Dimension free bounds for the Hardy--Littlewood maximal operator associated to convex sets
Luc Deleaval, Olivier Guédon, Bernard Maurey
备注：
Version v4 includes several suggestions from the reviewers; section 1.2 has been rewritten to a great extent. Lemma 4.7 (new) allows us to avoid some repetitions later on. Remark 7.14 completes Remark 7.13. Numerous minor edits have been done throughout, and some relevant references added
This survey is based on a series of lectures given by the authors at the working seminar "Convexité et Probabilités" 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 Müller 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--Strömberg between 2009 and 2013.
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.
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.
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 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.
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.
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 $1/p_{1}+\cdots+1/p_{m}\geq1.$
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.
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.
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.
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,\…
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.
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}.
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 …
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
