Hardy-Littlewood Maximal Operator Research Articles

Uniform boundedness of Kantorovich operators in Morrey spaces

In this paper, the Kantorovich operators $$K_n, n\in \mathbb {N}$$ are shown to be uniformly bounded in Morrey spaces on the closed interval [0, 1]. Also an upper estimate is obtained for the difference $$K_n(f)-f$$ for functions f of regularity of order 1 measured in Morrey spaces. One of the key tools is the pointwise inequality for the Kantorovich operators and the Hardy–Littlewood maximal operator, which is of interest on its own and can be applied to other problems related to the Kantorovich operators.

A remark on modified Morrey spaces on metric measure spaces

Morrey norms, which are originally endowed with two parameters, are considered on general metric measure spaces. Volberg, Nazarov and Treil showed that the modified Hardy-Littlewood maximal operator is bounded on Legesgue spaces. The modified Hardy-Littlewood maximal operator is known to be bounded on Morrey spaces on Euclidean spaces, if we introduce the third parameter instead of adopting a natural extension of Morrey spaces. When it comes to geometrically doubling, as long as an auxiliary parameter is introduced suitably, the Morrey norm does not depend on the third parameter and this norm extends naturally the original Morrey norm. If the underlying space has a rich geometric structure, there is still no need to introduce auxiliary parameters. However, an example shows that this is not the case in general metric measure spaces. In this paper, we present an example showing that Morrey spaces depend on the auxiliary parameters.

Weighted boundedness of the 2‐fold product of Hardy–Littlewood maximal operators

AbstractWe study new weighted estimates for the 2‐fold product of Hardy–Littlewood maximal operators defined by . This operator appears very naturally in the theory of bilinear operators such as the bilinear Calderón–Zygmund operators, the bilinear Hardy–Littlewood maximal operator introduced by Calderón or in the study of pseudodifferential operators. To this end, we need to study Hölder's inequality for Lorentz spaces with change of measures urn:x-wiley:0025584X:media:mana201700271:mana201700271-math-0002Unfortunately, we shall prove that this inequality does not hold, in general, and we shall have to consider a weaker version of it.

Level Set Estimates for the Discrete Frequency Function

We introduce the discrete frequency function as a possible new approach to understanding the discrete Hardy–Littlewood maximal function. Considering that the discrete Hardy–Littlewood maximal function is given at each integer by the supremum of averages over intervals of integer length, we define the discrete frequency function at that integer as the value at which the supremum is attained. After verifying that the function is well-defined, we investigate size and smoothness properties of this function.

On Morrey-type classes of harmonic functions

Weighted Morrey-type classes of functions that are harmonic in the unit disk and in the upper half plane are defined in this work. Under some conditions on the weight function, we study some properties of functions belonging to these classes. Maximum values of harmonic functions for a nontangential angle are estimated through the Hardy–Littlewood maximal function, and then the boundedness of Hardy–Littlewood operator is applied in the Morrey-type spaces. Weighted Morrey–Lebesgue type space is defined, where the shift operator is continuous with respect to shift, and its invariance with regard to the singular operator is proved. The validity of Minkowski inequality in Morrey–Lebesgue type spaces is also proved.

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.

Dimension free bounds for the Hardy–Littlewood maximal operator associated to convex sets

This survey is based on a series of lectures given by the authors at the working seminar Convexite et Probabilites at UPMC Jussieu, Paris, during the spring 2013. It is devoted to maximal inequalities associated to symmetric convex sets in high dimensional linear spaces, a topic mainly developed between 1982 and 1990 but recently renewed by further advances. The series focused on proving for these maximal functions inequalities in $L^p(\mathbb{R}^n)$ with bounds independent of the dimension $n$, for all $p \in (1, +\infty]$ in the best cases. This program was initiated in 1982 by Elias Stein, who obtained the first theorem of this kind for the family of Euclidean balls in arbitrary dimension. We present several results along this line, proved by Bourgain, Carbery and Muller during the period 1986--1990, and a new one due to Bourgain (2014) for the family of cubes in arbitrary dimension. We complete the cube case with negative results for the weak type $(1, 1)$ constant, due to Aldaz, Aubrun and Iakovlev--Stromberg between 2009 and 2013.

On relations between weak and strong type inequalities for maximal operators on non-doubling metric measure spaces

In this article we characterize all possible cases that may occur in the relations between the sets of p for which weak type (p, p) and strong type (p, p) inequalities for the Hardy–Littlewood maximal operators, both centered and non-centered, hold in the context of general metric measure spaces.

Pointwise estimates to the modified Riesz potential

In a smooth domain a function can be estimated pointwise by the classical Riesz potential of its gradient. Combining this estimate with the boundedness of the classical Riesz potential yields the optimal Sobolev–Poincare inequality. We show that this method gives a Sobolev–Poincare inequality also for irregular domains whenever we use the modified Riesz potential which arise naturally from the geometry of the domain. The exponent of the Sobolev–Poincare inequality depends on the domain. The Sobolev–Poincare inequality given by this approach is not sharp for irregular domains, although the embedding for the modified Riesz potential is optimal. In order to obtain the results we prove a new pointwise estimate for the Hardy–Littlewood maximal operator.

Multilinear Commutators of Singular Integral Operators in Variable Exponent Herz-Type Spaces

In this paper, we study the boundedness of multilinear commutators of Hardy–Littlewood maximal operators in variable exponent Herz and Herz–Morrey spaces, which in turn are used to obtain the boundedness for a large class of the multilinear commutators related to sublinear operators. Moreover, based on the atomic decomposition and on generalization of the BMO norm, we study the boundedness of multilinear commutators of singular integral operators with Calderon–Zygmund kernels in variable exponent Herz-type Hardy spaces.

The boundedness of generalized Bessel-Riesz operators on generalized Morrey spaces

The purpose of this paper is to prove the boundedness of generalized Bessel-Riesz operators on generalized Morrey spaces. The kernel of the operators contain some parameters, one of which is related to Bessel decay. As usual, we use the usual dyadic decomposition, Hölder’s inequality, a Hedberg-type inequality for the operators, and the boundedness of Hardy-Littlewood maximal operator in the proofs. In addition, we also exploit the relationship between the parameters of the kernel and of the space. We obtain that the norm of the operators is dominated by the norm of the kernels

Another characterizations of Muckenhoupt Ap class

This manuscript addresses Muckenhoupt Ap weight theory in connection to Morrey and BMO spaces. It is proved that ω belongs to Muckenhoupt Ap class, if and only if Hardy-Littlewood maximal function M is bounded from weighted Lebesgue spaces Lp(ω) to weighted Morrey spaces Mqp(ω) for 1 <q <p < ∞. As a corollary, if M is (weak) bounded on Mqp(ω), then ω ∈ Ap. The Ap condition also characterizes the boundedness of the Riesz transform Rj and convolution operators Tɛ on weighted Morrey spaces. Finally, we show that ω ∈ Ap if and only if ω ∈BMOp′(ω) for 1 ≤ p < ∞ and 1/p+1/p′=1.

Endpoint Regularity of Multisublinear Fractional Maximal Functions

AbstractIn this paper we investigate the endpoint regularity properties of the multisublinear fractional maximal operators, which include the multisublinear Hardy–Littlewood maximal operator. We obtain some new bounds for the derivative of the one-dimensional multisublinear fractional maximal operators acting on the vector-valued function with all ƒ j being BV-functions.

Endpoint Sobolev and BV continuity for maximal operators

In this paper we investigate some questions related to the continuity of maximal operators in W1,1 and BV spaces, complementing some well-known boundedness results. Letting M˜ be the one-dimensional uncentered Hardy–Littlewood maximal operator, we prove that the map f↦(M˜f)′ is continuous from W1,1(R) to L1(R). In the discrete setting, we prove that M˜:BV(Z)→BV(Z) is also continuous. For the one-dimensional fractional Hardy–Littlewood maximal operator, we prove by means of counterexamples that the corresponding continuity statements do not hold, both in the continuous and discrete settings, and for the centered and uncentered versions.

Weighted Lorentz and Lorentz–Morrey estimates to viscosity solutions of fully nonlinear elliptic equations

We prove a global weighted Lorentz and Lorentz–Morrey estimates of the viscosity solutions to the Dirichlet problem for fully nonlinear elliptic equation defined in a bounded domain. The oscillation of nonlinearity F with respect to x is assumed to be small in the -sense. Here, we employ the Lorentz boundedness of the Hardy–Littlewood maximal operators and an equivalent representation of weighted Lorentz norm.

Orlicz estimates for nondivergence linear elliptic equations with partially BMO coefficients

We prove a local Orlicz estimate of the Hessian of strong solutions to a class of nondivergence linear elliptic equations . The leading coefficients are assumed to be measurable in one variable and have small BMO-norms in the other variables. To attain our aim, an approximation approach, the Orlicz boundedness of the Hardy–Littlewood maximal functions and an equivalent representation of Orlicz norm are employed.

Sharp inequalities for one-sided Muckenhoupt weights

Let $A_\infty ^+$ denote the class of one-sided Muckenhoupt weights, namely all the weights $w$ for which $\mathsf M^+:L^p(w)\to L^{p,\infty}(w)$ for some $p>1$, where $\mathsf M^+$ is the forward Hardy-Littlewood maximal operator. We show that $w\in A_\infty ^+$ if and only if there exist numerical constants $\gamma\in(0,1)$ and $c>0$ such that $$ w(\{x \in \mathbb{R} : \, \mathsf M ^+\mathbf 1_E (x)>\gamma\})\leq c w(E) $$ for all measurable sets $E\subset \mathbb R$. Furthermore, letting $$ \mathsf C_w ^+(\alpha):= \sup_{0<w(E)<+\infty} \frac{1}{w(E)} w(\{x\in\mathbb R:\,\mathsf M^+\mathbf 1_E (x)>\alpha\}) $$ we show that for all $w\in A_\infty ^+$ we have the asymptotic estimate $\mathsf C_w ^+ (\alpha)-1\lesssim (1-\alpha)^\frac{1}{c[w]_{A_\infty ^+}}$ for $\alpha$ sufficiently close to $1$ and $c>0$ a numerical constant, and that this estimate is best possible. We also show that the reverse H\"older inequality for one-sided Muckenhoupt weights, previously proved by Mart\'in-Reyes and de la Torre, is sharp, thus providing a quantitative equivalent definition of $A_\infty ^+$. Our methods also allow us to show that a weight $w\in A_\infty ^+$ satisfies $w\in A_p ^+$ for all $p>e^{c[w]_{A_\infty ^+}}$.

Muckenhoupt–Wheeden conjectures for sparse operators

We provide an explicit example of a pair of weights and a dyadic sparse operator for which the Hardy–Littlewood maximal function is bounded from \(L^p(v)\) to \(L^p(u)\) and from \(L^{p'}(u^{1-p'})\) to \(L^{p'}(v^{1-p'})\) while the sparse operator is not bounded on the same spaces. Our construction also provides an example of a single weight for which the weak-type endpoint does not hold for sparse operators.

Hardy–Littlewood Maximal Operator in Weighted Grand Variable Exponent Lebesgue Space

The boundedness of the Hardy–Littlewood maximal operator is proved in weighted grand variable exponent Lebesgue space with power weights.

Maximal Operators in the Spherical Mean Setting

We define the Hardy–Littlewood maximal operator and the Poisson maximal operator associated with the spherical mean operator for which we prove the boundedness, also we show that these maximal operators are comparable.