maximal function}, Trans. Amer. Math. Soc., \textbf{165}(1972), 261-274.
  • Muckenhoupt, B., \textit{Weighted norm inequalities for fractional integrals}, Trans. Amer. Math. Soc., \textbf{192}(1974), 207-226.
  • Mustafayev, R., Kucukaslan, A., \textit{An extension of Muckenhoupt-Wheeden theorem to generalized weighted Morrey spaces}, Georgian Math. Journal, DOI: https://doi.org/10.1515/gmj-2020-2056.
  • Nakai, E., \textit{Hardy-Littlewood maximal operator, singular integral operators and Riesz potentials on generalized Morrey spaces}, Math. Nachr., \textbf{166}(1994), 95-103.
  • Nakai, E., On Generalized Fractional Integrals on The Weak Orlicz Spaces, $BMO^\varphi$, The Morrey Spaces and The Campanato Spaces, Function Spaces, Interpolation Theory and Related Topics (Lund, 2000), 389-401, de Gruyter, Berlin, 2002.
  • Nakai, E., \textit{Generalized fractional integrals on generalized Morrey spaces}, Math. Nachr., \textbf{287(2-3)}(2014), 339--351.
  • Nakamura, S., \textit{Generalized weighted Morrey spaces and classical operators}, Math. Nachr., \textbf{289}, No. 1718, (2016). DOI:10.1002/mana.201500260.
  • Peetre, J., \textit{On the theory of $M_{p,\lambda}$}, J. Funct. Anal., \textbf{4}(1969) 71-87.
  • Ruiz, A., Vega, L., \textit{Unique continuation for Schr\"{o}dinger operators with potentials in the Morrey class}, Publ. Math., \textbf{35(2)}(1991), 291-298, Conference of Mathematical Analysis (El Escorial, 1989).
  • Sawano, Y., Sugano, S., Tanaka, H., \textit{Generalized fractional integral operators and fractional maximal operators in the framework of Morrey spaces}, Trans. Amer. Math. Soc., \textbf{363(12)}(2011), 6481-6503.
  • Sawano, Y., \textit{Boundedness of the Generalized Fractional Integral Operators on Generalized Morrey Spaces over Metric Measure Spaces}, Journal of Analysis and its Applications, \textbf{36(2)}(2017) 159-190 DOI: 10.4171/ZAA/1584
  • Sugano, S., \textit{Some inequalities for generalized fractional integral operators on generalized Morrey spaces}, Math. Inequal. Appl., \textbf{14(4)}(2011), 849--865.
There are 36 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Abdulhamit Küçükaslan 0000-0002-9207-8977

Project Number 1059B191600675
Publication Date June 29, 2020
Published in Issue Year 2020 Volume: 12 Issue: 1

Cite

APA Küçükaslan, A. (2020). The Two-Type Estimates for The Boundedness of Generalized Fractional Maximal Operator on The Generalized Weighted Local Morrey Spaces. Turkish Journal of Mathematics and Computer Science, 12(1), 56-65.
AMA Küçükaslan A. The Two-Type Estimates for The Boundedness of Generalized Fractional Maximal Operator on The Generalized Weighted Local Morrey Spaces. TJMCS. June 2020;12(1):56-65.
Chicago Küçükaslan, Abdulhamit. “The Two-Type Estimates for The Boundedness of Generalized Fractional Maximal Operator on The Generalized Weighted Local Morrey Spaces”. Turkish Journal of Mathematics and Computer Science 12, no. 1 (June 2020): 56-65.
EndNote Küçükaslan A (June 1, 2020) The Two-Type Estimates for The Boundedness of Generalized Fractional Maximal Operator on The Generalized Weighted Local Morrey Spaces. Turkish Journal of Mathematics and Computer Science 12 1 56–65.
IEEE A. Küçükaslan, “The Two-Type Estimates for The Boundedness of Generalized Fractional Maximal Operator on The Generalized Weighted Local Morrey Spaces”, TJMCS, vol. 12, no. 1, pp. 56–65, 2020.
ISNAD Küçükaslan, Abdulhamit. “The Two-Type Estimates for The Boundedness of Generalized Fractional Maximal Operator on The Generalized Weighted Local Morrey Spaces”. Turkish Journal of Mathematics and Computer Science 12/1 (June 2020), 56-65.
JAMA Küçükaslan A. The Two-Type Estimates for The Boundedness of Generalized Fractional Maximal Operator on The Generalized Weighted Local Morrey Spaces. TJMCS. 2020;12:56–65.
MLA Küçükaslan, Abdulhamit. “The Two-Type Estimates for The Boundedness of Generalized Fractional Maximal Operator on The Generalized Weighted Local Morrey Spaces”. Turkish Journal of Mathematics and Computer Science, vol. 12, no. 1, 2020, pp. 56-65.
Vancouver Küçükaslan A. The Two-Type Estimates for The Boundedness of Generalized Fractional Maximal Operator on The Generalized Weighted Local Morrey Spaces. TJMCS. 2020;12(1):56-65.