An Extension of the Adams-type Theorem to the Vanishing Generalized Weighted Morrey Spaces

In this paper, we generalize Adams-type theorems given in [1,13] (which are the following Theorem A and Theorem B, respectively) to the vanishing generalized weighted Morrey spaces. We prove the Adams-type boundedness of the generalized fractional maximal operator $M_{\rho}$ from the vanishing generalized weighted Morrey spaces $\mathcal{\mathcal{VM}}_{p,\varphi^{\frac{1}{p}}}(\mathbb{R}^n, w)$ to another one $\mathcal{\mathcal{VM}}_{q,\varphi^{\frac{1}{q}}}(\mathbb{R}^n, w)$ with $w \in A_{p,q}$ for $1$$p$; and from the vanishing generalized weighted Morrey spaces $\mathcal{\mathcal{VM}}_{1,\varphi}(\mathbb{R}^n, w)$ to the vanishing generalized weighted weak Morrey spaces $\mathcal{\mathcal{VWM}}_{q,\varphi^{\frac{1}{q}}}(\mathbb{R}^n, w)$ with $w \in A_{1,q}$ for $p=1,\ 1$

Keywords:

Generalized fractional maximal operator, Vanishing generalized weighted Morrey space, Muckenhoupt-Weeden classes Muckenhoupt-Weeden class,

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Adams, D.R., A note on Riesz potentials, Duke Math., 42(4)(1975), 765–778.
Coifman, R.R., Fefferman, C., Weighted norm inequalities for maximal functions and singular integrals, Tamkang J. Math., Studia Math., 51(1974), 241–250.
Eridani, A., On the boundedness of a generalized fractional integral on generalized Morrey spaces, Tamkang J. Math., 33(4)(2002), 335–340.
Eridani, A., Gunawan, H., Nakai, E., Sawano, Y., Characterizations for the generalized fractional integral operators on Morrey spaces, Math. Inequal. Appl., 17(2)(2014), 761–777.
Gadjiev, A.D., On generalized potential-type integral operators, Dedicated to Roman Taberski on the occasion of his 70th birthday. Funct. Approx. Comment. Math., 25(1997), 37–44.
Garcia-Cuerva, J., Rubio de Francia, J.L., Weighted Norm Inequalities and Related Topics, North-Holland Math., 16, Amsterdam, 1985.
Guliyev, V.S., Integral operators on function spaces on the homogeneous groups and on domains in Rn. [in Russian], Diss. Steklov Mat. Inst., (1994), Moscow.
Guliyev, V.S., Boundedness of the maximal, potential and singular operators in the generalized Morrey spaces, J. Inequal. Appl., Art. ID 503948, 20 pp. (2009).
Guliyev, V.S., Generalized weighted Morrey spaces and higher order commutators of sublinear operators, Eurasian Math. J., 3(3)(2012), 33–61
Guliyev, V.S., Ismayilova, A.F., Kucukaslan, A., Serbetci, A., Generalized fractional integral operators on generalized local Morrey spaces, Journal of Function Spaces, Volume 2015, Article ID 594323, 8 pages.
Guliyev, V.S., Hasanov, J.J., Badalov, X.A., Commutators of Riesz potential in the vanishing generalized weighted Morrey spaces with variable exponent, Math. Inequal. Appl., 22(1)(2019), 331–351.
Komori, T.Y., Shirai, S., Weighted Morrey spaces and a singular integral operator, Math. Nachr., 282(2)(2009), 219–231.
Kucukaslan, A., Hasanov, S.G, Aykol, C., Generalized fractional integral operators on vanishing generalized local Morrey spaces, Int. J. of Math. Anal., 11(6)(2017), 277–291.
Kucukaslan, A., Guliyev, V.S., Serbetci, A., Generalized fractional maximal operators on generalized local Morrey spaces, Commun. Fac. Sci. Univ. Ank. Ser. A1. Math. Stat., 69(1)(2020), 73–87.
Kucukaslan, A., Equivalence of norms of the generalized fractional integral operator and the generalized fractional maximal operator on generalized weighted Morrey spaces, Ann. Funct. Anal. 11(2020), 1007–1026.
Kucukaslan, A., Two-type estimates for the boundedness of generalized fractional maximal operators on generalized weighted local Morrey spaces, Turk. J. Math. Comput. Sci., 12(1)(2020), 57–66.
Kucukaslan, A., Generalized fractional integrals in the vanishing generalized weighted local and global Morrey spaces, Filomat,(Accepted, 2022).
Mazzucato, A.L., Besov-Morrey spaces: function space theory and applications to non-linear PDE, Trans. Amer. Math. Soc., 355(4)(2003), 1297–1364.
Miranda, C., Sulle equazioni ellittiche del secondo ordine di tipo non variazionale, a coefficienti discontinui, Ann. Math. Pura E Appl. 63(4)(1963), 353–386.
Morrey, C.B., On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc., 43(1938), 126–166.
Muckenhoupt, B., Wheeden, R., Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc., 165(1972), 261–274.
Muckenhoupt, B., Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc., 192(1974), 207–226.
Nakai, E., Hardy–Littlewood maximal operator, singular integral operators and Riesz potentials on generalized Morrey spaces, Math. Nachr., 166(1994), 95–103.
Nakai, E., Hardy-Littlewood maximal operator, singular integral operators and the Riesz potentials on generalized Morrey spaces, Math. Nachr., 166(1994), 95–103.
Persson, L.E., Ragusa, M.A., Samko, N., Wall, P., Commutators of Hardy operators in vanishing Morrey spaces, AIP Conf. Proc. 1493(1)(2012), 859.
Ragusa, M.A., Commutators of fractional integral operators on vanishing-Morrey spaces, J. Global Optim., 40(1-3)(2008), 361–368.
Ruiz, A., Vega, L., Unique continuation for Schr¨odinger operators with potentials in the Morrey class, Publ. Math., 35(2)(1991), 291–298, Conference of Mathematical Analysis (El Escorial, 1989).
Samko, N., Weighted Hardy operators in the local generalized vanishing Morrey spaces, Positivity, 17(2013), 683–706.
Samko, N., Maximal, Potential and singular operators in vanishing generalized Morrey spaces, J. Global Optim., 57(2013), 1385–1399.
Vitanza, C., Functions with vanishing Morrey norm and elliptic partial differential equations, Proceedings of Methods of Real Analysis and Partial Differential Equations, Capri, Springer, (1990), 147–150.
Vitanza, C., Regularity results for a class of elliptic equations with coefficients in Morrey spaces, Ricerche di Matematica 42(2)(1993), 265–281.