Theresa C. Anderson, David E. Weirich
We state a version of Gehring's self improvement Theorem for reverse \Holder weights which is valid for dyadic cubes over spaces of homogeneous type and explore some of the consequences and applications.
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In this note we prove Scurry's testing conditions for the boundedness of a sequence-valued averaging positive dyadic operator from a weighted Lp space to a sequence-valued weighted Lp space by using parallel stopping cubes.
Bellman function for Hardy's inequality over dyadic trees
本文利用Bellman函数技术刻画了二元树上加权Hardy不等式成立的测度，证明了相关常数的最优性，并给出了Bellman函数的随机最优控制解释。
We give an alternative proof of a sharp generalization of an integral inequality for the dyadic maximal operator due to which the evaluation of the Bellman function of this operator with respect to two variables, is possible. This last mentioned inequality, which was first noticed in [3],also generalizes in a certain direction the results of [7].
We prove a sharp integral inequality for the dyadic maximal operator, connecting integrals of $ϕ$, and of the dyadic maximal function of $ϕ$.
In the dyadic case the union of the Reverse Hölder classes, $RH_p^d$ is strictly larger than the union of the Muckenhoupt classes $ A_p^d$. We introduce the $RH_1^d$ condition as a limiting case of the $RH_p^d$ inequalities as $p$ tends to 1 and show the sharp bound on $RH_1^d$ constant of the weight $w$ in terms of its $A_\infty^d$ constant. We also take a look at the summation conditions of the Buckley type for the dyadic Reverse Hölder and Muckenhoupt weights and deduce them from an intrinsic lemma which gives a summation representation of the bumped average of a weight. Our lemmata also al…
We find the exact Bellman function for the weak $L^1$ norm of local positive dyadic shifts. We also describe a sequence of functions, self-similar in nature, which in the limit extremize the local weak-type (1,1) inequality.
Evaluation of the Bellman functions is a difficult task. The exact Bellman functions of the dyadic Carleson Embedding Theorem 1.1 and the dyadic maximal operators are obtained in [3] and [4]. Actually, the same Bellman functions also work for the tree-like structure. In this paper, we give a self-complete proof of the coincidence of the Bellman functions on the more general infinitely refining filtered probability space, see Definition 1.3. The proof depends on a remodeling of the Bellman function of the dyadic Carleson Embedding Theorem.
We develop technical tools that enable the use of Bellman functions for BMO defined on $α$-trees, which are structures that generalize dyadic lattices. As applications, we prove the integral John--Nirenberg inequality and an inequality relating $L^1$- and $L^2$-oscillations for BMO on $α$-trees, with explicit constants. When the tree in question is the collection of all dyadic cubes in $\mathbb{R}^n,$ the inequalities proved are sharp. We also reformulate the John--Nirenberg inequality for the continuous BMO in terms of special martingales generated by BMO functions. The tools presented can be…
Let $σ$, $ω$ be measures on $\mathbb{R}^d$, and let $\{λ_Q\}_{Q\in\mathcal{D}}$ be a family of non-negative reals indexed by the collection $\mathcal{D}$ of dyadic cubes in $\mathbb{R}^d$. We characterize the two-weight norm inequality, \begin{equation*} \lVert T_λ(fσ)\rVert_{L^q(ω)}\le C \, \lVert f \rVert_{L^p(σ)}\quad \text{for every $f\in L^p(σ)$,} \end{equation*} for the positive dyadic operator \begin{equation*} T_λ(fσ):= \sum_{Q\in \mathcal{D}} λ_Q \, \Big(\frac{1}{σ(Q)} \int_Q f\mathrm{d}σ\Big) \, 1_Q \end{equation*} in the difficult range $0<q<1 \le p<\infty$ of integrability exponent…
Let M denote the dyadic Maximal Function. We show that there is a weight w, and Haar multiplier T for which the following weak-type inequality fails: $$ \sup_{t>0}t w\left\{x\in\mathbb R \mid |Tf(x)|>t\right\}\le C \int_{\mathbb R}|f|Mw(x)dx. $$ (With T replaced by M, this is a well-known fact.) This shows that a dyadic version of the so-called Muckenhoupt-Wheeden Conjecture is false. This accomplished by using current techniques in weighted inequalities to show that a particular $L^2$ consequence of the inequality above does not hold.
This exposition presents a self-contained proof of the $A_2$ theorem, the quantitatively sharp norm inequality for singular integral operators in the weighted space $L^2(w)$. The strategy of the proof is a streamlined version of the author's original one, based on a probabilistic Dyadic Representation Theorem for singular integral operators. While more recent non-probabilistic approaches are also available now, the probabilistic method provides additional structural information, which has independent interest and other applications. The presentation emphasizes connections to the David-Journé $…
Let $σ$ and $ω$ be locally finite Borel measures on $\mathbb{R}^d$, and let $p\in(1,\infty)$ and $q\in(0,\infty)$. We study the two-weight norm inequality $$ \lVert T(fσ) \rVert_{L^q(ω)}\leq C \lVert f \rVert_{L^p(σ)}, \quad \text{for all} \, \, f \in L^p(σ), $$ for both the positive summation operators $T=T_λ(\cdot σ)$ and positive maximal operators $T=M_λ(\cdot σ)$. Here, for a family $\{λ_Q\}$ of non-negative reals indexed by the dyadic cubes $Q$, these operators are defined by $$ T_λ(fσ):=\sum_Q λ_Q \langle f\rangle^σ_Q 1_Q \quad\text{ and } \quad M_λ(fσ):=\sup_Q λ_Q \langle f\rangle^σ_Q 1…
Two-weight dyadic Hardy's inequalities
本文研究无限树上的双权二进Hardy不等式，梳理已知迹测度刻画与证明，提出多种新刻画及新证明方法，引入共形不变版本，刻画算子紧性并列出开放问题。
We study trace inequalities of the type $$ \| T_k f\|_{L^q(dμ)}\leq C \|f\|_{L^p(dσ)}, \qquad f \in L^p(dσ), $$ in the ``upper triangle case'' $1 \leq q<p$ for integral operators $T_k$ with positive kernels, where $dσ$ and $dμ$ are positive Borel measures on $\R^n$. Our main tool is a generalization of Th. Wolff's inequality which gives two-sided estimates of the energy ${\mathcal E}_{k, σ} [μ]=\int_{\R^n} (T_k [μ])^{p'} d σ$ through the $L^1(dμ)$-norm of an appropriate nonlinear potential $W_{k, σ}[μ]$ associated with the kernel $k$ and measures $dμ$, $d σ$. We initially work with a dyadic in…
We give a self-contained proof of the $A_2$ conjecture, which claims that the norm of any Calderon-Zygmund operator is bounded by the first degree of the $A_2$ norm of the weight. The original proof of this result by the first author relied on a subtle and rather difficult reduction to a testing condition by the last three authors. Here we replace this reduction by a new weighted norm bound for dyadic shifts - linear in the $A_2$ norm of the weight and quadratic in the complexity of the shift -, which is based on a new quantitative two-weight inequality for the shifts. These sharp one- and two…
We give a new proof of the sharp weighted $L^2$ inequality ||T||_{L^2(w)} \leq c [w]_{A_2} where $T$ is the Hilbert transform, a Riesz transform, the Beurling-Ahlfors operator or any operator that can be approximated by Haar shift operators. Our proof avoids the Bellman function technique and two weight norm inequalities. We use instead a recent result due to A. Lerner to estimate the oscillation of dyadic operators.
Let $S$ be the Lie group $\mathrm{R}^n\ltimes \mathrm{R}^+$ endowed with the left-invariant Riemannian symmetric space structure and the right Haar measure $ρ$, which is a Lie group of exponential growth. Hebisch and Steger in [Math. Z. 245(2003), 37--61] proved that any integrable function on $(S,ρ)$ admits a Calderón--Zygmund decomposition which involves a particular family of sets, called Calderón--Zygmund sets. In this paper, we first show the existence of a dyadic grid in the group $S$, which has {nice} properties similar to the classical Euclidean dyadic cubes. Using the properties of th…
In this paper we extend dyadic shifts and the dyadic representation theorem to an operator-valued setting: We first define operator-valued dyadic shifts and prove that they are bounded. We then extend the dyadic representation theorem, which states that every scalar-valued Calderón-Zygmund operator can be represented as a series of dyadic shifts and paraproducts averaged over randomized dyadic systems, to operator-valued Calderón-Zygmund operators. As a corollary, we obtain another proof of the operator-valued, global T(1) theorem. We work in the setting of integral operators that have R-bound…
We consider boundedness of a certain positive dyadic operator $$ T^σ\colon L^p(σ; \ \! \ell^2) \to L^p(ω), $$ that arose during our attempts to develop a two-weight theory for the Hilbert transform in $L^p$. Boundedness of $T^σ$ is characterized when $p \in [2, \infty)$ in terms of certain testing conditions. This requires a new Carleson-type embedding theorem that is also proved.
In the Euclidean setting, the Fujii-Wilson-type $A_\infty$ weights satisfy a Reverse Hölder Inequality (RHI) but in spaces of homogeneous type the best known result has been that $A_\infty$ weights satisfy only a weak Reverse Hölder Inequality. In this paper, we compliment the results of Hytönen, Pérez and Rela and show that there exist both $A_\infty$ weights that do not satisfy an RHI and a genuinely weaker weight class that still satisfies a weak RHI. We also show that all the weights that satisfy a weak RHI have a self-improving property but the self-improving property of the strong Revers…
In a recent paper V. Vasyunin presented a proof of the reverse Hölder inequality with sharp constants for the weights satisfying the usual Muckenhoupt condition. In this paper we present the inverse, that is, we use the Bellman function technique to find the sharp $A_p$ constants for weights in a reverse-Hölder class on an interval; we also find the sharp constants for the higher-integrability result of Gehring. Additionally, we find bounds for the $A_p$ constants of reverse-Hölder-class weights defined on rectangles and on cubes in n dimensions.
The rectangular fractional integral operators
本文研究矩形分数次积分算子，在矩形倍增权重条件下证明其广义 Hardy–Littlewood–Sobolev 不等式，并体现二进矩形 $M$ 线性嵌入定理的应用。
We study the vector-valued positive dyadic operator \[T_λ(fσ):=\sum_{Q\in\mathcal{D}} λ_Q \int_Q f \mathrm{d}σ1_Q,\] where the coefficients $\{λ_Q:C\to D\}_{Q\in\mathcal{D}}$ are positive operators from a Banach lattice $C$ to a Banach lattice $D$. We assume that the Banach lattices $C$ and $D^*$ each have the Hardy--Littlewood property. An example of a Banach lattice with the Hardy--Littlewood property is a Lebesgue space. In the two-weight case, we prove that the $L^p_C(σ)\to L^q_D(ω)$ boundedness of the operator $T_λ( \cdot σ)$ is characterized by the direct and the dual $L^\infty$ testing …
Weighted Inequalities for $t$-Haar multipliers
本文给出三元权重下$t$-Haar乘子一致有界性的充要条件，推广了鞅变换的双权重结果，并简化了单权重情形，同时获得Sawyer型检验条件。
Weak reverse Hölder inequalities for doubling measures
本文证明了加倍测度下Bennewitz--Lewis定理的局部定量形式，通过二进工具和引导引理，同时推导出反向Hölder不等式与局部Gehring引理。
We characterize two weight inequalities for general positive dyadic operators. We consider both weak and strong type inequalities, and general (p,q) mapping properties. Special cases include Sawyers Fractional Integral operator results from 1988, and the bilinear embedding inequality of Nazarov-Treil-Volberg from 1999. The method of proof is an extension of Sawyer's argument.
We give a quantitative characterization of the pairs of weights $(w,v)$ for which the dyadic version of the one-sided Hardy-Littlewood maximal operator satisfies a restricted weak $(p,p)$ type inequality, for $1\leq p<\infty$. More precisely, given any measurable set $E_0$ the estimate \[w(\{x\in \mathbb{R}^n: M^{+,d}(\mathcal{X}_{E_0})(x)>t\})\leq \frac{C[(w,v)]_{A_p^{+,d}(\mathcal{R})}^p}{t^p}v(E_0)\] holds if and only if the pair $(w,v)$ belongs to $A_p^{+,d}(\mathcal{R})$, that is \[\frac{|E|}{|Q|}\leq [(w,v)]_{A_p^{+,d}(\mathcal{R})}\left(\frac{v(E)}{w(Q)}\right)^{1/p}\] for every dyadic …
We consider a two weight $L^{p}(μ) \to L^{q}(ν)$-inequality for well localized operators as defined and studied by F. Nazarov, S. Treil and A. Volberg when $p=q=2$. A counterexample of F. Nazarov shows that the direct analogue of these results fails for for $p=q\not=2$. Here a new square function testing condition is introduced and applied to characterize the two weight norm inequality. The use of the square function testing condition is also demonstrated in connection with certain positive dyadic operators.
Our main result is an abstract good-$λ$ inequality that allows us to consider three self-improving properties related to oscillation estimates in a very general context. The novelty of our approach is that there is one principle behind these self-improving phenomena. First, we obtain higher integrability properties for functions belonging to the so-called John-Nirenberg spaces. Second, and as a consequence of the previous fact, we present very easy proofs of some of the self-improving properties of the generalized Poincaré inequalities studied by B. Franchi, C. Pérez and R. Wheeden, and by P. …
Given a space of homogeneous type we give sufficient conditions on a variable exponent {p(.)} so that the fractional maximal operator {M_η} maps {L^{p(.)}(X)} to {L^{q(.)}(X)}, where {1/p(.) - 1/q(.) = η}. 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_η}. 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 po…
We give a characterization of the two-weight inequality for a simple vector-valued operator. Special cases of our result have been considered before in the form of the weighted Carleson embedding theorem, the dyadic positive operators of Nazarov, Treil, and Volberg in the square integrable case, and Lacey, Sawyer, Uriarte-Tuero in the L^p case. The main technique of this paper is a Sawyer-style argument and the characterization is for 1 < p < \infty. We are unaware of instances where this operator has been given attention in the two-weight setting before.
Two inequalities for commutators of singular integral operators satisfying Hörmander conditions of Young type
本文针对满足 Young 型 Hörmander 条件的奇异积分算子一般交换子，建立逐点稀疏控制并证明两类加权不等式，解耦交换子数量与 ε 指标，改进已有结果并适用于多类算子。
Characterizations of $A_\infty$ Weights in Ergodic Theory
该研究在遍历理论框架下从二进网格角度建立离散加权Calderón-Zygmund分解，借此得到离散$A_\infty$权的等价刻画，明确其与反向Hölder不等式在加倍条件下的互推关系。
Jackson and Bernstein Inequalities with Explicit Weight Dependence for Band-Limited Wavelet Approximation in Weighted Besov Spaces
本文在加权Besov空间中证明了带显式权重依赖的Jackson型不等式，误差衰减由光滑度与维度比决定，并验证了指数最优性与局部化必要性。
Weak-$A_\infty$ packing and estimates for fractional sparse forms on spaces of homogeneous type
研究齐型空间上分数次稀疏形式的双权估计，通过在放大球和二进方体上对弱反向Hölder不等式及稀疏检验进行归一化，引入\(\mathfrak d_Q^w\)，利用缺陷水平填充估计与双权检验得出混合弱\(A_{\infty}\)界。
In this article, the authors establish the wavelet characterization of Besov and Triebel--Lizorkin spaces on a given space $(X,d,μ)$ of homogeneous type in the sense of Coifman and Weiss. Moreover, the authors introduce almost diagonal operators on Besov and Triebel--Lizorkin sequence spaces on $X$, and obtain their boundedness. Using this wavelet characterization and this boundedness of almost diagonal operators, the authors obtain the molecular characterization of Besov and Triebel--Lizorkin spaces. Applying this molecular characterization, the authors further establish the Littlewood--Paley…
Let $({\mathcal X},ρ,μ)$ be a space of homogeneous type in the sense of Coifman and Weiss, and $Y({\mathcal X})$ a ball quasi-Banach function space on ${\mathcal X}$, which supports a Fefferman--Stein vector-valued maximal inequality, and the boundedness of the powered Hardy--Littlewood maximal operator on its associate space. The authors first introduce the Hardy space $H_{Y}^*({\mathcal X})$, associated with $Y({\mathcal X})$, via the grand maximal function, and then establish its various real-variable characterizations, respectively, in terms of radial or non-tangential maximal functions, a…
We show that the α-fractional Bilinear Indicator/Cube Testing Constant arising in arXiv:1906.05602 is controlled by the classical fractional Muckenhoupt constant, provided the product measure σ x ω is diagonally reverse doubling (in particular if it is reverse doubling) with exponent exceeding 2(n-α). Moreover, this control is sharp within the class of diagonally reverse doubling product measures. When combined with the main results in arXiv:1906.05602, 1907.07571 and 1907.10734, the above control of BICT_{T^{α}} for α>0 yields a two weight T1 theorem for doubling weights with appropriate diag…
We consider Coifman--Fefferman inequalities for rough homogeneous singular integrals $T_Ω$ and $C_p$ weights. It was recently shown by Li-Pérez-Rivera-Ríos-Roncal that $$ \|T_Ω\|_{L^p(w)} \le C_{p,T,w} \|Mf\|_{L^p(w)} $$ for every $0< p < \infty$ and every $w \in A_\infty$. Our first goal is to generalize this result for every $w \in C_q$ where $q > \max\{1,p\}$ without using extrapolation theory. Although the bounds we prove are new even in a qualitative sense, we also give the quantitative bound with respect to the $C_q$ characteristic. Our techniques rely on recent advances in sparse domina…
Sharp Brezis--Seeger--Van Schaftingen--Yung Formulae for Higher-Order Gradients in Ball Banach Function Spaces
本文在球Banach函数空间中证明了高阶梯度差分的sharp BSVY公式，并由此建立高阶齐次Sobolev空间的刻画及临界Gagliardo--Nirenberg和Sobolev型不等式，其新颖性在于稀疏刻画与加权不等式结合。
Let $(\mathbb{X},d,μ)$ be a space of homogeneous type in the sense of R. R. Coifman and G. Weiss, and $X(\mathbb{X})$ a ball quasi-Banach function space on $\mathbb{X}$. In this article, the authors introduce the weak Hardy space $WH_X(\mathbb{X})$ associated with $X(\mathbb{X})$ via the grand maximal function, and characterize $WH_X(\mathbb{X})$ by other maximal functions and atoms. The authors then apply these characterizations to obtain the real interpolation and the boundedness of Calderón--Zygmund operators in the critical case. The main novelties of this article exist in that the authors…
The Garnett-Jones Theorem on BMO spaces associated with operators and applications
本文将经典BMO空间上的Garnett-Jones定理推广到与算子相关的BMO_L空间，证明函数到L^∞的距离等价于John-Nirenberg不等式中的最优常数，并给出紧支集函数的分解应用。
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