Boundedness Of Fractional Maximal Operator Research Articles

Fractional operators and their commutators on generalized Orlicz spaces

In this paper we examine boundedness of fractional maximal operator. The main focus is on commutators and maximal commutators on generalized Orlicz spaces (also known as Musielak-Orlicz spaces) for fractional maximal functions and Riesz potentials. We prove their boundedness between generalized Orlicz spaces and give a characterization for functions of bounded mean oscillation.

Characterization of the boundedness of fractional maximal operator and its commutators in Orlicz and generalized Orlicz–Morrey spaces on spaces of homogeneous type

In this paper, we establish the necessary and sufficient conditions for the boundedness of fractional maximal operator $$M_{\alpha }$$ and the fractional maximal commutators $$M_{b,\alpha }$$ in Orlicz $$L^{\Phi }(X)$$ and generalized Orlicz–Morrey spaces $$\mathcal {M}^{\Phi ,\varphi }(X)$$ on spaces of homogeneous type $$X=(X,d,\mu )$$ in the sense of Coifman-Weiss.

Boundedness of fractional maximal operators for double phase functionals with variable exponents

In this paper, we study the boundedness of variable fractional maximal operators Mτ(⋅) for double phase functionalsΦp(⋅),q(⋅)(x,t)=tp(x)+(b(x)t)q(x), where τ(⋅) is a measurable function on RN satisfying 0≤infx∈RN⁡τ(x)≤supx∈RN⁡τ(x)≤N, p(⋅), q(⋅) are variable exponents satisfying log-Hölder condition and b(⋅) is a non-negative bounded function which is Hölder continuous of order θ∈(0,1]. We also discuss Sobolev's inequality for variable Riesz potentials Iτ(⋅)f of double phase functionals with variable exponents.

A Characterization of Boundedness of Fractional Maximal Operator with Variable Kernel on Herz-Morrey Spaces

A Characterization of Boundedness of Fractional Maximal Operator with Variable Kernel on Herz-Morrey Spaces

The boundedness of fractional maximal operators on variable Lebesgue spaces over spaces of homogeneous type

Given a space of homogeneous type we give sufficient conditions on a variable exponent {p(.)} so that the fractional maximal operator {M_{\eta}} maps {L^{p(.)}(X)} to {L^{q(.)}(X)}, where {1/p(.) - 1/q(.) = {\eta}}. In the endpoint case we also prove the corresponding weak type inequality. As an application we prove norm inequalities for the fractional integral operator {I_{\eta}}. Our proof for the fractional maximal operator uses the theory of dyadic cubes on spaces of homogeneous type, and even in the Euclidean setting it is simpler than existing proofs. For the fractional integral operator we extend a pointwise inequality of Welland to spaces of homogeneous type. Our work generalizes results from the Euclidean case and extends recent work by Adamowicz, et al. on the Hardy-Littlewood maximal operator on spaces of homogeneous type.

Fractional maximal function and its commutators on Orlicz spaces

In this paper, we find necessary and sufficient conditions for the boundedness of fractional maximal operator $$M_{\alpha }$$ on Orlicz spaces. As an application of this results we consider the boundedness of fractional maximal commutator $$M_{b,\alpha }$$ and nonlinear commutator of fractional maximal operator $$[b,M_{\alpha }]$$ on Orlicz spaces, when b belongs to the Lipschitz space, by which some new characterizations of the Lipschitz spaces are given.

On boundedness of fractional maximal operator in weighted L^p(.) spaces

On boundedness of fractional maximal operator in weighted L^p(.) spaces

Boundedness of Fractional Maximal Operator and its Commutators on Generalized Orlicz–Morrey Spaces

We consider generalized Orlicz–Morrey spaces \(M_{\Phi ,\varphi }({\mathbb {R}^n})\) including their weak versions \(WM_{\Phi ,\varphi }({\mathbb {R}^n})\). We find the sufficient conditions on the pairs \((\varphi _{1},\varphi _{2})\) and \((\Phi , \Psi )\) which ensures the boundedness of the fractional maximal operator \(M_{\alpha }\) from \(M_{\Phi ,\varphi _1}({\mathbb {R}^n})\) to \(M_{\Psi ,\varphi _2}({\mathbb {R}^n})\) and from \(M_{\Phi ,\varphi _1}({\mathbb {R}^n})\) to \(WM_{\Psi ,\varphi _2}({\mathbb {R}^n})\). As applications of those results, the boundedness of the commutators of the fractional maximal operator \(M_{b,\alpha }\) with \(b \in BMO({\mathbb {R}^n})\) on the spaces \(M_{\Phi ,\varphi }({\mathbb {R}^n})\) is also obtained. In all the cases the conditions for the boundedness are given in terms of supremal-type inequalities on weights \(\varphi (x,r)\), which do not assume any assumption on monotonicity of \(\varphi (x,r)\) on \(r\).

Boundedness of Fractional Maximal Operator and Their Higher Order Commutators in Generalized Morrey Spaces on Carnot Groups

In the article we consider the fractional maximal operator Mα,0≤α<Q on any Carnot group G (i.e., nilpotent stratified Lie group) in the generalized Morrey spaces Mp,φ(G), where Q is the homogeneous dimension of G. We find the conditions on the pair φ1, φ2) which ensures the boundedness of the operator Ma from one generalized Morrey space Mp,φ1(G) to another Mq,φ2(G), 1<p≤q<∞, 1/p−1/q=α/Q, and from the space Mp,φ1(G) to the weak space W Mq,φ2(G), 1≤q<∞, 1−1/q=α/Q. Also find conditions on the φ which ensure the Adams type boundedness of the Ma from Mp,φ1p(G) to Mp,φ1q(G) for 1 <p <q < ∞ and from M1,φ(G) to W Mq,φ1q(G) for 1 <q < ∞. In the case b∈BMO(G) and 1 <p <q < ∞, find the sufficient conditions on the pair (φ1, φ2) which ensures the boundedness of the kth-order commutator operator Mb, a, k from Mp,φ1(G) to Mp,φ2(G) with 1/p − 1/q = a/Q. Also find the sufficient conditions on the φ which ensures the boundedness of the operator Mb, α, k from Mp,φ1p(G) to Mq,φ1q(G) for 1 <p <q < ∞. In all the cases the conditions for the boundedness of Ma are given it terms of supremal-type inequalities on (φ1, φ2) and φ, which do not assume any assumption on monotonicity of (φ1, φ2) and φ in r. As applications we consider the Schrödinger operator −ΔG+V on G, where the nonnegative potential V belongs to the reverse Hölder class B∞( G). The Mp, φ1 − Mq, φ2 estimates for the operators Vγ(−ΔG+V)−β and Vγ∇G(−ΔG+V)−β are obtained.

Necessary and sufficient conditions for the boundedness of fractional maximal operators in local Morrey-type spaces

The problem of the boundedness of the fractional maximal operator M α , 0 < α < 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 fractional maximal operators between classical and weak-type Lorentz spaces

We derive conditions, which are both necessary and sufficient, for the boundedness of (power-logarithmic) fractional maximal operators between classical and weak-type Lorentz spaces.