Commutators of Fractional Maximal Operator on Orlicz Spaces

Acta Mathematica Sinica, English Series, 2010

For a Young function Θ and 0 ≤ α < 1, let M α,Θ be the fractional Orlicz maximal operator defined in the context of the spaces of homogeneous type (X, d, µ) by M α,Θ f (x) = sup x∈B µ(B) α ||f || Θ,B , where ||f || Θ,B is the mean Luxemburg norm of f on a ball B. When α = 0 we simply denote it by M Θ . In this paper we prove that if Φ and Ψ are two Young functions, there exists a third Young function Θ such that the composition M α,Ψ • M Φ is pointwise equivalent to M α,Θ . As a consequence we prove that for some Young functions Θ, if M α,Θ f < ∞ a.e. and δ ∈ (0, 1) then (M α,Θ f ) δ is an A 1 -weight.

2013

The problem of boundedness of the fractional maximal operator M α from complementary Morrey-type spaces to Morrey-type spaces is reduced to the problem of boundedness of the dual Hardy operator in weighted L pspaces on the cone of non-negative non-increasing functions, which allows obtaining sharp sufficient conditions for boundedness of M α .

Mediterranean Journal of Mathematics, 2018

In this paper, we give some new characterizations of the Lipschitz spaces via the boundedness of commutators associated with the fractional maximal operator, Riesz potential and Calderón-Zygmund operator on generalized Orlicz-Morrey spaces.

Annales Academiae Scientiarum Fennicae Mathematica, 2015

We prove the boundedness of the Hardy-Littlewood maximal operator and their commutators with BMO-coefficients in vanishing generalized Orlicz-Morrey spaces V M Φ,ϕ (R n) including weak versions of these spaces. The main advance in comparison with the existing results is that we manage to obtain conditions for the boundedness not in integral terms but in less restrictive terms of supremal operators involving the Young function Φ(u) and the function ϕ(x, r) defining the space. No kind of monotonicity condition on ϕ(x, r) in r is imposed.

Journal of Inequalities and Applications, 2019

Let 0 < α < n and M α be the fractional maximal function. The nonlinear commutator of M α and a locally integrable function b is given by [b, M α ](f) = bM α (f)-M α (bf). In this paper, we mainly give some necessary and sufficient conditions for the boundedness of [b, M α ] on variable Lebesgue spaces when b belongs to Lipschitz or BMO(R n) spaces, by which some new characterizations for certain subclasses of Lipschitz and BMO(R n) spaces are obtained.

Acta Mathematica Scientia, 2013

We consider generalized Orlicz-Morrey spaces M Φ,ϕ (R n) including their weak versions W M Φ,ϕ (R n). We find the sufficient conditions on the pairs (ϕ 1 , ϕ 2) and (Φ, Ψ) which ensures the boundedness of the fractional maximal operator M α from M Φ,ϕ 1 (R n) to M Ψ,ϕ 2 (R n) and from M Φ,ϕ 1 (R n) to W M Ψ,ϕ 2 (R n). As applications of those results, the boundedness of the commutators of the fractional maximal operator M b,α with b ∈ BM O(R n) on the spaces M Φ,ϕ (R n) is also obtained. In all the cases the conditions for the boundedness are given in terms of supremal-type inequalities on weights ϕ(x, r), which do not assume any assumption on monotonicity of ϕ(x, r) on r.

Analysis in Theory and Applications, 2020

A significant number of studies have been carried out on the generalized Lebesgue spaces L p(x) , Sobolev spaces W 1,p(x) and Herz spaces. In this paper, we demonstrated a characterization of boundedness of the fractional maximal operator with variable kernel on Herz-Morrey spaces.

arXiv (Cornell University), 2022

In this note, we provide various two-weight norm estimates of the multi-linear fractional maximal function and weighted maximal function between different Orlicz spaces. More precisely, we obtain Sawyer-type characterizations and norm estimates for these operators.

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.

Rocky Mountain Journal of Mathematics, 2008

In this work we find sharp conditions for boundedness on Orlicz spaces of the composition of j operators, each one being of restricted weak type (p, p) for some p > 1, and of strong type (∞, ∞). Particularly, we find necessary and sufficient conditions to obtain modular inequalities for the jtimes composition of the Cesàro Maximal function of order α. With this approach we treat a kind of strong maximal function related to Cesàro averages over n-dimensional rectangles.