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

Abdulhamit Küçükaslan

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 $r$.

Keywords

Generalized fractional maximal operator, Generalized weighted local Morrey spaces, Generalized weighted Morrey spaces, Muckenhoupt-Weeden classes

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.