Boundedness of the anisotropic fractional maximal operator in anisotropic local Morrey-type spaces (original) (raw)
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Burenkov V. I., Guliyev H. V., Guliyev V. S. Necessary and su- cient conditions for boundedness of the fractional maximal operator in the local Morrey-type spaces. J. Comput. Appl. Math. 208, no. 1 (2007), 280-301., 2007
The problem of boundedness of the fractional maximal operator M α , 0 < α < n in local and global Morrey-type spaces is reduced to the problem of boundedness of the Hardy operator in weighted L p-spaces on the cone of non-negative non-increasing functions. This allows obtaining sharp sufficient conditions for boundedness for all admissible values of the parameters. Moreover, in case of local Morrey-type spaces, for some values of the parameters, these sufficient conditions coincide with the necessary ones. Key Words: maximal operator, fractional maximal operator, local and global Morrey-type spaces, weak Morrey-type spaces, Hardy operator on the cone of monotonic functions.
Journal of Computational and Applied Mathematics, 2007
The problem of the boundedness of the fractional maximal operator M , 0 < < n, in local and global Morrey-type spaces is reduced to the problem of the boundedness of the Hardy operator in weighted L p-spaces on the cone of non-negative non-increasing functions. This allows obtaining sharp sufficient conditions for the boundedness for all admissible values of the parameters. Moreover, in case of local Morrey-type spaces, for some values of the parameters, these sufficient conditions coincide with the necessary ones.
Boundedness of the fractional maximal operator in local Morrey-type spaces
Complex Variables and Elliptic Equations, 2010
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Generalized fractional maximal operator on generalized local Morrey spaces
Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics, 2020
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 q for 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.
Parabolic Fractional Maximal Operator in Parabolic Local Morrey-Type Spaces
In this paper, we study the boundedness of the parabolic fractional maximal operator in parabolic local Morrey-type spaces. We reduce the problem of boundedness of the parabolic fractional maximal operator M α , 0 ≤ α < γ in general parabolic local Morrey-type spaces to the problem of boundedness of the supremal operator in weighted L p-spaces on the cone of non-negative non-decreasing functions.
Parabolic Fractional Maximal Operator in Modified Parabolic Morrey Spaces
Journal of Function Spaces and Applications, 2012
We prove that the parabolic fractional maximal operatorMαP,0≤α<γ, is bounded from the modified parabolic Morrey spaceM̃1,λ,P(ℝn)to the weak modified parabolic Morrey spaceWM̃q,λ,P(ℝn)if and only ifα/γ≤1-1/q≤α/(γ-λ)and fromM̃p,λ,P(ℝn)toM̃q,λ,P(ℝn)if and only ifα/γ≤1/p-1/q≤α/(γ-λ). Hereγ=trPis the homogeneous dimension onℝn. In the limiting case(γ-λ)/α≤p≤γ/αwe prove that the operatorMαPis bounded fromM̃p,λ,P(ℝn)toL∞(ℝn). As an application, we prove the boundedness ofMαPfrom the parabolic Besov-modified Morrey spacesBM̃pθ,λs(ℝn)toBM̃qθ,λs(ℝn). As other applications, we establish the boundedness of some Schrödinger-ype operators on modified parabolic Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class.