Detlef Mueller, Christoph Thiele
Let L be the distinguished Laplacian on certain semidirect products of R by R^n which are of ax+b type. We prove pointwise estimates for the convolution kernels of spectrally localized wave operators for arbitrary time and scaling parameter.
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Let S be a Damek-Ricci space and L be a distinguished left invariant Laplacian on S. We prove pointwise estimates for the convolution kernels of spectrally localized wave operators associated with L. This generalizes previous results proved by D. Mueller and C. Thiele on ax+b-groups. We also prove pointwise estimates of the gradient of these convolution kernels. As a corollary we reprove basic multiplier estimates from previous papers of W. Hebisch and T. Steger and M. Vallarino by different methods. Finally we derive Sobolev estimates for the solution to the wave equation associated with L.
Exploiting continuity properties of Fourier multipliers on modulation spaces and Wiener amalgam spaces, we study the Cauchy problem for the NLW equation. Local wellposedness for rough data in modulation spaces and Wiener amalgam spaces is shown. The results formulated in the framework of modulation spaces refine those in [3]. The same arguments may apply to obtain local wellposedness for the NLKG equation.
The problem whether weighted estimates for multilinear Fourier multipliers with Sobolev regularity hold under weak condition on weights is considered.
We prove a Hörmander type multiplier theorem for multilinear Fourier multipiers with multiple weights. We also give weighted estimates for their commutators with vector $BMO$ functions.
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.
Let $P(D)$ be the Laplacian $Δ,$ or the wave operator $\square$. The following type of Carleman estimate is known to be true on a certain range of $p,q$: \[ \|e^{v\cdot x}u\|_{L^q(\mathbb{R}^d)} \le C\|e^{v\cdot x}P(D)u\|_{L^p(\mathbb{R}^d)} \] with $C$ independent of $v\in \mathbb{R}^d$. The estimates are consequences of the uniform Sobolev type estimates for second order differential operators due to Kenig-Ruiz-Sogge \cite{KRS} and Jeong-Kwon-Lee \cite{JKL}. The range of $p,q$ for which the uniform Sobolev type estimates hold was completely characterized for the second order differential ope…
In this article mixed norm estimates are obtained for some integral operators, from which those for the Hermite semigroup and the Bochner Riesz means associated with the Hermite expansions are deduced. Also, mixed norm estimates for the Littlewood Paley g functions and g* functions for the Hermite expansions are obtained, which lead to those for Hermite multipliers
We establish Littlewood-Paley decompositions for Muckenhoupt weights in the setting of UMD spaces. As a consequence we obtain two-weight variants of the Mikhlin multiplier theorem for operator-valued multipliers. We also show two-weight estimates for multipliers satisfying Hörmander type conditions.
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。
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 $1<p\le 2$.
We consider bilinear oscillatory integrals, i.e. pseudo-product operators whose symbol involves an oscillating factor. Lebesgue space inequalities are established, which give decay as the oscillation becomes stronger ; this extends the well-known linear theory of oscillatory integral in some directions. The proof relies on a combination of time-frequency analysis of Coifman-Meyer type with stationary and non-stationary phase estimates. As a consequence of this analysis, we obtain Lebesgue estimates for new bilinear multipliers defined by non-smooth symbols.
Recently, Nagel and Stein studied the $\square_b$-heat equation, where $\square_b$ is the Kohn Laplacian on the boundary of a weakly-pseudoconvex domain of finite type in $\C^2$. They showed that the Schwartz kernel of $e^{-t\square_b}$ satisfies good "off-diagonal" estimates, while that of $e^{-t\square_b}-π$ satisfies good "on-diagonal" estimates, where $π$ is the Szegö projection. We offer a simple proof of these results, which easily generalizes to other, similar situations. Our methods involve adapting the well-known relationship between the heat equation and the finite propagation speed …
The main purpose of this paper is to study $L^r$ Hölder type estimates for a bi-parameter trilinear Fourier multiplier with flag singularity, and the analogous pseudo-differential operator, when the symbols are in a certain product form. More precisely, for $f,g,h\in \mathcal{S}(\mathbb{R}^{2})$, the bi-parameter trilinear flag Fourier multiplier operators we consider are defined by $$ T_{m_1,m_2}(f,g,h)(x):=\int_{\mathbb{R}^{6}}m_1(ξ,η,ζ)m_2(η,ζ)\hat f(ξ) \hat g(η)\hat h(ζ)e^{2πi(ξ+η+ζ)\cdot x}dξdηdζ, $$ when $m_1,m_2$ are two bi-parameter symbols. We will show that our problem can be reduced…
Estimates for maximal Fourier multiplier operators on $\Bbb R^2$ via square functions
本文通过平方函数估计，推导出Bochner-Riesz型傅里叶乘子在R²上的最大函数L^p有界性，扩展了Carbery 1983年的结果。
We consider bilinear multipliers that appeared as a distinguished particular case in the classification of two-dimensional bilinear Hilbert transforms by Demeter and Thiele [9]. In this note we investigate their boundedness on Sobolev spaces. Furthermore, we study structurally similar operators with symbols that also depend on the spatial variables. The new results build on the existing L^p estimates for a paraproduct-like operator previously studied by the authors in [5] and [10]. Our primary intention is to emphasize the analogies with Coifman-Meyer multipliers and with bilinear pseudodiffer…
For $ 0< λ< \frac{1}2$, let $ B_{λ}$ be the Bochner-Riesz multiplier of index $ λ$ on the plane. Associated to this multiplier is the critical index $1 < p_λ= \frac{4} {3+2 λ} < \frac{4}3$. We prove a sparse bound for $ B_{λ}$ with indices $ (p_λ, q)$, where $ p_λ' < q < 4$. This is a further quantification of the endpoint weak $L^{p_λ}$ boundedness of $ B_{λ}$, due to Seeger. Indeed, the sparse bound immediately implies new endpoint weighted weak type estimates for weights in $ A_1 \cap RH_{ρ}$, where $ ρ> \frac4 {4 - 3 p_{λ}}$.
In this paper, we study the $L^p$ boundedness of a class of oscillating multiplier operator for the Dunkl transform, $T_{m_α}=\mathcal{F}_k^{-1}(m_α\mathcal{F}_k(f))$ with $m(ξ)=|ξ|^{-α}e^{\pm i|ξ|}ϕ(ξ)$. We obtain an $L^p$-bound result for the corresponding maximal functions. As a specific applications, we give an extension of the $L^p$ estimate for the wave equation and of Stein's theorem for the analytic family of maximal spherical means \cite{Stein}
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.
In this paper, we prove some uniform estimates between Lebesgue and Hardy spaces for operators closely related to the multilinear paraproducts on R^d. We are looking for uniformity with respect to parameters, which allow us to disturb the geometry and the metric on $R^d$.
We present a simple proof of the resolvent estimates of elliptic Fourier multipliers on the Euclidean space, and apply them to the analysis of time-global and spatially-local smoothing estimates of a class of dispersive equations. For this purpose we study in detail the properties of the restriction of Fourier transform on the unit cotangent sphere associated with the symbols of multipliers.
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 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.
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。
In this paper we study some estimates of norms in variable exponent Lebesgue spaces for maximal multiplier operators.We will consider the case when multiplier is the Fourier transform of a compactly supported Borel measure
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。
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 prove $L^p$ estimates for various multi-parameter bi- and trilinear operators with symbols acting on fibers of the two-dimensional functions. In particular, this yields estimates for the general bi-parameter form of the twisted paraproduct studied in arXiv:1011.6140.
Let $\varrho\in C^{\infty} ({\Bbb R}^d\setminus\{0\})$ be a non-radial homogeneous distance function satisfying $\varrho(tξ)=t\varrho(ξ)$. For $f\in\frak S ({\Bbb R}^{d+1})$ and $δ>0$, we consider convolution operator ${\Cal T}^δ$ associated with the smooth cone type multipliers defined by $$\hat {{\Cal T}^δ f}(ξ,τ)= (1-\frac{\varrho(ξ)}{|τ|} )^δ_+\hat f (ξ,τ), (ξ,τ)\in {\Bbb R}^d \times \Bbb R.$$ If the unit sphere $Σ_{\varrho}\fallingdotseq\{ξ\in {\Bbb R}^d : \varrho(ξ)=1\}$ is a convex hypersurface of finite type and $\varrho$ is not radial, then we prove that ${\Cal T}^{δ(p)}$ maps from $H…
In this paper, estimates are proven for convolution kernels associated to multipliers from a reasonably general class of compactly supported two-dimensional functions constructed out of real-analytic functions. These estimates are both for overall decay rate and decay rate in specific directions. The estimates are sharp for a certain range of exponents appearing in the theorems.
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…
We discuss the work of Birman and Solomyak on the singular numbers of integral operators from the point of view of modern approximation theory, in particular with the use of wavelet techniques. We are able to provide a simple proof of norm estimates for integral operators with kernel in $B^{\frac{1}{p}-\frac{1}{2}}_{p,p}(\mathbb R,L_2(\mathbb R))$. This recovers, extends and sheds new light on a theorem of Birman and Solomyak. We also use these techniques to provide a simple proof of Schur multiplier bounds for double operator integrals, with bounded symbol in $B^{\frac{1}{p}-\frac{1}{2}}_{\fr…
Spectral multipliers and wave equation for sub-Laplacians: lower regularity bounds of Euclidean type
本文证明在括号生成条件下，次拉普拉斯算子的谱乘子与波动传播子估计的有效范围不超过欧氏拉普拉斯算子的对应范围，结果适用于任意步数的 Carnot 群和次黎曼流形。
Endpoint estimates for higher order Marcinkiewicz multipliers
本文研究了任意缺项阶的Marcinkiewicz乘子，通过更一般的乘子端点估计方法，证明了其最优端点界，并推广了高阶Hörmander-Mihlin乘子的映射性质及空间$L\log^{τ/2}L$的弱平方函数刻画。
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 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)}…
On some operator-valued Fourier pseudo-multipliers associated to Grushin operators
本文延续先前对Grushin伪乘子稀疏控制和定量加权估计的研究，进一步将分析扩展到与Grushin算子相关的一族算子值Fourier伪乘子的类似估计。
On wavelet polynomials and Weyl multipliers
基于小波正交系统，建立新的界，证明log n是几乎处处收敛的韦伊尔乘子，且为最优序列。
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.
Several estimates for singular integrals, maximal functions and the spherical summation operator are given in the spaces $L^p_{\text{rad}}L^2_{\text{ang}}(\mathbb{R}^n)$, $n\geq 2$.
Non-Degenerate Multilinear Singular Multipliers with Fractional Rank
本研究针对满足m维线性子空间外符号估计的多线性乘子，建立了其作用于ℝᵈ上(n-1)元函数组的Lᵖ估计，重点分析了分数阶秩情形下的相关性质，为多线性奇异乘子的有界性研究提供了新结果。
$L^p$-estimates for FIO-cone multipliers
本文在 $\Bbb R^3$ 上引入并研究 FIO-锥乘子，利用 Guth-Wang-Zhang 方法在 $4/3\le p\le 4$ 范围得到更强的 $L^p$-估计，并据此解决了一类例外光滑解析曲面极大平均的临界 Lebesgue 指数猜想。
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.
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.
Endpoint estimates for multiparameter multipliers of Marcinkiewicz type
本文研究多参数马尔钦维奇乘法子的端点估计，证明其最佳性并扩展了一维结果到任意多参数。
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…
We prove L^p estimates for a large class of multi-linear operators, which includes the multi-linear paraproducts studied by Coifman and Meyer, as well as the bilinear Hilbert transform.
In this paper, we give explicit expressions of differential-difference operators appeared in the hypothesis of the general Fourier multiplier theorem associated to the Heisenberg groups proved by Mauceri and De Micheal for one dimension and C. Lin for higher dimension. We also give a much shorter proof of the above-mentioned theorem. Then we obtain a sharp weighted estimate for Fourier multipliers on the Heisenberg groups.
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。
The Interplay of Shifted Square and Maximal Function Estimates in the Context of Multilinear Fourier Multipliers
本文通过移位正方形和最大函数的结合，改进了多线性傅里叶乘子的有界性定理，并应用于奇异积分在Orlicz空间中的有界性研究。
We present a new criterion for the weighted $L^p-L^q$ boundedness of multiplier operators for Laguerre and Hermite expansions that arise from a Laplace-Stieltjes transform. As a special case, we recover known results on weighted estimates for Laguerre and Hermite fractional integrals with a unified and simpler approach.
Muscalu, Tao, and Thiele prove $L^p$ estimates for the "Biest" operator defined on Schwartz functions by the map \begin{align*} \hspace{5mm} C^{1,1,1}:& (f_1, f_2, f_3) \mapsto \int_{ξ_1 < ξ_2< ξ_3} \left[ \prod_{j=1}^3 \hat{f}_j (ξ_j) e^{2 πi x ξ_j } \right] d \vecξ \end{align*} via a time-frequency argument that produces bounds for all multipliers with non-degenerate trilinear simplex symbols. In this article we prove $L^p$ estimates for a pair of simplex multipliers for which the non-degeneracy condition fails and which are defined on Schwartz functions by the maps \begin{align*} C^{1,1,-2}…
The $L^p$ boundedness theory of convolution operators is \linebreak based on an initial $L^2\to L^2$ estimate derived from the Fourier transform. The corresponding theory of multilinear operators lacks such a simple initial estimate in view of the unavailability of Plancherel's identity in this setting, and up to now it has not been clear what a natural initial estimate might be. In this work we achieve exactly this goal, i.e., obtain an initial $L^2\times\cdots\times L^2\to L^{2/m}$ estimate for general building blocks of $m$-linear multiplier operators. We apply this result to deduce analogo…
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.
Quantitative estimates for the absolute convergence of wavelet-type series
本文研究了小波型dyadic结构函数系统的绝对收敛性，提出新的定量估计，证明了Weyl乘子的最优增长，并展示了去除正交性假设的通用结果。
Timelimited functions and bandlimited functions play a fundamental role in signal and image processing. But by the uncertainty principles, a signal cannot be simultaneously time and bandlimited. A natural assumption is thus that a signal is almost time and almost bandlimited. The aim of this paper is to prove that the set of almost time and almost bandlimited signals is not excluded from the uncertainty principles. The transforms under consideration are integral operators with bounded kernels for which there is a Parseval Theorem. Then we define the wavelet multipliers for this class of operat…
We establish endpoint estimates for a class of oscillating spectral multipliers on Lie groups of Heisenberg type. The analysis follows an earlier argument due to the second and fourth author but requires the detailed analysis of the wave equation on these groups due to Müller and Seeger. We highlight and develop the connection between sharp bounds for oscillating multipliers and the problem of determining the minimal amount of smoothness required for Mihlin-Hörmander multipliers, a problem that was solved for groups of Heisenberg type but remains open for other groups.
We prove various extensions of the Coifman-Rubio de Francia-Semmes multiplier theorem to operator-valued multipliers on Banach function spaces. Our results involve a new boundedness condition on sets of operators which we call $\ell^{r}(\ell^{s})$-boundedness, which implies $\mathcal{R}$-boundedness in many cases. The proofs are based on new Littlewood-Paley-Rubio de Francia-type estimates in Banach function spaces which were recently obtained by the authors.
Estimates for some bilinear wave operators
本文研究双线性傅里叶乘子算子，建立 Seeger–Sogge–Stein 结果的双线性版本，改进 Rodríguez-López–Rule–Staubach 的相应结果，并讨论估计的尖锐性。
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。
Directional square functions
本文提出基于方向 Carleson 嵌入与多参数时频分析的平方函数估计框架，得到锥形及 $N$ 方向矩形乘子的定量界，并改进 $N$ 边形 Fourier 限制的最优对数界与加权估计。
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