It is well known that the weak ($1,1$) bounds doesn't hold for the strong maximal operators, but it still enjoys certain weak $L\log L$ type norm inequality. Let $Φ_n(t)=t(1+(\log^+t)^{n-1})$ and the space $L_{Φ_n}({\mathbb R^{n}})$ be the set of all measurable functions on ${\mathbb R^{n}}$ such that $\|f\|_{L_{Φ_n}({\mathbb R^{n}})} :=\|Φ_n(|f|)\|_{L^1({\mathbb R^{n}})}<\infty$. In this paper, we introduce a new weak norm space $L_{Φ_n}^{1,\infty}({\mathbb R^{n}})$, which is more larger than $L^{1,\infty}({\mathbb R^{n}})$ space, and establish the correspondng limiting weak type behaviors of the strong maximal operators. As a corollary, we show that $ \max\{{2^n}{((n-1)!)^{-1}},1\}$ is a lower bound for the best constant of the $L_{Φ_n}\to L_{Φ_n}^{1,\infty}$ norm of the strong maximal operators. Similar results have been extended to the multilinear strong maximal operators.
By a reduction method, the limiting weak-type behaviors of factional maximal operators and fractional integrals are established without any smoothness assumption on the kernel, which essentially improve and extend previous results. As a byproduct, we characterize the boundedness of several operators by the membership of their kernel in Lebesgue space on sphere.
In this paper, we explore the limiting weak-type behaviors of some integral operators including maximal operators, singular and fractional integral operators and maximal truncated singular integrals et al. Some optimal limiting weak-type behaviors are given, which essentially improve and extend the previous results in this topics.
The natural maximal and minimal functions commute pointwise with the logarithm on $A_\infty$. We use this observation to characterize the spaces $A_1$ and $RH_\infty$ on metric measure spaces with a doubling measure. As the limiting cases of Muckenhoupt $A_p$ and reverse Hölder classes, respectively, their behavior is remarkably symmetric. On general metric measure spaces, an additional geometric assumption is needed in order to pass between $A_p$ and reverse Hölder descriptions. Finally, we apply the characterization to give simple proofs of several known properties of $A_1$ and $RH_\infty$, …
We estimate in Lp the maximal Riesz transform in terms of the Riesz transform itself for p greater than 1. In the limiting case p=1 the weak L1 inequality is shown to fail. Surprisingly, the weak L1 inequality for the maximal Beurling transform in terms of the Beurling transform does hold.
A set subset of Euclidean space whose indicator function has maximal Gowers norm, among all sets of equal measure, is an ellipsoid up to Lebesgue null sets. If the indicator function has nearly maximal Gowers norm then the set nearly coincides with an ellipsoid.
We provide lower $L^q$ and weak $L^p$-bounds for the localized dyadic maximal operator on $R^n$, when the local $L^1$ and the local $L^p$ norm of the function are given. We actually do that in the more general context of homo- geneous tree-like families in probability spaces.
For a local maximal function defined on a certain family of cubes lying ``well inside'' of $Ω$, a proper open subset of $\mathbb R ^n$, we characterize the couple of weights $(u,v)$ for which it is bounded from $L^p(v)$ on $L^q(u)$.
We give weighted norm inequalities for the maximal fractional operator $ \mathcal M_{q,β}$ of Hardy-Littlewood and the fractional integral $I_γ$. These inequalities are established between $(L^{q},L^{p}) ^α(X,d,μ)$ spaces (which are super spaces of Lebesgue spaces $L^α(X,d,μ)$, and subspaces of amalgams $(L^{q},L^{p})(X,d,μ)$) and in the setting of space of homogeneous type $(X,d,μ)$. The conditions on the weights are stated in terms of Orlicz norm.
Given a space of homogeneous type $(X,μ,d)$, we prove strong-type weighted norm inequalities for the Hardy-Littlewood maximal operator over the variable exponent Lebesgue spaces $L^\pp$. We prove that the variable Muckenhoupt condition $\App$ is necessary and sufficient for the strong type inequality if $\pp$ satisfies log-Hölder continuity conditions and $1 < p_- \leq p_+ < \infty$. Our results generalize to spaces of homogeneous type the analogous results in Euclidean space proved by Cruz-Uribe, Fiorenza and Neugebuaer (2012).
This paper focuses on the operator norm of the truncated Hardy-Littlewood maximal operator $M^b_a$ and the strong truncated Hardy-Littlewood maximal operator $\tilde{M}^{\boldsymbol{b}}_{\boldsymbol{a}}$, respectively. We first present the $L^1$-norm of $M^b_a$, and then the $L^1$-norm of $\tilde{M}^{\boldsymbol{b}}_{\boldsymbol{a}}$ is given. Our study may have some enlightening significance for the research on sharp constant for the classical Hardy-Littlewood maximal inequality.
We extend the theory of weighted norm inequalities on variable Lebesgue spaces to the case of bilinear operators. We introduce a bilinear version of the variable $\A_\pp$ condition, and show that it is necessary and sufficient for the bilinear maximal operator to satisfy a weighted norm inequality. Our work generalizes the linear results of the first author, Fiorenza and Neugebauer \cite{dcu-f-nPreprint2010} in the variable Lebesgue spaces and the bilinear results of Lerner {\em et al.} \cite{MR2483720} in the classical Lebesgue spaces. As an application we prove weighted norm inequalities for…
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.
We obtain the explicit upper Bellman function for the natural dyadic maximal operator acting from ${\rm BMO}(\mathbb{R}^n)$ into ${\rm BLO}(\mathbb{R}^n).$ As a consequence, we show that the ${\rm BMO}\to{\rm BLO}$ norm of the natural operator equals 1 for all $n,$ and so does the norm of the classical dyadic maximal operator. The main result is a partial corollary of a theorem for the so-called $α$-trees, which generalize dyadic lattices. The Bellman function in this setting exhibits an interesting quasi-periodic structure depending on $α,$ but also allows a majorant independent of $α,$ hence…
In this work we obtain boundedness on weighted variable Lebesgue spaces of some maximal functions that come from the localized analysis considering a critical radius function. This analysis appears naturally in the context of the Schrödinger operator $\mathcal{L}=-Δ+V$ in $\mathbb{R}^d$, where $V$ a non-negative potential satisfying some specific reverse Hölder condition. We consider new classes of weights that locally behave as the Muckenhoupt class for Lebesgue spaces with variable exponents considered in Cruz Uribe et al. (J. Math. Anal. Appl. 394(2):744-760, 2012) and actually include them…
We characterize two-weight inequalities for certain maximal truncations of the Hilbert transform in terms of testing conditions on simpler functions. For 1<p<2 and two positive Borel measures u, v on R, we assume that u is doubling, and we consider maximal truncations T_# of the Hilbert transform. The norm estimate || T(f u) ||_{L^p(v)} < C || f ||_{L^p(u)} is characterized in terms of an A_p condition on the weights and two testing conditions. The first is the norm condition above, but the function f varies over bounded functions supported on a cube. The second is a dual weak-type condition, …
Let $Ω_1,Ω_2$ be functions of homogeneous of degree $0$ and $\vecΩ=(Ω_1,Ω_2)\in L\log L(\mathbb{S}^{n-1})\times L\log L(\mathbb{S}^{n-1})$. In this paper, we investigate the limiting weak-type behavior for bilinear maximal function $M_{\vecΩ}$ and bilinear singular integral $T_{\vecΩ}$ associated with rough kernel $\vecΩ$. For all $f,g\in L^1(\mathbb{R}^n)$, we show that $$\lim_{λ\to 0^+}λ|\big\{ x\in\mathbb{R}^n:M_{\vecΩ}(f_1,f_2)(x)>λ\big\}|^2 = \frac{\|Ω_1Ω_2\|_{L^{1/2}(\mathbb{S}^{n-1})}}{ω_{n-1}^2}\prod\limits_{i=1}^2\| f_i\|_{L^1}$$ and $$\lim_{λ\to 0^+}λ|\big\{ x\in\mathbb{R}^n:| T_{\ve…
Let $v,~ω_1, ~ω_2$ be weights and $1<p_1, ~p_2<\infty.$ Suppose that $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}$ and $(ω_1, ω_2)\in RH(p_1, p_2).$ For the multisublinear maximal operator $\mathfrak{M}$ in martingale spaces, we characterize the weights for which $\mathfrak{M}$ is bounded from $L^{p_1}(ω_1)\times L^{p_2}(ω_2)$ to $L^{p,\infty}(v)\hbox{or}L^p(v).$ If $v=ω_2^{\frac{p}{p_2}}ω_2^{\frac{p}{p_2}},$ we partially give the bilinear version of one-weight theory.
We prove weak-type (1,1) estimates for compositions of maximal operators with singular integrals. Our main object of interest is the operator $Δ^*Ψ$ where $Δ^*$ is Bourgain's maximal multiplier operator and $Ψ$ is the sum of several modulated singular integrals; here our method yields a significantly improved bound for the $L^q$ operator norm when $1 < q < 2$. We also consider associated variation-norm estimates.
We study the Hardy-Littlewood maximal operator defined via an unconditional norm, acting on block decreasing functions. We show that the uncentered maximal operator maps block decreasing functions of special bounded variation to functions with integrable distributional derivatives, thus improving their regularity. In the special case of the maximal operator defined by the l_infty-norm, that is, by averaging over cubes, the result extends to block decreasing functions of bounded variation, not necessarily special.
We characterize two-weight norm inequalities for potential type integral operators in terms of Sawyer-type testing conditions. Our result is stated in a space of homogeneous type with no additional geometric assumptions, such as group structure or non-empty annulus property, which appeared in earlier works on the subject. One of the new ingredients in the proof is the use of a finite collection of adjacent dyadic systems recently constructed by the author and T. Hytönen. We further extend the previous Euclidean characterization of two-weight norm inequalities for fractional maximal functions i…
We answer a special case of a question of T. Hytonen regarding the two weight norm inequality for the maximal function M in the affirmative, namely that there is a constant D > 1, depending only on dimension n, such that the two weight norm inequality for M holds if and only if the A_2 condition holds, and the testing condition holds for all doubling cubes.
W prove a dimension-free estimate for the $L^2(\mathbb{R}^d)$ norm of the maximal truncated Riesz transform in terms of the $L^2(\mathbb{R}^d)$ norm of the Riesz transform. Consequently, the vector of maximal truncated Riesz transforms has a dimension-free estimate on $L^2(\mathbb{R}^d)$. We also show that the maximal function of the vector of truncated Riesz transforms has a dimension-free estimate on all $L^p(\mathbb{R}^d)$ spaces, $1<p<\infty.$
In recent articles it was proved that when $μ$ is a finite, radial measure in $\real^n$ with a bounded, radially decreasing density, the $L^p(μ)$ norm of the associated maximal operator $M_μ$ grows to infinity with the dimension for a small range of values of $p$ near 1. We prove that when $μ$ is Lebesgue measure restricted to the unit ball and $p<2$, the $L^p$ operator norms of the maximal operator are unbounded in dimension, even when the action is restricted to radially decreasing functions. In spite of this, this maximal operator admits dimension-free $L^p$ bounds for every $p>2$, when res…
We prove some Sawyer-type characterizations for multilinear fractional maximal function for the upper triangle case. We also provide some two-weight norm estimates for this operator. As one of the main tools, we use an extension of the usual Carleson Embedding that is an analogue of the P. L. Duren extension of the Carleson Embedding for measures.
For 1<p< \infty, weight w \in A_p, and any L ^2 -bounded Calderón-Zygmund operator T, we show that there is a constant C(T,P) so that we prove the sharp norm dependence on T_#, the maximal truncations of T, in both weak and strong type L^p(w) norms. Namely, for the weak type norm, T_# maps L^p(w) to weak-L^p(w) with norm at most \|w\|_{A_p}. And for the strong type norm, the norm estimate is \|w\|_{A_p}^{\max(1, (p-1) ^{-1})}. These estimates are not improvable in the power of \lVert w\rVert_{A_p}. Our argument follows the outlines of the arguments of Lacey-Petermichl-Reguera (Math.\ Ann.\ 201…
We prove an extension of the Walsh-analog of the Carleson-Hunt theorem, where the $L^\infty$ norm defining the Carleson maximal operator has been replaced by an $L^q$ maximal-multiplier-norm. Additionally, we consider certain associated variation-norm estimates.
In this paper, we show that there exists a positive density subsequence of orthonormal spherical harmonics which achieves the maximal Lp norm growth for 2<p<=6, therefore giving an example of a Riemannian surface supporting such subsequence of eigenfunctions. This answers the question proposed by Sogge and Zelditch (arXiv:1011.0215). Furthermore, we provide an explicit lower bound on the density in this example.
We give a new characterisation of inner product spaces amongst normed vector spaces in terms of the maximal cirumradius of spheres. It turns out that a normed vector space $(X,\norm{\cdot})$ with $\dim X\geq 2$ is an inner product space if and only if all spheres are not degenerate, i.e. the maximal circumradius of points on the sphere equals the radius of the sphere.
We study the limiting eigenvalue distribution of $n\times n$ banded Toeplitz matrices as $n\to \infty$. From classical results of Schmidt-Spitzer and Hirschman it is known that the eigenvalues accumulate on a special curve in the complex plane and the normalized eigenvalue counting measure converges weakly to a measure on this curve as $n\to\infty$. In this paper, we characterize the limiting measure in terms of an equilibrium problem. The limiting measure is one component of the unique vector of measures that minimes an energy functional defined on admissible vectors of measures. In addition,…
The aim of this paper is to study two-weight norm inequalities for fractional maximal functions and fractional Bergman operator defined on the upper-half space. Namely, we characterize those pairs of weights for which these maximal operators satisfy strong and weak type inequalities. Our characterizations are in terms of Sawyer and Békollé-Bonami type conditions. We also obtain a $Φ$-bump characterization for these maximal functions, where $Φ$ is a Orlicz function. As a consequence, we obtain two-weight norm inequalities for fractional Bergman operators. Finally, we provide some sharp weighted…
We study classical weighted $L^p\to L^q$ inequalities for the fractional maximal operators on $\R^d$, proved originally by Muckenhoupt and Wheeden in the 70's. We establish a slightly stronger version of this inequality with the use of a novel extension of Bellman function method. More precisely, the estimate is deduced from the existence of a certain special function which enjoys appropriate majorization and concavity. From this result and an explicit version of the ``$A_{p-\varepsilon}$ theorem," derived also with Bellman functions, we obtain the sharp inequality of Lacey, Moen, Pérez and To…
Dimension-free estimates for discrete maximal functions related to normalized sampled gaussians
Rutgers University（罗格斯大学）；Instytut Matematyczny Uniwersytet Wrocławski（弗罗茨瓦夫大学数学研究所）；University of Georgia（佐治亚大学）；Instytut Matematyczny Polskiej Akademii Nauk（波兰科学院数学研究所）
In a filtered measure space, a characterization of weights for which the trace inequality of a positive operator holds is given by the use of discrete Wolff's potential. A refinement of the Carleson embedding theorem is also introduced. Sawyer type characterization of weights for which a two-weight norm inequality for a generalized Doob's maximal operator holds is established by an application of our Carleson embedding theorem. Moreover, Hytönen-Pérez type one-weight norm estimate for Doob's maximal operator is obtained by the use of our two-weight characterization.
Let $T$ be a finite tree graph, $T^N$ be the Cartesian power graph of $T$, and $d^N$ be the graph distance metric on $T^N$. Also let \[ \mathbb S_r^N(x) := \{v \in T^N: d^N(x,v) = r\} \] be the sphere of radius $r$ centered at $x$ and $M$ be the spherical maximal averaging operator on $T^N$ given by \[ Mf(x) := \sup_{\substack{r \geq 0 \\ \mathbb S_r^N(x) \neq \emptyset}} \frac{1}{|\mathbb S_r^N(x)|} |\sum_{\mathbb S_r^N(x)} f(y)|. \] We will show that for any fixed $1 \leq p \leq \infty$, the $L^p$ operator norm of $M$, i.e. \[ \|M\|_p := \sup_{\|f\|_p = 1} \|Mf\|_p, \] grows exponentially in…
