L. Grafakos, Stephen J. Montgomery-Smith, O. Motrunich
The best constant in the usual Lp norm inequality for the centered Hardy-Littlewood maximal function on R1 is obtained for the class of all ``peak-shaped'' functions. A positive function on the line is called ``peak-shaped'' if it is positive and convex except at one point. The techniques we use include convexity and an adaptation of the standard Euler-Langrange variational method.
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In this paper we get the sharp estimates of the $p$-adic Hardy and Hardy-Littlewood-Pólya operators on $L^q(|x|^α_pdx)$. Also, we prove that the commutators generated by the $p$-adic Hardy operators (Hardy-Littlewood-Pólya operators) and the central BMO functions are bounded on $L^q(|x|^α_pdx)$, more generally, on Herz spaces.
We study the mapping properties of the Hardy--Littlewood fractional maximal operator between Lorentz spaces of the homogeneous tree and discuss the optimality of all the results.
Generalized affine Hardy-Littlewood-Sobolev inequalities
该研究建立了涉及两个不同函数、强于经典形式的仿射Hardy-Littlewood-Sobolev不等式，并证明了对数凹函数下该新不等式的反向不等式。
Using elementary techniques, we prove sharp anisotropic Hardy-Littlewood inequalities for positive multilinear forms. In particular, we recover an inequality proved by F. Bayart in 2018.
In this paper, we prove strong type, weak type inequalities of Hardy-Littlewood maximal operator and fractional Hardy-Littlewood maximal operator on variable sequence spaces lp(Z). This is achieved using Calderon-Zygmund decomposition for sequences, properties of modular functional and Log Holder continuity.
It is shown that the Hardy-Littlewood maximal function associated to the cube in $\mathbb R^n$ obeys dimensional free bounds in $L^p$ fir $p>1$. Earlier work only covered the range $p>\frac 32$.
Two-weight criteria of various type for the Hardy-Littlewood maximal operator and singular integrals in variable exponent Lebesgue spaces defined on the real line are established.
A Direct Proof of Hardy-Littlewood Maximal Inequality for Operator-valued Functions
本文针对$2<p<\infty$的情形，给出了算子值Hardy-Littlewood极大不等式的一个直接证明。
The Hardy-Littlewood-Pólya inequality of majorization is extended to the framework of ordered Banach spaces. Several applications illustrating our main results are also included.
We investigate the growth of the constants of the polynomial Hardy-Littlewood inequality.
We prove that supermodularity is a necessary condition for the generalized Hardy- Littlewood and Riesz rearrangement inequalities. We also show the necessity of the monotonicity of the kernels involved in the Riesz{type integral.
We obtain in this short article the non-asymptotic exact estimations for the norm of (generalized) weighted Hardy-Littlewood average integral operator in the so-called Bilateral Grand Lebesgue Spaces. We also give examples to show the sharpness of these inequalities.
We consider estimates of Hardy and Littlewood for norms of operators on sequence spaces, and we apply a factorization result of Maurey to obtain improved estimates and simplified proofs for the special case of a positive operator.
We show that the optimal constants for the Hardy--Littlewood inequalities for $2$-homogeneous polynomials on $\ell_{p}(\mathbb{R}^{2})$ are precisely $2^{2/p}$ for all $2<p<4.$
We discuss the Hardy-Littlewood maximal operator on discrete Morrey spaces of arbitrary dimension. In particular, we obtain its boundedness on the discrete Morrey spaces using a discrete version of the Fefferman-Stein inequality. As a corollary, we also obtain the boundedness of some Riesz potentials on discrete Morrey spaces.
The notion of multipolynomials was recently introduced and explored by T. Velanga in [10] as an attempt to encompass the theories of polynomials and multi- linear operators. In the present paper we push this subject further, by proving Hardy- Littlewood inequalities for multipolynomials and, en passant, a variant of the Kahane- Salem-Zygmund inequality in this framework.
On the operators of Hardy-Littlewood-Pólya type
该研究引入并分析新型Hardy-Littlewood-Pólya型算子，推导其有界性与紧性的充要条件，并给出部分特殊情形下算子范数的精确取值，丰富了相关算子理论。
We study the mapping property of the commutator of bilinear Hardy-Littlewood maximal operator in homogeneous Triebel-Lizorkin space. We also show that the commutator of bilinear Hardy-Littlewood maximal operator is a compact operator acting on product of Lebesgue spaces. The results are new even in the linear case.
In this article we obtain the non - asymptotical low estimations for bilinear Riesz's functional through the Lebesgue spaces norms by means of building of some examples.
In this paper we investigate the boundedness of classical operators, namely the Hardy-Littlewood maximal operator, fractional integral operators, and Calderon-Zygmund operators, on generalized weighted Morrey spaces and generalized weighted weak Morrey spaces. We prove that the operators are bounded on these spaces, under certain assumptions.
The Hardy-Littlewood Maximal Operator on Discrete Weighted Morrey Spaces
本文引入离散加权Morrey空间，借助加权$l^1$-序列的离散Calderón-Zygmund分解，证明了离散Hardy-Littlewood极大算子在该空间及离散加权Lebesgue空间上的有界性。
Let $\mathcal{M}$ be the bilinear Hardy-Littlewood maximal function and $\vec{b}=(b,b)$ be a collection of locally integrable functions. In this paper, the authors establish characterizations of the weighted {\rm BMO} space in terms of several different commutators of bilinear Hardy-Littlewood maximal function, respectively; these commutators include the maximal iterated commutator $\mathcal{M}_{Π\vec{b}}$, the maximal linear commutator $\mathcal{M}_{Σ\vec{b}}$, the iterated commutator $[Π\vec{b},\mathcal{M}]$ and the linear commutator $[Σ\vec{b},\mathcal{M}]$.
The Hardy--Littlewood inequality for $m$-homogeneous polynomials on $\ell_{p}$ spaces is valid for $p>m.$ In this note, among other results, we present an optimal version of this inequality for the case $p=m.$ We also show that the optimal constant, when restricted to the case of $2$-homogeneous polynomials on $\ell_{2}(\mathbb{R}^{2})$ is precisely $2$. In an Appendix we justify why, curiously, the optimal exponents of the Hardy--Littlewood inequality do not behave smoothly.
The multilinear Hardy--Littlewood inequalities provide estimates for the sum of the coefficients of multilinear forms $T:\ell_{p_{1}}^{n}\times\cdots \times\ell_{p_{m}}^{n}\rightarrow\mathbb{R}$ (or $\mathbb{C}$) when $1/p_{1}+\cdots+1/p_{m}<1.$ In this paper we investigate the critical and super-critical cases; i.e., when $1/p_{1}+\cdots+1/p_{m}\geq1.$
Hardy-Sobolev-Rellich, Hardy-Littlewood-Sobolev and Caffarelli-Kohn-Nirenberg inequalities on general Lie groups
该研究在一般连通李群的次椭圆微分算子框架下，建立了多类几何不等式及其临界形式，覆盖不同拓扑与模性情形，并推导了对应的不确定型原理。
A Matrix Generalization of the Hardy-Littlewood-Pólya Rearrangement Inequality and Its Applications
本文通过分析正交矩阵上的优化问题，将Hardy-Littlewood-Pólya重排不等式推广到正定矩阵与矩形矩阵，并据此推导了信号处理、机器学习领域类距离函数的新不等式。
Vector-Valued Hardy--Littlewood Maximal Inequalities on Ball Banach Function Spaces
本文通过向量值极大不等式刻画非中心哈代-利特尔伍德极大算子在球巴拿赫函数空间及其Köthe伴随空间上的有界性，所得结果补充了Nieraeth的近期研究成果。
We introduce abstract net spaces on directed sets and prove their embedding and interpolation properties. Typical examples of interest are lattices of irreducible unitary representations of compact Lie groups and of class I representations with respect to a subgroup. As an application, we prove Hardy-Littlewood type inequalities and their converses on compact Lie groups and on compact homogeneous manifolds.
Littlewood-type theorems for Hardy spaces in infinitely many variables
本文将开单位圆盘上随机解析函数的 Littlewood 定理推广至无穷多变量 Hardy 空间，证明了相应的类似结论，且结果适用于有限变量情形及 Dirichlet 级数。
In this paper we provide a family of inequalities, extending a recent result due to Albuquerque et al.
In this short note we obtain new lower bounds for the constants of the real Hardy--Littlewood inequality for $m$-linear forms on $\ell_{p}^{2}$ spaces when $p=2m$ and for certain values of $m$. The real and complex cases for the general case $\ell_{p}^{n}$ were recently investigated by G. Araujo et al. When $n=2$ our results improve the best known estimates for these constants.
The best known upper estimates for the constants of the Hardy--Littlewood inequality for $m$-linear forms on $\ell_{p}$ spaces are of the form $\left(\sqrt{2}\right) ^{m-1}.$ We present better estimates which depend on $p$ and $m$. An interesting consequence is that if $p\geq m^{2}$ then the constants have a subpolynomial growth as $m$ tends to infinity.
This note has a twofold purpose. To improve the best known lower estimates of the Hardy-Littlewood inequality for $m$-linear forms in $\ell_{p}$ spaces and to provide a closed formula encompassing the cases $p>2m$ and $% p=2m.$ Our approach is quite simpler than the previous and uses the best known constants of the Clarkson's inequalities.
In this paper we obtain quite general and definitive forms for Hardy-Littlewood type inequalities. Moreover, when restricted to the original particular cases, our approach provides much simpler and straightforward proofs and we are able to show that in most cases the exponents involved are optimal. The technique we used is a combination of probabilistic tools and of an interpolative approach; this former technique is also employed in this paper to improve the constants for vector-valued Bohnenblust--Hille type inequalities.
In this paper we prove a noncommutative version of Hardy-Littlewood inequalities relating a function and its Fourier coefficients on the group $SU(2)$. As a consequence, we use it to obtain lower bounds for the $L^p-L^q$ norms of Fourier multipliers on the group $SU(2)$, for $1 < p \leq 2 \leq q < 1$. In addition, we give upper bounds of a similar form, analogous to the known results on the torus, but now in the noncommutative setting of $SU(2)$.
We investigate the growth of the polynomial and multilinear Hardy--Littlewood inequalities. Analytical and numerical approaches are performed and, in particular, among other results, we show that a simple application of the best known constants of the Clarkson inequality improves a recent result of Araujo et al. We also obtain the optimal constants of the generalized Hardy--Littlewood inequality in some special cases.
We present unified approach to obtain sharp mean-squared and multiplicative inequalities of Hardy-Littlewood-Polyá and Taikov types for multiple closed operators acting on Hilbert space. We apply our results to establish new sharp inequalities for the norms of powers of the Laplace-Beltrami operators on compact Riemmanian manifolds and derive the well-known Taikov and Hardy-Littlewood-Polyá inequalities for functions defined on $d$-dimensional space in the limit case. Other applications include the best approximation of unbounded operators by linear bounded ones and the best approximation of o…
In this paper, we study the best constant of the following discrete Hardy-Littlewood-Sobolev inequality, \begin{equation} \sum_{i,j,i\neq j}\frac{f_{i}g_{j}}{\mid i-j\mid^{n-α}}\leq C_{r,s,α} |f|_{l^r} |g|_{l^s}, \end{equation}where $i,j\in \mathbb Z^n$, $r,s>1$, $0<α<n$, and $\frac 1r+\frac 1s+\frac {n-α}n\geq 2$. Indeed, we can prove that the best constant is attainable in the supercritical case $\frac 1r+\frac 1s+\frac {n-α}n> 2$.
For a real-valued function $f$ on a metric measure space $(X,d,μ)$ the Hardy-Littlewood maximal-function of $f$ is given by the following `supremum-norm': $$Mf(x):=\sup_{r>0}\frac{1}{μ(\mathcal{B}_{x,r})}\int_{\mathcal{B}_{x,r}}|f|dμ.$$ In this note, we replace the supremum-norm on parameters $r$ by $\mathcal{L}_p$-norm with weight $w$ on parameters $r$ and define Hardy-Littlewood integral-function $I_{p,w}f$. It is shown that $I_{p,w}f$ converges pointwise to $Mf$ as $p\to\infty$. Boundedness of the sublinear operator $I_{p,w}$ and continuity of the function $I_{p,w}f$ in case that $X$ is a L…
We analyse Morrey spaces, generalised Morrey spaces and Campanato spaces on homogeneous groups. The boundedness of the Hardy-Littlewood maximal operator, Bessel-Riesz operators, generalised Bessel-Riesz operators and generalised fractional integral operators in generalised Morrey spaces on homogeneous groups is shown. Moreover, we prove the boundedness of the modified version of the generalised fractional integral operator and Olsen type inequalities in Campanato spaces and generalised Morrey spaces on homogeneous groups, respectively. Our results extend results known in the isotropic Euclidea…
We study new weighted estimates for the 2-fold product of Hardy-Littlewood maximal operators defined by $M^{\otimes}(f,g):= MfMg$. 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 $$ \Vert fg \Vert_{L^{p,\infty}\left(w_1^{p/p_1} w_2^{p/p_2}\right)} \le C \Vert f \Vert_{L^{p_1,\infty}(w_1 )} \Vert g \Vert_{L^{p_2,\…
Dimension-free estimates for semi-commutative discrete Hardy-Littlewood maximal operators
本文在半交换$L_{p}$空间上对离散二进Hardy-Littlewood极大算子建立了无维数估计，证明了半径充分大时对所有$1<p<\infty$均成立，并导出了极大遍历不等式与双边几乎一致收敛。
For the Hardy-Littlewood maximal and Calderón-Zygmund operators, the weighted boundedness on the Lebesgue spaces are well known. We extend these to the Orlicz-Morrey spaces. Moreover, we prove the weighted boundedness on the weak Orlicz-Morrey spaces. To do this we show the weak-weak modular inequality. The Orlicz-Morrey space and its weak version contain weighted Orlicz, Morrey and Lebesgue spaces and their weak versions as special cases. Then we also get the boundedness for these function spaces as corollaries.
We extend the recent boundedness result of Kurka for Hardy-Littlewood maximal function to discrete setting.
Boundedness and norm of certain p-adic Hardy-Littlewood-Pólya-type operators
本文研究了p进Hardy-Littlewood-Pólya型算子的有界性和范数，通过参数引入，刻画了其在p进加权Lebesgue空间上的L^q-L^r有界性并给出了特殊情形下的精确范数估计。
The Hardy-Littlewood-Polya inequality of majorization is extended for the ω-m-star-convex functions to the framework of ordered Banach spaces. Several open problems which seem of interest for further extensions of the Hardy-Littlewood-Polya inequality are also included.
In this paper we prove that the Hardy-Littlewood maximal operator is bounded on Morrey spaces $\mathcal{M}_{1,λ}(\rn)$, $0 \le \la < n$ for radial, decreasing functions on $\rn$
The purpose of this text is twofold. We present a review of the existing stability results for Sobolev, Hardy-Littlewood-Sobolev (HLS) and related inequalities. We also contribute to the topic with some observations on constructive stability estimates for (HLS).
In this short article we obtain the non-asymptotic upper and low estimations for linear and bilinear weight Riesz's functional through the Lebesgue spaces.
This paper is a second one following our work [CLZ13] in series, considering sharp Hardy- Littlewood-Sobolev inequalities on groups of Heisenberg type. The first important breakthrough was made by Frank and Lieb in [FL12]. In this paper, analogous results are obtained for octonionic Heisenberg group.
In this paper, we investigate the Hardy-Littlewood maximal function on non-commutative symmetric spaces. We complete the results of T. Bekjan and J. Shao. Moreover, we refine the main results of the papers \cite{Bek} and \cite{Sh}.
The Hardy-Littlewood property and maximal operators associated with the inverse Gauss measure
本文利用伴随逆Gauss测度的算子半群定义的极大算子，刻画了具有Hardy-Littlewood性质的Banach格。
Boundedness of the Hardy-Littlewood maximal operator on generalized fofana spaces
本文引入了广义Fofana空间并研究了其基本性质，这些空间是广义Morrey空间的推广，进而证明了Hardy-Littlewood极大算子在这些新空间上的有界性。
This is a survey article about recent developments in dimension-free estimates for maximal functions corresponding to the Hardy--Littlewood averaging operators associated with convex symmetric bodies in $\mathbb R^d$ and $\mathbb Z^d$.
The purpose of this paper is to develop some methods to study Riesz type inequalities, Hardy-Littlewood type theorems and smooth moduli of holomorphic, pluriharmonic and harmonic functions in high-dimensional cases. Initially, we prove some sharp Riesz type inequalities of pluriharmonic functions on bounded symmetric domains. The obtained results extend the main results in (\textit{Trans. Amer. Math. Soc.} {\bf 372} (2019)~ 4031--4051). Furthermore, some Hardy-Littlewood type theorems of holomorphic and pluriharmonic functions on John domains are established. Additionally, we also discuss the …
Affine Hardy--Littlewood--Sobolev inequalities
本文建立了比经典结果更强的仿射 Hardy--Littlewood--Sobolev 不等式，并推导出对数凹函数的反向不等式及仿射分数阶 Sobolev 不等式。
We investigate the class $\mathcal{B}^{loc}(\mathbb{R}^{n})$ of exponents $p(\cdot)$ for with local Hardy-Littlewood maximal operator is bounded in $L^{p(\cdot)}(\mathbb{R}^{n})$ space. Littlewood-Paley square-function characterization of $L^{p(\cdot)}(\mathbb{R}^{n})$ spaces with the above class of exponent are also obtained.
We show that the discrete Hardy-Littlewood maximal functions associated with the Euclidean balls in $\mathbb Z^d$ with dyadic radii have bounds independent of the dimension on $\ell^p(\mathbb Z^d)$ for $p\in[2, \infty]$.
We characterize the rearrangement invariant spaces for which there exists a non-constant fixed point, for the Hardy-Littlewood maximal operator (the case for the spaces $L^p(\mathbb{R}^{n})$ was first considered by Korry in \cite{Ko}). The main result that we prove is that the space $L^{\frac{n}{n-2},\infty}(\mathbb{R}^{n})\cap L^{\infty}(\mathbb{R}^{n})$ is minimal among those having this property
We show that the sharp constant in the Hardy-Littlewood-Sobolev inequality can be derived using the method that we employed earlier for a similar inequality on the Heisenberg group. The merit of this proof is that it does not rely on rearrangement inequalities; it is the first one to do so for the whole parameter range.
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