Fractional Maximal Functions Research Articles

Characterizations for boundedness of fractional maximal function commutators in variable Lebesgue spaces on stratified groups

Characterizations for boundedness of fractional maximal function commutators in variable Lebesgue spaces on stratified groups

Gradient estimates for mixed local and nonlocal parabolic problems with measure data

We obtain a nonlinear Calderón-Zygmund type estimate for the spatial gradient of SOLA (solutions obtained as limits of approximations) to a mixed local and nonlocal parabolic problem involving measure data. Specifically, we consider a non-homogeneous problem with a source term as a Radon measure, where the leading nonlinear local operator has linear growth and the nonlinearity is allowed to be merely measurable. Moreover, the nonlocal operator is of the type fractional Laplacian with only measurable kernel coefficients. Under some smallness assumption on the local operator, we prove that the gradient of SOLA has the same integrability as that of a fractional maximal function of order 1 of the Radon measure. The proof relies on the use of a new type of maximal functions and novel parabolic tail estimates, which to the best of our knowledge, have not been observed in the previous works. Furthermore, to complement the aforementioned gradient estimates, we prove the existence of SOLA to an initial value problem and derive some gradient estimates with respect to the L1-norm.

Some new characterizations of boundedness of commutators of $p$-adic maximal-type functions on $p$-adic Morrey spaces in terms of Lipschitz spaces

<abstract><p>In this note, we investigate some new characterizations of the $p$-adic version of Lipschitz spaces via the boundedness of commutators of the $p$-adic maximal-type functions, including $p$-adic sharp maximal functions, $p$-adic fractional maximal functions, and $p$-adic fractional maximal commutators on $p$-adic Morrey spaces, when a symbol function $b$ belongs to the Lipschitz spaces.</p></abstract>

Some estimates for commutators of the fractional maximal function on stratified Lie groups

In this paper, the main aim is to consider the boundedness of the nonlinear commutator [b, M_{alpha}] and the maximal commutator M_{alpha ,b} on the Lebesgue spaces over some stratified Lie group mathbb{G} when the symbol b belongs to the Lipschitz space. As a result, some new characterizations of the Lipschitz spaces on Lie group via [b, M_{alpha}] and M_{alpha ,b} are given.

Gradient bounds for non-uniformly quasilinear elliptic two-sided obstacle problems with variable exponents

We are concerned with a class of quasilinear elliptic variational inequalities, the so-called two-obstacle problems, that are driven by the double-phase operators with variable exponents. In this paper, by using the terminology of fractional maximal distribution functions, we prove decay estimates for level sets of the gradient of weak solutions in turn implying regularity estimates in some generalized weighted function spaces.

Fractional maximal functions and mean oscillation on bounded doubling metric measure spaces

Let (X,d,μ) be a doubling metric measure space. We consider the behavior of the fractional maximal function Mα for 0≤α<Q, where Q is the doubling dimension, acting on functions of bounded mean oscillation (BMO) and vanishing mean oscillation (VMO). For α>0, we additionally assume that the space is bounded. We show that Mα is bounded from BMO to BLO, a subclass of BMO, and maps VMO to itself when μ has the annular decay property. We also show by means of examples that the action of Mα is not continuous on these function spaces.

Cones generated by a generalized fractional maximal function

The paper considers the space of generalized fractional-maximal function, constructed on the basis of a rearrangement-invariant space. Two types of cones generated by a nonincreasing rearrangement of a generalized fractional-maximal function and equipped with positive homogeneous functionals are constructed. The question of embedding the space of generalized fractional-maximal function in a rearrangement invariant space is investigated. This question reduces to the embedding of the considered cone in the corresponding rearrangement-invariant spaces. In addition, conditions for covering a cone generated by a generalized fractional-maximal function by the cone generated by generalized Riesz potentials are given. Cones from non-increasing rearrangements of generalized potentials were previously considered in the works of M. Goldman, E. Bakhtigareeva, G. Karshygina and others.

Weak and strong type inequalities criteria for fractional maximal functions and fractional integrals associated with Gegenbauer differential operator

Abstract This paper deals with the introduction and study of a new, Orlicz type space associated with the Gegenbauer differential operator and briefly called here a G-Orlicz space. We prove that such spaces are Banach function spaces. The necessary and sufficient conditions are established which ensure the validity of weak and strong type inequalities for fractional maximal functions and fractional integrals of G-Orlicz spaces.

Two-weight inequalities for multilinear maximal functions in Orlicz spaces

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.

Characterization of Lipschitz Space via the Commutators of Fractional Maximal Functions on Variable Lebesgue Spaces

We obtain some new characterizations of a variable version of Lipschitz spaces in terms of the boundedness of commutators of sharp maximal functions, fractional maximal functions or fractional maximal commutators in the context of the variable Lebesgue spaces, where the symbols of the commutators belong to the variable Lipschitz space. A useful tool is that a symbol b belongs a variable Lipschitz space of pointwise type if and only if b belongs to a variable Lipschitz space of integral type under certain assumptions.

Weighted Inequalities for Fractional Maximal Functions on the Infinite Rooted k-Ary Tree

In this article, we introduce the fractional Hardy–Littlewood maximal function on the infinite rooted k-ary tree and study its weighted boundedness. We also provide examples of weights for which the fractional Hardy–Littlewood maximal function satisfies strong type (p, q) estimates on the infinite rooted k-ary tree.

Gradient estimates via Riesz potentials and fractional maximal operators for quasilinear elliptic equations with applications

In this paper, the aim of our work is to establish global weighted gradient estimates via fractional maximal functions and the point-wise regularity estimates of Dirichlet problem for divergence elliptic equations of the type div(A(x,∇u))=div(f)inΩ,andu=gon∂Ω,that related to Riesz potentials. Here, in our setting, Ω⊂Rn, n≥2 is a bounded Reifenberg flat domain (that its boundary is sufficiently flat in sense of Reifenberg) and the small-BMO condition (small bounded mean oscillations) is assumed on the nonlinearity A. Further, the emphasis of the paper is the existence of weak solution to a class of quasilinear elliptic equations containing Riesz potential of the gradient term, as an application of the global point-wise bound. And regarding this study, we also analyze the necessary and sufficient conditions that guarantee the existence of solution to such nonlinear elliptic problems.

Endpoint Sobolev boundedness and continuity of multilinear fractional maximal functions

In this paper, we explore the endpoint Sobolev regularity of the fractional maximal operators in the multilinear setting. Some new bounds and continuity for the derivatives of the multilinear fractional maximal functions are established, both in the centered and in the uncentered cases.

Endpoint Sobolev bounds for fractional Hardy–Littlewood maximal operators

AbstractLet $$0&lt;\alpha &lt;d$$ 0 &lt; α &lt; d and $$1\le p&lt;d/\alpha $$ 1 ≤ p &lt; d / α . We present a proof that for all $$f\in W^{1,p}({\mathbb {R}}^d)$$ f ∈ W 1 , p ( R d ) both the centered and the uncentered Hardy–Littlewood fractional maximal operator $${\mathrm {M}}_\alpha f$$ M α f are weakly differentiable and $$ \Vert \nabla {\mathrm {M}}_\alpha f\Vert _{p^*} \le C_{d,\alpha ,p} \Vert \nabla f\Vert _p , $$ ‖ ∇ M α f ‖ p ∗ ≤ C d , α , p ‖ ∇ f ‖ p , where $$ p^* = (p^{-1}-\alpha /d)^{-1} . $$ p ∗ = ( p - 1 - α / d ) - 1 . In particular it covers the endpoint case $$p=1$$ p = 1 for $$0&lt;\alpha &lt;1$$ 0 &lt; α &lt; 1 where the bound was previously unknown. For $$p=1$$ p = 1 we can replace $$W^{1,1}({\mathbb {R}}^d)$$ W 1 , 1 ( R d ) by $$\mathrm {BV}({\mathbb {R}}^d)$$ BV ( R d ) . The ingredients used are a pointwise estimate for the gradient of the fractional maximal function, the layer cake formula, a Vitali type argument, a reduction from balls to dyadic cubes, the coarea formula, a relative isoperimetric inequality and an earlier established result for $$\alpha =0$$ α = 0 in the dyadic setting. We use that for $$\alpha &gt;0$$ α &gt; 0 the fractional maximal function does not use certain small balls. For $$\alpha =0$$ α = 0 the proof collapses.

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.

Sparse Brudnyi and John–Nirenberg Spaces

A generalization of the theory of Y. Brudnyi [7], and A. and Y. Brudnyi [5, 6], is presented. Our construction connects Brudnyi’s theory, which relies on local polynomial approximation, with new results on sparse domination. In particular, we find an analogue of the maximal theorem for the fractional maximal function, solving a problem proposed by Kruglyak–Kuznetsov. Our spaces shed light on the structure of the John–Nirenberg spaces. We show that SJN p (sparse John–Nirenberg space) coincides with L p ,1<p<∞. This characterization yields the John–Nirenberg inequality by extrapolation and is useful in the theory of commutators.

Riesz Potential and Fractional Maximal Function

In this article, we begin with Riesz potential. We then discuss some properties of the Riesz potential. Finally we discuss a relation of Riesz Potential with fractional maximal function in the sense that fractional maximal function can be controlled by Riesz potential and the fractional maximal function maps the space Lp to Lq whenever the Riesz potential does.

A note on maximal operators related to Laplace-Bessel differential operators on variable exponent Lebesgue spaces

Abstract In this paper, we consider the maximal operator related to the Laplace-Bessel differential operator ( B B -maximal operator) on L p ( ⋅ ) , γ ( R k , + n ) {L}_{p\left(\cdot ),\gamma }\left({{\mathbb{R}}}_{k,+}^{n}) variable exponent Lebesgue spaces. We will give a necessary condition for the boundedness of the B B -maximal operator on variable exponent Lebesgue spaces. Moreover, we will obtain that the B B -maximal operator is not bounded on L p ( ⋅ ) , γ ( R k , + n ) {L}_{p\left(\cdot ),\gamma }\left({{\mathbb{R}}}_{k,+}^{n}) variable exponent Lebesgue spaces in the case of p − = 1 {p}_{-}=1 . We will also prove the boundedness of the fractional maximal function associated with the Laplace-Bessel differential operator (fractional B B -maximal function) on L p ( ⋅ ) , γ ( R k , + n ) {L}_{p\left(\cdot ),\gamma }\left({{\mathbb{R}}}_{k,+}^{n}) variable exponent Lebesgue spaces.

Existence and regularity of solutions for nonlinear measure data problems with general growth

We investigate the existence of distributional solutions to nonlinear elliptic problems with general growth when the right-hand side is a bounded Radon measure. An optimal global Calderon-Zygmund type estimate for such problems is obtained by using mapping property of the fractional maximal function of the measure.

Global regularity for degenerate/singular parabolic equations involving measure data

We consider degenerate and singular parabolic equations with $p$-Laplacian structure in bounded nonsmooth domains when the right-hand side is a signed Radon measure with finite total mass. We develop a new tool that allows global regularity estimates for the spatial gradient of solutions to such parabolic measure data problems, by introducing the (intrinsic) fractional maximal function of a given measure.