ON THE BOUNDEDNESS OF A GENERALIZED FRACTIONAL-MAXIMAL OPERATOR IN LORENTZ SPACES
DOI:
https://doi.org/10.26577/JMMCS.2023.v118.i2.01Keywords:
fractional-maximal function, non-increasing rearrangement, generalized fractional-maximal operator, weighted Lorentz spaces, supremal operatorAbstract
In this paper considers a generalized fractional-maximal operator, a special case of which is the classical fractional-maximal function. Conditions for the function Φ, which defines a generalized fractional-maximal function, and for the weight functions w and v, which determine the weighted Lorentz spaces Λp(v) and Λq(w) (1 < p ≤ q < ∞) under which the generalized maximal-fractional operator is bounded from one Lorentz space Λp(v) to another Lorentz space Λq(w) are obtained. For the classical fractional maximal operator and the classical maximal Hardy-Littlewood function such results were previously known. When proving the main result, we make essential use of an estimate for a nonincreasing rearrangement of a generalized fractional-maximal operator. In addition, we introduce a supremal operator for which conditions of boundedness in weighted Lebesgue spaces are obtained. This result is also essentially used in the proof of the main theorem.
References
[2] Bokayev N.A., Gogatishvili A., Abek A.N., "Cones generated by a generalized fractional maximal function" , Bulletin of the Karaganda university Mathematics series V.110, No.2 (2023): (in print)
[3] Stein E.M., Singular integrals and differentiability properties of functions (M.:MIR, 1973).
[4] Stein E.M., Weiss G., Introduction to fourier analysis on euclidean spaces (Princeton University Press (New Jersey),1971).66 On the boundedness of a generalized fractional-maximal operator in Lorentz spaces. . .
[5] Bennett C., Sharpley R., Interpolation of operators (Pure and Applied Mathematics 129, Academic Press, Boston, MA,1988).
[6] Grafakos L., Classical Fourier Analysis (Second edition) (Springer, 2008).
[7] Sawyer E., "Boundedness of classical operators on classical lorentz spaces" , Studia Mathematica T. XCVI (1990): 145-158.
[8] Cianchi A., Kerman R., Opic B., Pick L., "A sharp rearrangement inequality for fractional maximal operator", Studia Math. 138 (2000): 277-284.
[9] Edmunds D.E., Opic B., "Boundedness of fractional maximal operators between classical and weak-type Lorentz spaces" , Institute of Mathematics, Polish Academy of Sciences in Dissertationes Mathematicae Dissertationes Mathematicae V.410 (2002): 1-50.
[10] Burenkov, V.I., Gogatishvili, A., Guliyev, V.S., Mustafaev, R., "Boundedness of the fractional maximal operator in local Morrey-type spaces" , Complex Analysis and Elliptic Equations V.55, No.8-10 (2010): 739-758.
[11] Gogatishvili, A., Pick, L., Opic, B., "Weighted inequalities for Hardy-type operators involving suprema" , Collect. Math. V.57, No.3 (2006): 227-255.
[12] Hakim, D.I., Nakai, E., Sawano, Y., "Generalized fractional maximal operators and vector-valued inequalities on generalized Orlicz-Morrey spaces" , Revista Matematica Complutense V.29 (2016): 59-90.
[13] Mustafayev, R.Ch., Bilgicli, N., "Generalized fractional maximal functions in Lorentz spaces" , Journal of Mathematical Inequalities V.12, No.3 (2018): 827-851.
[14] Kucukaslan, A., "Equivalence of norms of the generalized fractional integral operator and the generalized fractional maximal operator on the generalized weighted Morrey spaces" , Annals of Functional Analysis V.11 (2020): 1007-1026.
[15] Mustafayev, R.Ch., Bilgicli, N., Yilmaz, M., "Norms of maximal functions berween generalized classical Lorentz spaces" ,arXiv:2110.13698v1 (2021).
[16] Bari N.K., Stechkin S.B., "Best approximations and differential properties of two conjugate functions" , Tr. Mosk. Mat.Obs. V.5 (1956): 483–522 (in Russian)
