Generalized fractional maximal operator on generalized local Morrey spaces

Abstract

In this paper, we study the boundedness of generalized fractional maximal operator M on generalized local Morrey spaces LM fx 0 g p;' and generalized Morrey spaces Mp;', including weak estimates. Firstly, we prove the Spanne type boundedness of M from the space LM fx 0 g p;' 1 to another LM fx 0 g q;' 2 , 1 < p < q < 1 and from LM fx 0 g 1;' 1 to the weak space W LM fx 0 g q;' 2 for p = 1 and 1 < q < 1. Secondly, we prove the Adams type boundedness of M from the space M p;' 1 p to another M q;' 1 q for 1 < p < q < 1 and from M 1;' to the weak space W M q;' 1 qfor p = 1 and 1 < q < 1. In all cases the conditions for the boundedness of M are given in terms of supremal-type integral inequalities on (' 1 ; ' 2 ; ) and ('; ), which do not assume any assumption on monotonicity of ' 1 (x; r), ' 2 (x; r) and '(x; r) in r.

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

“…t n ≤ C ρ(s) s n for s < t. In this case, there is a close and strong relation between the operators M ρ and I ρ (see, [14]) such that…”

“…Nakai [24] introduced the the generalized Morrey spaces M p,φ and proved the boundedness of the generalized fractional integral operator I ρ in these spaces. Nowadays many authors have been culminating important observations about these twooperators M ρ and I ρ especially in connection with Morrey-type spaces (see [3,10,[14][15][16]).…”

Maximal and fractional maximal operators in the Lorentz Morrey spaces and their applications to the Bochner Riesz and Schrodinger type operators

“…Morrey spaces appeared to be useful in the study of local behavior properties of the solutions of second order elliptic PDEs. For more information about Morrey-type spaces see [5,6,10,13] and [14].…”

Two type Estimates for the Boundedness of Generalized Riesz Potential Operator in the Generalized Weighted Local Morrey Spaces

“…During the last decades, the theory of boundedness of classical operators of the harmonic analysis in the generalized Morrey spaces M p,ϕ (R n ) have been well studied by now, we refer the readers to [9,14,15,21] and [23].…”

“…Nakai [22] proved the boundedness of I ρ from the generalized Morrey spaces M 1,ϕ (R n ) to the spaces M 1,ψ (R n ) for suitable functions ϕ and ψ. The boundedness of I ρ from the generalized Morrey spaces M p,ϕ (R n ) to the spaces M q,ψ (R n ) is studied by Eridani [5], Guliyev et al [9], Kucukaslan et al [14,15], Kucukaslan [16,17], Nakai [23] and Nakamura [24].…”