Chih-Chieh Hung, Chun-Yen Shen
We provide elementary proofs for the terms that are left in the work of Kelly Bickel, Sandra Pott, Maria C. Reguera, Eric T. Sawyer, Brett D. Wick who proved the sharp weighted $A_2$ bound for Haar shifts and Haar multiplier. Our proofs use weighted square function estimate, Carleson embedding and Wilson's system.
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In this paper we are proving that Sawyer type condition for boundedness work for the two weight estimates of individual Haar multipliers, as well as for the Haar shift and other "well localized" operators.
C. Stockdale, P. Villarroya, and B. Wick introduced the $ε$-maximal operator to prove the Haar multiplier is bounded on the weighted spaces $L^p(w)$ for a class of weights larger than $A_p$. We prove the $ε$-maximal operator and Haar multiplier are bounded on variable Lebesgue spaces $\Lpp(\R^n)$ for a larger collection of exponent functions than the log-Holder continuous functions used to prove the boundedness of the maximal operator on $\Lpp(\R^n)$. We also prove that the Haar multiplier is compact when restricted to a dyadic cube $Q_0$.
A sufficient condition for Haar multipliers in Triebel-Lizorkin spaces
针对 Haar 系统非无条件的 Triebel-Lizorkin 空间，本文研究仅依赖 Haar 频率的乘子，用变差范数给出算子有界的充分条件，并证明其在 $p,q>1$ 时关于参数 $u$ 基本最优。
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。
We prove that $\log n$ is an almost everywhere convergence Weyl multiplier for the orthonormal systems of non-overlapping Haar polynomials. Moreover, it is done for the general systems of martingale difference polynomials.
We study the natural resolution of the conjugated Haar multiplier $M_{w^{\frac{1}{2}}}T_σM_{w^{-\frac{1}{2}}},$ where the multiplication operators $M_{w^{\pm\frac{1}{2}}}$ are decomposed into their canonical paraproduct decompositions. We prove that each constituent operator obtained from this resolution has a linear bound on $L^2(\mathbb{R}^d;w)$ in terms of the $A_{2}$ characteristic of $w$. The main tools used are a product formula for Haar coefficients, the Carleson Embedding Theorem, the linear bound for the square function, and the well-known linear bound of $T_σ$ on $L^2(\mathbb{R}^d,w)…
We introduce multilinear analogues of dyadic paraproduct operators and Haar Multipliers, and study boundedness properties of these operators and their commutators. We also characterize dyadic BMO functions via the boundedness of certain paraproducts and also via the boundedness of the commutators of multilinear Haar Multipliers and paraproduct operators.
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。
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…
As a corollary to our main theorem we give a new proof of the result that the norm of the Hilbert transform on L^2(w) has norm bounded by a the A_2 characteristic of a weight to the first power, a theorem of one of us. This new proof begins as the prior proofs do, by passing to Haar shifts. Then, we apply a deep two-weight T1 theorem of Nazarov-Treil-Volberg, to reduce the matter to checking a certain carleson measure condition. This condition is checked with a corona decomposition of the weight. Prior proofs of this type have used Bellman functions, while this proof is flexible enough to addr…
As a corollary to our main result we deduce sharp A_p$ inequalities for T being either the Hilbert transform in dimension d=1, the Beurling transform in dimension d=2, or a Riesz transform in any dimension d\ge 2. For T_{\ast} the maximal truncations of these operators, we prove the sharp A_p weighted weak and strong-type L ^{p} (w) inequalities, for all 1<p<\infty. Key elements of the proof are (1) extrapolation (2) a recent argument for the A_2 bound in the untruncated case, an argument of Lacey-Petermichl-Reguera. (3) a weak-L^1 estimate for duals of maximal truncations. And (4) recent char…
In connection with proving the A_2 conjecture in 2010, T. Hytönen obtained a representation of general Caldéron-Zygmund operators in terms of simpler operators known as Haar shifts. In this note, we prove that the result is sharp in the sense that Haar shift representations of Hytönen's type are only available for Calderón-Zygmund operators.
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…
We provide an improvement of the Hörmander multiplier theorem in which the Sobolev space $L^r_s(\mathbb R^n)$ with integrability index $r$ and smoothness index $s>n/r$ is replaced by the Sobolev space with smoothness $s$ built upon the Lorentz space $L^{n/s,1}(\mathbb R^n)$.
A sharp Hörmander estimate for multi-parameter and multi-linear Fourier multiplier operators
本文研究多线性多参数Fourier乘子的Hörmander型定理，给出了基于L^u的Sobolev范数刻画下的光滑性假设结果，在多参数双线性情形下精确、多线性情形下几乎精确，且改进了单参数多线性情形的已有结论。
A sharp Hörmander condition for bilinear Fourier multipliers with Lipschitz singularities
该研究针对带有限Lipschitz曲线奇异集的双线性Fourier乘子，提出关于Sobolev指数精确的Hörmander条件，证明其局部L²范围内的L^p有界性，推广了非退化双线性Hilbert变换并规避退化附近一致有界性难题。
Sharp bounds in Ionescu--Wainger multiplier theorem for canonical fractions
本文证明了典型分数Ionescu--Wainger乘子定理中损失阶$N^{c_p/\log\log N}$的最优性。
Let $W$ denote a matrix $A_2$ weight. In this paper, we implement a scalar argument using the square function to deduce square-function type results for vector-valued functions in $L^2(\mathbb{R},\mathbb{C}^d)$. These results are then used to study the boundedness of the Hilbert transform and Haar multipliers on $L^2(\mathbb{R},\mathbb{C}^d)$. Our proof shortens the original argument by Treil and Volberg and improves the dependence on the $A_2$ characteristic. In particular, we prove that the Hilbert transform and Haar multipliers map $L^2(\mathbb{R},W,\mathbb{C}^d)$ to itself with dependence …
We establish sharp (H^1, L^{1,q}) and local (L \log^r L, L^{1,q}) mapping properties for rough one-dimensional multipliers. In particular, we show that the multipliers in the Marcinkiewicz multiplier theorem map H^1 to L^{1,\infty} and L \log^{1/2} L to L^{1,\infty}, and that these estimates are sharp.
We study the boundedness on the Wiener amalgam spaces $W^{p,q}_s$ of Fourier multipliers with symbols of the type $e^{iμ(ξ)}$, for some real-valued functions $μ(ξ)$ whose prototype is $|ξ|^β$ with $β\in (0,2]$. Under some suitable assumptions on $μ$, we give the characterization of $W^{p,q}_s\rightarrow W^{p,q}$ boundedness of $e^{iμ(D)}$, for arbitrary pairs of $0< p,q\leq \infty$. Our results are an essential improvement of the previous known results, for both sides of sufficiency and necessity, even for the special case $μ(ξ)=|ξ|^β$ with $1<β<2$.
We provide an improvement of Calderón and Torchinsky's version of the Hörmander multiplier theorem on Hardy spaces $H^p$ ($0<p<\infty$), by replacing the Sobolev space $L_s^2(A_0)$ by the Lorentz-Sobolev space $L_s^{τ^{(s,p)} ,\min(1,p) }(A_0)$, where $τ^{(s,p)} =\frac{n}{s-(n/\min{(1,p)}-n)}$ and $A_0$ is the annulus $\{ξ\in \mathbb{R}^n: 1/2<|ξ|<2\}$. Our theorem also extends that of Grafakos and Slavíková to the range $0<p\le 1$. Our result is sharp in the sense that the preceding Lorentz-Sobolev space cannot be replaced by a smaller Lorentz-Sobolev space $L^{r,q}_s(A_0)$ with $r< τ^{(s,p)}…
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。
The Hilbert transform has a linear bound in the $A_{2}$ characteristic on weighted $L^{2}$, \begin{equation*} \left\Vert H\right\Vert _{L^{2}(w)\rightarrow L^{2}(w)}\lesssim \left[ w \right] _{A_{2}}, \end{equation*} and we extend this linear bound to the nine constituent operators in the natural weighted resolution of the conjugation $M_{w^{\frac{1}{2}}}\mathcal{S }M_{w^{-\frac{1}{2}}}$ induced by the canonical decomposition of a multiplier into paraproducts:% \begin{equation*} M_{f}=P_{f}^{-}+P_{f}^{0}+P_{f}^{+}. \end{equation*} The main tools used are composition of paraproducts, a product …
We show that multipliers of second order Riesz transforms on products of discrete abelian groups enjoy the $L^{p} $ estimate $p^{\ast} -1$, where $p^{\ast} = \max \{ p,q \}$ and $p$ and $q$ are conjugate exponents. This estimate is sharp if one considers all multipliers of the form $\sum_i σ_{i} R_{i} R^{\ast}_{i}$ with $| σ_{i} | \leqslant 1$ and infinite groups. In the real valued case, we obtain better sharp estimates for some specific multipliers, such as $\sum_{i} σ_{i} R_{i} R^{\ast}_{i}$ with $0 \leqslant σ_{i} \leqslant 1$. These are the first known precise $L^{p} $ estimates for discr…
In this paper we prove sharp weak type 1 estimates for spherical Fourier multipliers on symmetric spaces of the noncompact type. This complements earlier results of J.-Ph. Anker and A.D. Ionescu.
Necessary conditions for weighted estimates of Multilinear Multipliers and Pseudo-Differential Operators
本文研究了多线性傅里叶乘法算子和伪微分算子加权估计的必要条件，证明了多重权重假设的最佳性，并改进了经典线性情况下的结果。
Mollified disc multipliers
该研究证明了圆盘乘子磨光化在二维勒贝格空间$L^4$上的最优有界性，属于调和分析中的乘子理论问题。
The Bochner-Riesz multipliers $ B_{δ}$ on $ \mathbb R ^{n}$ are shown to satisfy a range of sparse bounds, for all $0< δ< \frac {n-1}2 $. The range of sparse bounds increases to the optimal range, as $ δ$ increases to the critical value, $ δ=\frac {n-1}2$, even assuming only partial information on the Bochner-Riesz conjecture in dimensions $ n \geq 3$. In dimension $n=2$, we prove a sharp range of sparse bounds. The method of proof is based upon a `single scale' analysis, and yields the sharpest known weighted estimates for the Bochner-Riesz multipliers in the category of Muckenhoupt weights.
We generalize the idea of a multiplier in two different ways and generalize a recent result of Geiss, Montomery-Smith and Saksman. First of all, we consider multipliers in the form of a vector acting on a scalar function. Using this technique we compute the sharp lower bound estimate for $L^p$ operator norm of a quadratic perturbation of the real part of the Ahlfors-Beurling operator. Secondly, we consider matrix-valued multipliers to obtain a new proof showing that the $L^p$ operator norm of the Ahlfors-Beurling operator is bounded below by p^*-1.
We provide asymptotically sharp bounds for the $L_p$ norms of the Fourier multipliers with the symbols $e^{iλφ(ξ/|ξ|)}$, where $λ\in \mathbb{R}$ is a large parameter.
Balanced measures, sparse domination and complexity-dependent weight classes
本文研究一般测度空间上原子滤子定义的Haar位移算子，证明弱正则性条件足以推出稀疏控制型定理，并据此刻画了依赖算子复杂度的权类，且结果在定性意义下是精确的。
We prove endpoint bounds for the square function associated with radial Fourier multipliers acting on $L^{p}$ radial functions. This is a consequence of endpoint bounds for a corresponding square function for Hankel multipliers. We obtain a sharp Marcinkiewicz-type multiplier theorem for multivariate Hankel multipliers and $L^p$ bounds of maximal operators generated by Hankel multipliers as corollaries. The proof is built on techniques developed by Garrigós and Seeger for characterizations of Hankel multipliers.
Quantitative weighted estimates are obtained for the Littlewood-Paley square function $S$ associated with a lacunary decomposition of ${\mathbb R}$ and for the Marcinkiewicz multiplier operator. In particular, we find the sharp dependence on $[w]_{A_p}$ for the $L^p(w)$ operator norm of $S$ for $1<p\le 2$.
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.
Endpoint estimates for Haar shift operators with balanced measures
本文证明了带平衡测度的Haar移位算子的H^1和BMO端点不等式，简化了L^p结果的证明，并证明了在鞅Lipschitz空间上的正则性，且测度类是最优的。
Given a bounded measurable function $σ$ on $\mathbb{R}^n$, we let $T_σ$ be the operator obtained by multiplication on the Fourier transform by $σ$. Let $0<s_1\le s_2\le \cdots \le s_n<1$ and $ψ$ be a Schwartz function on the real line whose Fourier transform $\widehatψ$ is supported in $[-2,-1/2]\cup[1/2,2]$ and which satisfies $\sum_{j \in \mathbb{Z}} \widehatψ\left(2^{-j} ξ\right)=1$ for all $ξ\neq 0$. In this work we sharpen the known forms of the Marcinkiewicz multiplier theorem by finding an almost optimal function space with the property that, if the function \begin{equation*} (ξ_1,\dots…
In this work, we describe several results exhibited during a talk at the El Escorial 2012 conference. We aim to pursue the development of a multi-frequency Calderon-Zygmund analysis introduced in [9]. We set a definition of general multi-frequency Calderon-Zygmund operator. Unweighted estimates are obtained using the corresponding multi-frequency decomposition of [9]. Involving a new kind of maximal sharp function, weighted estimates are obtained.
Weighted spectral cluster bounds and a sharp multiplier theorem for ultraspherical Grushin operators
本文研究d维球面上在k维球面处奇异的Grushin型退化椭圆算子，通过建立加权谱簇界和超球多项式精确估计，证明了在2k≤d时最优的Mihlin-Hörmander型谱乘子定理及Bochner-Riesz可求和性结果。
Let $L$ be a non-negative self-adjoint operator on $L^2(\mathbb{R}^n)$. By spectral theory, we can define the operator $F(L)$, which is bounded on $L^2(X)$, for any bounded Borel function $F$. In this paper, we study the sharp weighted $L^p$ estimates for spectral multipliers $F(L)$ and their commutators $[b, F(L)]$ with BMO functions $b$. We would like to emphasize that the Gaussian upper bound condition on the heat kernels associated to the semigroups $e^{-tL}$ is not assumed in this paper.
We prove a multiplier theorem of Mihlin-Hörmander type for operators of the form $-Δ_x - V(x) Δ_y$ on $\mathbb{R}^{d_1}_x \times \mathbb{R}^{d_2}_y$, where $V(x) = \sum_{j=1}^{d_1} V_j(x_j)$, the $V_j$ are perturbations of the power law $t \mapsto |t|^{2σ}$, and $σ\in (1/2,\infty)$. The result is sharp whenever $d_1 \geq σd_2$. The main novelty of the result resides in its robustness: this appears to be the first sharp multiplier theorem for nonelliptic subelliptic operators allowing for step higher than two and perturbation of the coefficients. The proof hinges on precise estimates for eigenv…
We prove that the operator norm on weighted Lebesgue space L2(w) of the commutators of the Hilbert, Riesz and Beurling transforms with a BMO function b depends quadratically on the A2-characteristic of the weight, as opposed to the linear dependence known to hold for the operators themselves. It is known that the operator norms of these commutators can be controlled by the norm of the commutator with appropriate Haar shift operators, and we prove the estimate for these commutators. For the shift operator corresponding to the Hilbert transform we use Bellman function methods, however there is n…
We investigate the unconditional basis property of martingale differences in weighted $L^2$ spaces in the non-homogeneous situation (i.e. when the reference measure is not doubling). Specifically, we prove that finiteness of the quantity $[w]_{A_2}=\sup_I \, < w>_I < w^{-1}>_I$, defined through averages $ <\cdot >_I$ relative to the reference measure $ν$, implies that each martingale transform relative to $ν$ is bounded in $L^2(w\, dν)$. Moreover, we prove the linear in $[w]_{A_2}$ estimate of the unconditional basis constant of the Haar system. Even in the classical case of the standard dyadi…
Quantitative formulations of Fefferman's counterexample for the ball multiplier are naturally linked to square function and vector-valued estimates for directional singular integrals. The latter are usually referred to as Meyer-type lemmas and are traditionally attacked by combining weighted inequalities with sharp estimates for maximal directional averaging operators. This classical approach fails to give sharp bounds. In this article we develop a novel framework for square function estimates, based on directional Carleson embedding theorems and multi-parameter time-frequency analysis, which …
A sharp spectral multiplier theorem on the Engel group
该文证明Engel群上次拉普拉斯算子在局部H^s(s>2)条件下的谱乘子定理，确定尖锐阈值为2，结合加权Plancherel估计与秩二谱子空间分解，并给出局部化核的渐近展开。
Sparse domination results for compactness on weighted spaces
本文利用稀疏界方法，证明了Calderón-Zygmund算子和Haar乘子在加权$L^p$空间上的紧性，并扩展了权重类范围。
We prove endpoint-type sparse bounds for Walsh-Fourier Marcinkiewicz multipliers and Littlewood-Paley square functions. These results are motivated by conjectures of Lerner in the Fourier setting. As a corollary, we obtain novel quantitative weighted norm inequalities for these operators. Among these, we establish the sharp growth rate of the $L^p$ weighted operator norm in terms of the $A_p$ characteristic in the full range $1<p<\infty$ for Walsh-Littlewood-Paley square functions, and a restricted range for Marcinkiewicz multipliers. Zygmund's $L{(\log L)^{\frac12}}$ inequality is the core of…
It is well known that the Stein-Tomas $L^2$ Fourier restriction theorem can be used to derive sharp $L^p$ bounds for radial Fourier multipliers such as the Bochner-Riesz means. In a similar manner, $L^p \to L^2$ estimates for spectral projection operators have been utilized in order to obtain sharp $L^p$ bounds for spectral multipliers of self-adjoint elliptic pseudo-differential operators on compact manifolds. In this paper, we refine an endpoint result for spectral multipliers due to Seeger, providing endpoint bounds in terms of Besov spaces. Our proof is based on the ideas from the recent w…
We provide necessary and sufficient conditions for multilinear multiplier operators with symbols in $L^r$-based product-type Sobolev spaces uniformly over all annuli to be bounded from products of Hardy spaces to a Lebesgue space. We consider the case $1<r\leq 2$ and we characterize boundedness in terms of inequalities relating the Lebesgue indices (or Hardy indices), the dimension, and the regularity and integrability indices of the Sobolev space. The case $r>2$ cannot be handled by known techniques and remains open. Our result not only extends but also establishes the sharpness of previous r…
The bilinear cone multiplier on $\mathbb{R}^2\times \mathbb{R}^2$
本文研究了二维空间上的双线性锥乘法子，通过将双线性算子分解为与线性锥乘法子及其变体相关的平方函数，并利用合适的强极大值估计和几何方法，得到了双线性锥乘法子的$ L^p $有界性。
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 prove a sharp Mihlin-Hormander multiplier theorem for Schroedinger operators $H$ on $\R^n$. The method, which allows us to deal with general potentials, improves Hebisch's method relying on heat kernel estimates for positive potentials. Our result applies to, in particular, the negative Poeschl-Teller potential $V(x)= -ν(ν+1) \sech^2 x $, $ν\in \N$, for which $H$ has a resonance at zero.
In this paper, the index groups for which the weighted Young's inequalities hold in both continuous case and discrete case are characterized. As applications, the index groups for the product inequalities on modulation spaces are characterized, we also obtain the weakest conditions for the boundedness of bilinear Fourier multipliers on modulation spaces in some sense. For the fractional integral operator, the sharp conditions for the boundedness of power weighted Lp-Lq estimates in both continuous case and discrete case are obtained. By a quite different approach from others, our theorems opti…
We consider generalized Bochner-Riesz multipliers of the form $(1-ρ(ξ))_+^λ$ where $ρ(ξ)$ is the Minkowski functional of a convex domain in $\mathbb{R}^2$, with emphasis on domains for which the usual Carleson-Sjölin $L^p$ bounds can be improved. We produce convex domains for which previous results due to Seeger and Ziesler are not sharp. We identify two key properties of convex domains that lead to improved $L^p$ bounds for the associated Bochner-Riesz operators. First, we introduce the notion of the "additive energy" of the boundary of a convex domain. Second, we associate a set of direction…
Spectral multipliers III: Endpoint bounds, intertwining operators, and twisted Hardy spaces
该研究将自由拉普拉斯谱乘子的基本估计推广到带实标量势的受扰哈密顿量，得到Mihlin乘子精确界等结果，部分证实相关猜想并刻画扭曲Hardy空间。
A sharp multiplier theorem for solvable extensions of Heisenberg and related groups
本文针对Heisenberg型群及Métivier群与阿贝尔群直积的可解扩张，将谱乘子定理的光滑性条件从齐次维数阈值改进为拓扑维数阈值，达到尖锐最优结果。
Sparse bounds for pseudo-multipliers associated to Grushin operators, I
本文利用稀疏算子的逐点控制方法，证明了满足Hörmander条件的Grushin伪乘子的尖锐定量加权$L^p$估计。
We prove a sharp Hörmander multiplier theorem for Schrödinger operators $H=-Δ+V$ on $\mathbb{R}^n$. The result is obtained under certain condition on a weighted $L^\infty$ estimate, coupled with a weighted $L^2$ estimate for $H$, which is a weaker condition than that for nonnegative operators via the heat kernel approach. Our approach is elaborated in one dimension with potential $V$ belonging to certain critical weighted $L^1$ class. Namely, we assume that $\int (1+|x|) |V(x)|dx$ is finite and $H$ has no resonance at zero. In the resonance case we assume $\int (1+|x|^2) |V(x)| dx$ is finite.
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 Hilbert transform norms on Schatten classes: classification and optimal spectral corrections
本文证明Hilbert变换在Schatten类上的范数为$C_p$，解决多个猜想，并通过谱缺陷恒等式分类迹不等式，提出最优谱修正。
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