Alexander Reznikov, Sergei Treil, Alexander Volberg
A simple shortcut to proving sharp weighted estimates for the Martingale Transform and for the dyadic shift of order 1 (and so for the Hilbert transform) is presented. It is a unified proof for these both transforms. Key words: Calder\on--Zygmund operators, $A_2$ weights, $A_1$ weights, Carleson embedding theorem, Bellman function, dyadic shifts, nonhomogeneous Harmonic Analysis.
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In this article, we investigate the boundedness properties of the multilinear dyadic paraproduct operators in the weighted setting. We also obtain weighted estimates for the multilinear Haar multipliers and their commutators with dyadic BMO functions.
We give a short and simple polynomial estimate of the norm of weighted dyadic shift on metric space with geometric doubling, which is linear in the norm of the weight. Combined with the existence of special probability space of dyadic lattices built in A. Reznikov, A. Volberg, "Random "dyadic" lattice in geometrically doubling metric space and $A_2$ conjecture", arXiv:1103.5246, and with decomposition of Calderón-Zygmund operators to dyadic shifts from Hytönen's "The sharp weighted bound for general Calderón-Zygmund operators", arXiv:1007.4330 (and later T. Hytönen, C. Pérez, S. Treil, A. Volb…
The following deals with the $T(b)$ theorems of David, Journé, and Semmes \cite{DJS} considered in a dyadic setting. We find sharp growth estimates for a global and a local dyadic $T(b)$ Theorem. We use multiscale analysis and Haar wavelets in the local case.
We prove a sharp integral inequality for the dyadic maximal operator, connecting integrals of $ϕ$, and of the dyadic maximal function of $ϕ$.
We define a generalized dyadic maximal operator involving the infinite product and discuss weighted inequalities for the operator. A formulation of the Carleson embedding theorem is proved. Our results depend heavily on a generalized Hölder's inequalities.
In this paper we refine the recent sparse domination of the integrated $p = 2$ matrix weighted dyadic square function by T. Hytonen, S. Petermichl, and A. Volberg to prove a pointwise sparse domination of general matrix weighted dyadic square functions. We then use this to prove sharp two matrix weighted strong type inequalities for matrix weighted dyadic square functions when $1 < p \leq 2$.
In this paper, we show that dyadic paraproducts $π_b$ with $b$ in dyadic BMO are bounded on matrix weighted $L^p(W)$ if $W$ is a matrix $\text{A}_p$ weight.
We prove sharp $L^1$ inequalities for the dyadic maximal function $M_Tϕ$ when $ϕ$ satisfies certain $L^1$ and $L^{\infty}$ conditions
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.
We show sufficient conditions on matrix weights $U$ and $V$ for the martingale transforms to be uniformly bounded from $L^2(V)$ to $L^2(U)$. We also show that these conditions imply the uniform boundedness of the dyadic shifts as well as the boundedness of the Hilbert transform between these spaces.
Sharp upper bounds for integral quantities related to the Bellman function of three variables of the dyadic maximal operator
本文研究二进极大算子三积分变量 Bellman 函数相关积分量，给出其精确上界。
The trace spaces of Sobolev spaces and related fractional smoothness spaces have been an active area of research since the work of Nikolskii, Aronszajn, Slobodetskii, Babich and Gagliardo among others in the 1950's. In this paper we review the literature concerning such results for a variety of weighted smoothness spaces. For this purpose, we present a characterization of the trace spaces (of fractional order of smoothness), based on integral averages on dyadic cubes, which is well adapted to extending functions using the Whitney extension operator.
This paper presents a new proof of the results regarding the continuity of weighted estimates with respect to the characteristic of the weight. Here we first prove the result in the dyadic case which is "easier" and then by the use of the Bellman function technique we pass to the continuous setting which is harder in general. To be more precise, as far as we know, this passage from the Martingale transform to the Hilbert transform described in this note is new.
The convex body maximal operator is a natural generalisation of the Hardy Littlewood maximal operator. In the presence of a matrix weight it is not bounded.
Dyadic Martingale Transforms and Weighted Walsh-Carleson Operators
本文研究了由双侧马丁格尔变换产生的加权瓦尔什-卡拉森极大算子，证明了在满足均匀二进制变化条件和顶部二进制尺度上均匀界的情况下，这些算子在子序列上的弱型(1,1)估计，并给出了与权重在顶部二进制尺度附近行为相关的发散准则。
On a sharp upper bound related to the Bellman function of three integral variables of the dyadic maximal operator
该研究聚焦于二进极大算子的三积分变量Bellman函数相关积分量的精确上界，探究其性质，该上界已由文献[11]确定。
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.
In this paper, we introduce a dyadic structure on convex domains of finite type via the so-called dyadic flow tents. This dyadic structure allows us to establish weighted norm estimates for the Bergman projection $P$ on such domains with respect to Muckenhoupt weights. In particular, this result gives an alternative proof of the $L^p$ boundedness of $P$. Moreover, using extrapolation, we are also able to derive weighted vector-valued estimates and weighted modular inequalities for the Bergman projection.
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.
We extend the definitions of dyadic paraproduct and t-Haar multipliers to dyadic operators that depend on the complexity (m,n), for m and n positive integers. We will use the ideas developed by Nazarov and Volberg to prove that the weighted L^2(w)-norm of a paraproduct with complexity (m,n) associated to a function b\in BMO, depends linearly on the A_2-characteristic of the weight w, linearly on the BMO-norm of b, and polynomially in the complexity. This argument provides a new proof of the linear bound for the dyadic paraproduct (the one with complexity (0,0)). Also we prove that the L^2-norm…
We prove that the class of Muckenhoupt A_p weights coincides with the intersection of finitely many suitable translates of dyadic A_p, in both the one-parameter and multiparameter cases, and that the analogous results hold for the reverse Hölder class RH_p, for doubling measures, and for the space VMO of functions of vanishing mean oscillation. We extend to the multiparameter (product) space BMO of functions of bounded mean oscillation the corresponding one-parameter BMO result due to T. Mei, by means of the Carleson-measure characterization of multiparameter BMO. Our results hold in both the …
This book is a short introduction into dyadic analysis with applications to classical weighted norm inequalities.
In this paper, we study the behavior of the weighted composition operators acting on Bergman spaces defined on strictly pseudoconvex domains via the sparse domination technique from harmonic analysis. As a byproduct, we also prove a weighted type estimate for the weighted composition operators which is adapted to Sawyer-testing conditions. Our results extend the work by the first author, Li, Shi and Wick under a much more general setting.
Using Wilson's Haar basis in $\R^n$, which is different than the usual tensor product Haar functions, we define its associated dyadic paraproduct in $\R^n$. We can then extend "trivially" Beznosova's Bellman function proof of the linear bound in $L^2(w)$ with respect to $[w]_{A_2}$ for the 1-dimensional dyadic paraproduct. Here trivial means that each piece of the argument that had a Bellman function proof has an $n$-dimensional counterpart that holds with the same Bellman function. The lemma that allows for this painless extension we call the good Bellman function Lemma. Furthermore the argum…
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].
On the weighted weak-type constant for the dyadic square function
本文通过显式构造例子，证明了二进平方函数的加权弱型估计在A_2特征上是最优的。
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 show that the two-weight estimate for the dyadic square function proved by Lacey--Li in [2] is sharp.
Dyadic harmonic analysis and weighted inequalities: the sparse revolution
本文探讨了双变量调和分析在获得经典Calderón-Zygmund奇异积分算子及其交换子加权估计中的应用，并总结了稀疏算子主导方法的发展历程及主要贡献。
In this paper, we prove a sharp Mei's Lemma with assuming the bases of the underlying general dyadic grids are different. As a byproduct, we specify all the possible cases of adjacent general dyadic systems with different bases. The proofs have connections with certain number-theoretic properties.
Sharp Lower Bounds for Dyadic Square Functions of indicator functions of sets
本文研究了dyadic square functions的下界，针对S₂和S₁，给出了基于布朗运动首次退出时间和Takagi函数的精确下界，改进了经典Burkholder-Davis-Gundy下界。
We prove a pointwise estimate for positive dyadic shifts of complexity $m$ which is linear in the complexity. This can be used to give a pointwise estimate for Calderón-Zygmund operators and to answer a question posed by A. Lerner. Several applications to weighted estimates for both multilinear Calderón-Zygmund operators and square functions are discussed.
For a general Calderon-Zygmund operator $T$ on $R^N$, it is shown that $\|Tf\|_{L^2(w)}\leq C(T)\|w\|_{A_2}\|f\|_{L^2(w)}$ for all Muckenhoupt weights $w\in A_2$. This optimal estimate was known as the $A_2$ conjecture. A recent result of Perez-Treil-Volberg reduced the problem to a testing condition on indicator functions, which is verified in this paper. The proof consists of the following elements: (i) a variant of the Nazarov-Treil-Volberg method of random dyadic systems with just one random system and completely without bad parts; (ii) a resulting representation of a general Calderon-Zygm…
The theory of (Muckenhoupt) weights arises in many areas of analysis, for example in connection with bounds for singular integrals and maximal functions on weighted spaces. We prove that a certain averaging process gives a method for constructing A_p weights from a measurably varying family of dyadic A_p weights. This averaging process is suggested by the relationship between the A_p weight class and the space of functions of bounded mean oscillation. The same averaging process also constructs weights satisfying reverse Holder (RH_p) conditions from families of dyadic RH_p weights, and extends…
It is well-known that dyadic martingale transforms are a good model for Calderón-Zygmund singular integral operators. In this paper we extend some results on weighted norm inequalities to vector-valued functions. We prove that, if $W$ is an $A_2$ matrix weight, then the weighted $L^2$-norm of a Calderón-Zygmund operator with cancellation has the same dependence on the $A_2$ characteristic of $W$ as the weighted $L^2$-norm of the martingale transform. Thus the question of the dependence of the norm of matrix-weighted Calderón-Zygmund operators on the $A_2$ characteristic of the weight is reduce…
For cylindrically symmetric functions dyadically supported on the paraboloid, we obtain a family of sharp linear and bilinear adjoint restriction estimates. As corollaries, we first extend the ranges of exponents for the classical \textit{linear or bilinear adjoint restriction conjectures} for such functions and verify the \textit{linear adjoint restriction conjecture} for the paraboloid. We also interpret the restriction estimates in terms of solutions to the Schrödinger equation and establish the analogous results when the paraboloid is replaced by the lower third of the sphere.
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…
Commutators for the maximal and sharp functions with weighted Lipschitz functions on weighted Morrey spaces
研究加权Morrey空间上Hardy-Littlewood极大函数与sharp极大函数交换子的有界性，并据此给出加权Lipschitz函数的新刻画。
We prove sharp estimates for the Bergman projection in weighted Bergman spaces in terms of the Bekolle constant. Our main tools are a dyadic model dominating the operator and an adaptation of a method of Cruz-Uribe, Martell and Perez.
Using the Calderón-Zygmund decomposition, we give a novel and simple proof that $L^2$ bounded dyadic shifts admit a domination by positive sparse forms with linear growth in the complexity of the shift. Our estimate, coupled with Hytönen's dyadic representation theorem, upgrades to a positive sparse domination of the class $\mathcal U$ of singular integrals satisfying the assumptions of the classical $T(1)$-theorem of David and Journé, with logarithmic-Dini type smoothness of the integral kernel. Furthermore, our proof extends rather easily to the $\mathbb R^n$-valued case, yielding as a corol…
We give a new proof of the sharp one weight $L^p$ inequality for any operator $T$ that can be approximated by Haar shift operators such as the Hilbert transform, any Riesz transform, the Beurling-Ahlfors operator. 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. Our method is flexible enough to prove the corresponding sharp one-weight norm inequalities for some operators of harmonic analysis: the maximal singular integrals associated to $T$, Dyadic square functions …
Using modern techniques of dyadic harmonic analysis, we are able to prove sharp estimates for the Bergman projection and Berezin transform and more general operators in weighted Bergman spaces on the unit ball in $\mathbb{C}^n$. The estimates are in terms of the Bekolle-Bonami constant of the weight.
In this note we give a sharp weighted estimate for square function from $L^2(w)$ to $L^2(w)$, $w\in A_2$. This has been known. But we also give a sharpening of this weighted estimate in the spirit of $T1$-type testing conditions. Finally we show that for any weight $w\in A^d_2$ and any characteristic function of a measurable set $\|S_wχ_E\|_{L^{2, \infty}(w^{-1})} \le C \sqrt{[w]_{A^d_2}}\, \|χ_E\|_w$, and this estimate is sharp. So on characteristic functions of measurable sets at least, no logarithmic correction is needed for the weak type of the dyadic square function.The sharp estimate for…
For an $A_p$ weight $w$ the norm of the Hilbert Transform in $L^p(w)$, $1<p<\infty$ is estimated by $[w]_{A_p}^{s}$, where $[w]_{A_p}$ is the $A_p$ characteristic of the weight $w$ and $s = \max(1,1/(p-1))$; as simple examples with power weights show, these estimates are sharp. A natural question to ask, is whether it is possible to improve the exponent $s$ in the above estimate if one replaces the $A_p$ characteristic by its "fattened" version, where the averages are replaced by Poisson-like averages. For power weights (for example with $p=2$ and Poisson averages) one can see that there is in…
We prove sharp weak type weighted estimates for a class of sparse operators that includes majorants of standard $α$-fractional singular integrals, fractional integral operators, Marcinkiewicz integral operators, and square functions. These bounds are knows to be sharp in many cases, and our main new result is the optimal bound $$[w]_{A_{p,q}}^{\frac{1}{q}}[w^{q}]_{A_{\infty}}^{\frac{1}{2}-\frac{1}{p}}\lesssim[w]_{A_{p,q}}^{\frac{1}{2}-\fracα{d}}$$ for proper conditions which satisfy that three index $p$, $q$ and $α$ ensure weak type norm of fractional square functions on $L^{q}(w^{q})$ with $p…
We prove weighted and vector-valued variational estimates for ergodic averages on $\mathbb{R}^d$. The weighted square function estimate relating ergodic averages to the dyadic martingale is obtained using an $\ell^r$ version of a reverse Hölder inequality for variation seminorms.
We prove the matrix $A_2$ conjecture for the dyadic square function, that is, a norm estimate of the matrix weighted square function, where the focus is on the sharp linear dependence on the matrix $A_2$ constant in the estimate. Moreover, we give a mixed estimate in terms of $A_2$ and $A_{\infty}$ constants. Key is a sparse domination of a process inspired by the integrated form of the matrix--weighted square function.
Two-weight estimates for the square function and $t$-Haar multipliers
本文研究了权三元组$(u,v,w)$下二进加权平方函数$S_w$与$t$-Haar乘子从$L^2(u)$到$L^2(v)$的有界性，给出了两者的充要条件并建立了基于平方函数有界性的刻画。
Sharp mixed $A_p$-$A_\infty$ estimates for sparse operators on filtered and nonhomogeneous measure spaces
本文在两类非倍测度背景下证明稀疏算子的混合$A_p$-$A_\infty$估计，获得与经典情形相同幂次的强、弱型界，并解决Rubio de Francia平方函数及Haar平移的开放问题。
We consider a general family of Carleson sequences associated with dyadic $A_2$ weights and find sharp -- or, in one case, simply best known -- upper and lower bounds for their Carleson norms in terms of the $A_2$-characteristic of the weight. The results obtained make precise and significantly generalize earlier estimates by Wittwer, Vasyunin, Beznosova, and others. We also record several corollaries, one of which is a range of new characterizations of dyadic $A_2.$ Particular emphasis is placed on the relationship between sharp constants and optimizing sequences of weights; in most cases exp…
Sharp higher order Admas' inequality with exact growth condition on weighted Sobolev spaces
本文在加权Sobolev空间上建立了具有精确增长条件的尖锐高阶Adams不等式，证明其临界常数与指数的最优性，并将其应用于一类常微分方程，得到了弱解概念和正则性理论。
For a matrix A_2 weight W on R^p, we introduce a new notion of W-Calderón-Zygmund matrix kernels, following earlier work in by Isralowitz. We state and prove a T1 theorem for such operators and give a representation theorem in terms of dyadic W-Haar shifts and paraproducts, in the spirit of Hytönen's Representation Theorem. Finally, by means of a Bellman function argument, we give sharp bounds for such operators in terms of bounds for weighted matrix martingale transforms and paraproducts.
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…
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。
Any Calderon-Zygmund operator T is pointwise dominated by a convergent sum of positive dyadic operators. We give an elementary self-contained proof of this fact, which is simpler than the probabilistic arguments used for all previous results in this direction. Our argument also applies to the q-variation of certain Calderon-Zygmund operators, a stronger nonlinearity than the maximal truncations. As an application, we obtain new sharp weighted inequalities.
Given a general dyadic grid ${\mathscr{D}}$ and a sparse family of cubes ${\mathcal S}=\{Q_j^k\}\in {\mathscr{D}}$, define a dyadic positive operator ${\mathcal A}_{\mathscr{D},{\mathcal S}}$ by $${\mathcal A}_{\mathscr{D},{\mathcal S}}f(x)=\sum_{j,k}f_{Q_j^k}χ_{Q_j^k}(x).$$ Given a Banach function space $X({\mathbb R}^n)$ and the maximal Calderón-Zygmund operator $T_{\natural}$, we show that $$\|T_{\natural}f\|_X\le c(n,T)\sup_{\mathscr{D},{\mathcal S}}\|{\mathcal A}_{\mathscr{D},{\mathcal S}}f\|_{X}.$$ This result is applied to weighted inequalities. In particular, it implies: (i) the "two-w…
We characterize the Hilbert--Schmidt class membership of commutator with the Hilbert transform in the two weight setting. The characterization depends upon the symbol of the commutator being in a new weighted Besov space. This follows from a Schatten class $S_p$ result for dyadic paraproducts, where $1< p < \infty $. We discuss the difficulties in extending the dyadic result to the full range of Schatten classes for the Hilbert transform.
In this paper, we study the weighted inequality for multilinear fractional maximal operators and fractional integrals. We give sharp weighted estimates for both operators.
Using Bellman function approach, we present new proofs of weighted $L^2$ inequalities for square functions, with the optimal dependence on the $A_2$ characteristics of the weight and further explicit constants. We study the estimates both in the analytic and probabilistic context, and, as application, obtain related estimates for the classical Lusin and Littlewood-Paley square functions.
Commutators of the maximal and sharp functions with weighted Lipschitz functions
本文研究了Hardy-Littlewood极大函数与加权Lipschitz函数形成的极大交换子和非线性交换子在加权Lebesgue空间上的有界性，并给出了加权Lipschitz空间的新刻画，同时将类似结果推广到锐函数的非线性交换子。
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