Stephanie Salomone
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.
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We prove a sharp integral inequality for the dyadic maximal operator, connecting integrals of $ϕ$, and of the dyadic maximal function of $ϕ$.
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.
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 $1 < p \leq 2$.
We prove sharp $L^1$ inequalities for the dyadic maximal function $M_Tϕ$ when $ϕ$ satisfies certain $L^1$ and $L^{\infty}$ conditions
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 show that the two-weight estimate for the dyadic square function proved by Lacey--Li in [2] is sharp.
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。
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…
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.
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 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.
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 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…
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…
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。
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。
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 $…
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…
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 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.
Vector-valued estimates for shifted operators
该研究针对移位算子估计的对数增长最优性未完全解决的问题，给出广义Peetre极大函数的精确向量值移位极大不等式，以新方式推导出多类移位算子带最优对数增长的改进估计，还得到移位二进Hardy-Littlewood极大算子的向量值极大不等式。
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…
I give a mini-survey of several approaches to the $A_2$ theorem, biased towards the "corona" rather than the "Bellman" side of the coin. There are two new results (a streamlined form of Lerner's local oscillation formula, and the sharpness of the linear-in-complexity weak $(1,1)$ bound for dyadic shifts) and two new proofs of known results (the $A_p$--$A_\infty$ testing conditions, and the two-weight $T1$ theorem for positive dyadic operators).
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.
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.
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.
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 …
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。
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…
Sharp $L^p$ estimates for the strong spherical maximal operator
本文证明了强球面极大算子在 $n\geq4$ 且 $p>(n+1)/(n-1)$ 时的 $L^p$ 有界性，解决了 Hickman 和 Zahl 的猜想，方法基于二进分解和 $TT^*$ 论证。
For continuous boundary data, including data of polynomial growth, modified Poisson integrals are used to write solutions to the half space Dirichlet and Neumann problems in $\mathbb{R}^{n}$. Pointwise growth estimates for these integrals are given and the estimates are proved sharp in a strong sense. For decaying data, a new type of modified Poisson integral is introduced and used to develop asymptotic expansions for solutions of these half space problems.
Bilinear Rough Singular Integrals near the Critical Integrability via Sharp Fourier Multiplier Criteria
本文研究了双线性粗糙奇异积分算子在临界可积性附近的有界性问题，提出了一种新的傅里叶乘子算子有界性准则，改进了已有条件并接近线性理论推测的端点估计。
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.
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.
Sharp bilinear estimates for maximal singular integrals with kernels in weighted $L^q$ spaces
本文研究$\u211d$上带粗糙齐次核的（二进）极大双线性算子的有界性，在完整拟巴拿赫指数范围内建立了精确的双线性$L^p$估计，通过允许核角分量属于$\u2120^1$上的加权$L^q$空间，拓展并统一了该领域的多项近期研究成果。
We obtain sharp bounds for the monotonic rearrangement operator from "dyadic-type" classes to "continuous". In particular, for the $\mathrm{BMO}$ space and Muckenhoupt classes. The idea is to connect the problem with a simple geometric construction named $α$-extension.
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…
This paper concerns the finite-time blow-up and asymptotic behaviour of solutions to nonlinear Volterra integrodifferential equations. Our main contribution is to determine sharp estimates on the growth rates of both explosive and nonexplosive solutions for a class of equations with nonsingular kernels under weak hypotheses on the nonlinearity. In this superlinear setting we must be content with estimates of the form $\lim_{t\toτ}A(x(t),t) = 1$, where $τ$ is the blow-up time if solutions are explosive or $τ= \infty$ if solutions are global. Our estimates improve on the sharpness of results in …
We prove sharp weak and strong type weighted estimates for a class of dyadic operators that includes majorants of both standard singular integrals and square functions. Our main new result is the optimal bound $[w]_{A_p}^{1/p}[w]_{A_\infty}^{1/2-1/p}\lesssim [w]_{A_p}^{1/2}$ for the weak type norm of square functions on $L^p(w)$ for $p>2$; previously, such a bound was only known with a logarithmic correction. By the same approach, we also recover several related results in a streamlined manner.
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。
We use the Bellman function method to give an elementary proof of a sharp weighted estimate for the Haar shifts, which is linear in the $A_2$ norm of the weight and in the complexity of the shift. Together with the representation of a general Calderón--Zygmund operator as a weighted average (over all dyadic lattices) of Haar shifts, (cf. arXiv:1010.0755v2[math.CA], arXiv:1007.4330v1[math.CA]) it gives a significantly simpler proof of the so-called the $A_2$ conjecture. The main estimate is a very general fact about concave functions, which can be very useful in other problems of martingale Har…
Let $X$ be a standard Gaussian random variable. For any $p \in (1, \infty)$, we prove the existence of a universal constant $C_{p}>0$ such that the inequality $$(\mathbb{E} |h'(X)|^{p})^{1/p} \geq C_{p} \sqrt{d} (\mathbb{E} |h(X)|^{p})^{1/p}$$ holds for all $d\geq 1$ and all polynomials $h : \mathbb{R} \to \mathbb{C}$ whose spectrum is supported on frequencies at least $d$, that is, $\mathbb{E} h(X) X^{k}=0$ for all $k=0,1, \ldots, d-1$. As an application of this optimal estimate, we obtain an affirmative answer to the Gaussian analogue of a question of Mendel and Naor (2014) concerning the gr…
Wiman-Valiron method for fractional derivatives and sharp growth estimates of $α$-analytic solutions for linear fractional differential equations
本文通过扩展Wiman-Valiron理论，研究了分数微分方程中α-解析解的精确增长估计，并推广了Kochubei的结果。
We prove a sharp Bernstein-type inequality for complex polynomials which are positive and satisfy a polynomial growth condition on the positive real axis. This leads to an improved upper estimate in the recent work of Culiuc and Treil on the weighted martingale Carleson embedding theorem with matrix weights. In the scalar case this new upper bound is optimal.
Curvature and sharp growth rates of log-quasimodes on compact manifolds
该研究在截面曲率非正的紧黎曼流形上得到谱投影算子L^q范数的最优估计，证明估计在平坦或负曲率流形上饱和，据此可对q∈(2,q_c]时的紧空间形式分类，弥补了q>q_c时无法区分零/负曲率的局限。
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