Bilinear multipliers on Lorenzt spaces
Francisco Villarroya
本文给出了与双线性希尔伯特变换密切相关的几类双线性乘子算子在Lebesgue或Lorentz空间之间有界的一个充分条件和两个必要条件.
We give one sufficient and two necessary conditions for boundedness between Lebesgue or Lorentz spaces of several classes of bilinear multiplier operators closely connected with the bilinear Hilbert transform.
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We consider the bilinear Fourier multiplier operator with the multiplier written as a linear combination of a fixed bump function. For those operators we prove two transference theorems, one in amalgam spaces and the other in Wiener amalgam spaces.
In this paper, the $L^2 \times L^{\infty} \to L^2$ and $L^2 \times L^2 \to L^1$ boundedness of bilinear Fourier multiplier operators is discussed under weak smoothness conditions on multipliers. As an application, we prove the $L^2 \times BMO \to L^2$ and $L^2 \times L^2 \to H^1$ boundedness of bilinear operators with multipliers of limited smoothness satisfying vanishing conditions.
Transference of bilinear multiplier operators on Lorentz spaces
本文证明了调制不变乘子算子在由实直线和环面定义的群之间的有界性迁移的DeLeeuw型定理。
This work is devoted to studying the boundedness on Lebesgue spaces of bilinear multipliers on $\R$ whose symbol is narrowly supported around a curve (in the frequency plane). We are looking for the optimal decay rate (depending on the width of this support) for exponents satisfying a sub-Hölder scaling. As expected, the geometry of the curve plays an important role, which is described. This has applications for the bilinear Bochner-Riesz problem (in particular, boundedness of multipliers whose symbol is the characteristic function of a set), as well as for the bilinear restriction-extension p…
Paraproducts for bilinear multipliers associated with convex sets
本文研究与凸集相关的双线性乘子，证明奇异副乘积在局部 $L^2$ 范围内的有界性，并分别在指数边界曲线和高阶空隙条件下给出结果。
We use wavelets of tensor product type to obtain the boundedness of bilinear multiplier operators on $\mathbb R^n\times \mathbb R^n$ associated with Hörmander multipliers on $\mathbb R^{2n}$ with minimal smoothness. We focus on the local $L^2$ case and we obtain boundedness under the minimal smoothness assumption of $n/2$ derivatives. We also provide counterexamples to obtain necessary conditions for all sets of indices.
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…
We use the very recent approach developed by Lacey in [23] and extended by Bernicot-Frey-Petermichl in [3], in order to control Bochner-Riesz operators by a sparse bilinear form. In this way, new quantitative weighted estimates, as well as vector-valued inequalities are deduced.
In this paper we investigate the boundedness properties of bilinear multiplier operators associated with unimodular functions of the form $m(ξ,η)=e^{i ϕ(ξ-η)}$. We prove that if $ϕ$ is a $C^1(\mathbb R^n)$ real-valued non-linear function, then for all exponents $p,q,r$ lying outside the local $L^2-$range and satisfying the Hölder's condition $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$, the bilinear multiplier norm $$\|e^{iλϕ(ξ-η)}\|_{\mathcal M_{p,q,r}(\mathbb R^n)}\rightarrow \infty,~ λ\in \mathbb R,~ |λ|\rightarrow \infty.$$ For exponents in the local $L^2-$range, we give examples of unimodular fu…
Results analogous to those proved by Rubio de Francia are obtained for a class of maximal functions formed by dilations of bilinear multiplier operators of limited decay. We focus our attention to $L^2\times L^2\to L^1$ estimates. We discuss two applications: the boundedness of the bilinear maximal Bochner-Riesz operator and of the bilinear spherical maximal operator. For the latter we improve the known results by reducing the dimension restriction from $n\ge 8$ to $n\ge 4$.
We investigate two types of boundedness criteria for bilinear Fourier multiplier operators with symbols with bounded partial derivatives of all (or sufficiently many) orders. Theorems of the first type explicitly prescribe only a certain rate of decay of the symbol itself while theorems of the second type require, in addition, the same rate of decay of all derivatives of the symbol. We show that even though these two types of bilinear multiplier theorems are closely related, there are some fundamental differences between them which arise in limiting cases. Also, since theorems of the latter ty…
A note on bilinear multipliers with convex singularities
本文通过为奇异仿积建立估计，证明了符号为凸曲线上方区域示性函数的双线性乘子在局部 L^2 范围内的有界性，并进一步将其推广到凸多边形曲线情形下超出局部 L^2 范围的有界性。
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。
Weighted Bilinear Multiplier Theorems in Dunkl Setting via Singular Integrals
本文在Dunkl背景下，通过证明Littlewood-Paley型定理和多线性Calderón-Zygmund算子的加权不等式新结果，建立了具有多个Muckenhoupt权的双线性乘子算子的单权与双权不等式。
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。
A locally integrable function $m(ξ,η)$ defined on $\mathbb R^n\times \mathbb R^n$ is said to be a bilinear multiplier on $\mathbb R^n$ of type $(p_1,p_2, p_3)$ if $$ B_m(f,g)(x)=\int_{\mathbb R^n} \int_{\mathbb R^n}\hat f(ξ)\hat g(η)m(ξ,η)e^{2πi(<ξ+η,x>} dξdη$$ defines a bounded bilinear operator from $L^{p_1}(\mathbb R^n)\times L^{p_2}(\mathbb R^n) $ to $L^{p_3}(\mathbb R^n)$. The study of the basic properties of such spaces is investigated and several methods of constructing examples of bilinear multipliers are provided. The special case where $m(ξ,η)= M(ξ-η)$ for a given $M$ defined on $\ma…
We provide sufficient normal curvature conditions on the boundary of a domain $D \subset \BBR^4$ to guarantee unboundedness of the bilinear Fourier multiplier operator $\T_D$ with symbol $χ_D$ outside the local $L^2$ setting, \textit{i.e}. from $L^{p_1} (\BBR^2) \times L^{p_2} (\BBR^2) \to L^{p_3'} (\BBR^2)$ with $\sum \frac{1}{p_j} = 1$ and $p_j <2$ for some $j$. In particular, these curvature conditions are satisfied by any domain $D$ that is locally strictly convex at a single boundary point.
In this paper we introduce Stein's square function associated with bilinear Bochner-Riesz means and investigate its $L^p$ boundedness properties. Further, we discuss several applications of the square function in the context of bilinear multipliers. In particular, we obtain results for maximal function associated with generalised bilinear Bochner-Riesz means. This extends the results proved in~\cite{JS}. Another application concerns the $L^p$ estimates for bilinear fractional Schrödinger multipliers. Finally, we improve upon a result of Grafakos, He and Honzik~\cite{GHH} in the context of bili…
Relations between bilinear multipliers on $ \mathbb R^n, \mathbb{T}^n$ and $\mathbb{Z}^n$
本文证明了de Leeuw关于周期双线性乘子的结果的双线性模拟和一些Jodeit型推广结果。
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。
We prove that bilinear fractional integral operators and similar multipliers are smoothing in the sense that they improve the regularity of functions. We also treat bilinear singular multiplier operators which preserve regularity and obtain several Leibniz-type rules in the contexts of Lebesgue and mixed Lebesgue spaces.
Boundedness of bilinear radial Fourier multipliers
本文研究了双线性径向傅里叶乘法子的有界性条件，并通过光滑性条件证明其在L^2到L^1的有界性，同时给出了多线性Bochner-Riesz算子的另一种证明及广义双线性Bochner-Riesz算子的估计。
In this paper, we study properties of the bilinear multiplier space. We give a necessary condition for a continuous integrable function to be a bilinear multiplier on variable exponent Lebesgue spaces. And we prove the localization theorem of multipliers on variable exponent Lebesgue spaces. Moreover, we present a Mihlin-Hörmander type theorem for multilinear Fourier multipliers on weighted variable Lebesgue spaces and give some applications.
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.
On some bilinear Fourier multipliers with oscillating factors, I
本文研究了带振荡因子的双线性 Fourier 乘子，证明了当符号属于特定 Hörmander 类时相应双线性算子的有界性，改进了已有结果。
Bilinear oscillatory Fourier multipliers
本文研究含振荡因子的双线性 Fourier 乘子，建立其在 Lebesgue 空间间（含端点）的有界性，并讨论所得结果的尖锐性。
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。
On some bilinear Fourier multipliers with oscillating factors, II
本文研究带振荡因子 $e^{i(|ξ|^s+|η|^s+|ξ+η|^s)}$ 且符号属于 Hörmander 类 $S^{m}_{1,0}$ 的双线性 Fourier 乘子，给出保证其在若干端点型 $L^p\times L^q\to L^r$ 空间上有界的 $m$ 的判据。
Given a smooth bump function, we consider the multiplier formed by taking the linear combination of the translations of the bump function and the corresponding bilinear Fourier multiplier operator. Under certain condition on the bump function, we give a complete characterization of the coefficients of the linear combination for which the corresponding bilinear operator defines a bounded operator from $L^2\times L^2$ to $L^2$-based amalgam spaces.
We develop a special multilinear complex interpolation theorem that allows us to prove an optimal version of the bilinear Hörmander multiplier theorem concerning symbols that lie in the Sobolev space $L^r_s(\mathbb R^{2n})$, $2\le r<\infty$, $rs>2n$, uniformly over all annuli. More precisely, given a smoothness index $s$, we find the largest open set of indices $(1/p_1,1/p_2 )$ for which we have boundedness for the associated bilinear multiplier operator from $L^{p_1}(\mathbb R^{ n})\times L^{p_2} (\mathbb R^{ n})$ to $ L^p(\mathbb R^{ n})$ when $1/p=1/p_1+1/p_2$, $1<p_1,p_2<\infty$.
Motivated by the problem of spherical summability of products of Fourier series, we study the boundedness of the bilinear Bochner-Riesz multipliers $(1-|ξ|^2-|η|^2)^δ_+$ and we make some advances in this investigation. We obtain an optimal result concerning the boundedness of these means from $L^2\times L^2 $ into $L^1$ with minimal smoothness, i.e., any $δ>0$, and we obtain estimates for other pairs of spaces for larger values of $δ$. Our study is broad enough to encompass general bilinear multipliers $m(ξ,η)$ radial in $ξ$ and $η$ with minimal smoothness, measured in Sobolev space norms. Our…
We obtain a sharp $L^2\times L^2 \to L^1$ boundedness criterion for a class of bilinear operators associated with a multiplier given by a signed sum of dyadic dilations of a given function, in terms of the $L^q$ integrability of this function; precisely we show that boundedness holds if and only if $q<4$. We discuss applications of this result concerning bilinear rough singular integrals and bilinear dyadic spherical maximal functions. Our second result is an optimal $L^2\times L^2\to L^1$ boundedness criterion for bilinear operators associated with multipliers with $L^\infty$ derivatives. Thi…
Let $n\ge 1$ and $\mathfrak{T}_{m}$ be the bilinear square Fourier multiplier operator associated with a symbol $m$, which is defined by $$ \mathfrak{T}_{m}(f_1,f_2)(x) = \biggl( \int_{0}^\infty\Big|\int_{(\mathbb{R}^n)^2} e^{2πix\cdot (ξ_1 +ξ_2) }m(tξ_1,tξ_2) \hat{f}_{1}(ξ_1)\hat{f}_{2}(ξ_2)dξ_1 dξ_2\Big|^2\frac{dt}{t } \biggr)^{\frac 12}. $$ Let $s$ be an integer with $s\in[n+1,2n]$ and $p_0$ be a number satisfying $2n/s\le p_0\le 2$. Suppose that $ν_{\vecω}=\prod_{i=1}^2ω_i^{p/ p_i}$ and each $ω_i$ is a nonnegative function on $\mathbb{R}^n$. In this paper, we show that $\mathfrak{T}_{m}$ i…
Uniform bounds for bilinear symbols with linear K-quasiconformally embedded singularity
本文研究了具有线性K-准共形嵌入奇异性的双线性符号的统一界，通过非退化奇异性和准共形矩阵推动的乘子，恢复了单维双线性Hilbert变换的统一界，并推导了二维双线性Beurling变换的统一界。
On the bilinear cone multiplier
本文研究双线性锥乘子算子在$R\to\infty$时的逐点几乎处处收敛问题，通过建立对应极大算子的加权$L^2\times L^2\to L^1$估计，证明了满足Hölder关系的大范围$L^{p_1}\times L^{p_2}$函数对的收敛性。
The reappearance of a sometimes called exotic behavior for linear and multilinear pseudodifferential operators is investigated. The phenomenon is shown to be present in a recently introduced class of bilinear pseudodifferential operators which can be seen as more general variable coefficient counterparts of the bilinear Hilbert transform and other singular bilinear multipliers operators. The unboundedness on product of Lebesgue spaces but the boundedness on spaces of smooth functions (which is the exotic behavior referred to) of such operators is obtained. In addition, by introducing a new way…
On plate decompositions of cone multipliers
一个重要的不等式，由于沃尔夫板分解锥乘子是已知的后果，各种问题的调和分析。我们观察到，在沃尔夫不等式的范围内，圆锥形和球形版本，可以通过使用双线性限制的结果。我们也利用这个不等式给出了一些改进的估计平方函数的锥乘子在低维分解。这给出了三维空间中锥乘子算子的一个新的L^4界。
Bilinear and Fractional Leibniz Rules Beyond Euclidean Spaces: Weighted Besov and Triebel--Lizorkin Estimates
本文提出了一种适用于非欧几里得空间的双线性与分数勒贝格法则框架，扩展了经典Kato-Ponce不等式，并在加权空间中建立了新的估计方法。
An alternate approach to bilinear rough singular integrals
本文提出一种新方法，通过输入函数局部傅里叶级数展开得到带衰减的三线性估计，证明核远离对角线时极大截断双线性粗糙奇异积分的最优$L^p$估计，并给出其全维度最优指数范围$L^p$有界性的自包含新证明。
Extrapolation for bilinear compact operators in the variable exponent setting
本文研究了可变指数设定下双线性紧算子的外推问题，推导了交换子的新紧性结果，并统一了之前的相关研究成果。
Endpoint Criteria for One-Dimensional Bilinear Rough Singular Integrals
研究一维双线性粗糙奇异积分的端点定理，通过刻画相关角乘子结构，在临界核假设下建立两个有界性准则，包括\(L\log L\)结果及临界方向指标下的结果，且两类临界核不可比，方法涉及多种数学工具。
In a previous paper, we obtained several "compact versions" of Rubio de Francia's weighted extrapolation theorem, which allowed us to extrapolate the compactness of linear operators from just one space to the full range of weighted Lebesgue spaces, where these operators are bounded. In this paper, we study the extrapolation of compactness for bilinear operators in terms of bilinear Muckenhoupt weights. As applications, we easily recover and improve earlier results on the weighted compactness of commutators of bilinear Calderón-Zygmund operators, bilinear fractional integrals and bilinear Fouri…
Bilinear rough singular integrals under a fractional geometric condition
在分数阶几何条件下，证明了双线性粗糙奇异积分算子及其极大和极大截断形式的Banach范围有界性，该条件弱于L^q和Orlicz空间条件，证明避免小波分解而采用局部傅里叶级数展开。
Restricted weighted weak boundedness for product type operators
本文针对满足 Dini 条件的奇异积分算子乘积型双线性算子，证明了受限加权弱型不等式，并由此解决了一类有界变差双线性 Fourier 乘子的有界性问题。
On the best constants of Schur multipliers of second order divided difference functions
本文基于双线性转移和Hörmander-Mikhlin-Schur乘子定理，给出了二阶差商函数双线性Schur乘子有界性的新证明，显著改进了其渐近界的上下界并证明了其在特定条件下的最优性。
We establish a uniform domination of the family of trilinear multiplier forms with singularity over a one-dimensional subspace by positive sparse forms involving $L^p$-averages. This class includes the adjoint forms to the bilinear Hilbert transforms. Our result strengthens the $L^p$-boundedness proved in \cite{MTT} and entails as a corollary a rich multilinear weighted theory. In particular, we obtain $L^{q_1}(v_1) \times L^{q_2}(v_2)$-boundedness of the bilinear Hilbert transform when the weights $v_j$ belong to the class $A_{\frac{q+1}{2}}\cap RH_2$. Our proof relies on a stopping time cons…
We extend the theory of weighted norm inequalities on variable Lebesgue spaces to the case of bilinear operators. We introduce a bilinear version of the variable $\A_\pp$ condition, and show that it is necessary and sufficient for the bilinear maximal operator to satisfy a weighted norm inequality. Our work generalizes the linear results of the first author, Fiorenza and Neugebauer \cite{dcu-f-nPreprint2010} in the variable Lebesgue spaces and the bilinear results of Lerner {\em et al.} \cite{MR2483720} in the classical Lebesgue spaces. As an application we prove weighted norm inequalities for…
We extend and improve the known results about the boundedness of the bilinear pseudo-differential operators with symbols in the bilinear Hörmander class $BS^{m}_{0,0}(\mathbb{R}^n)$. We consider wider classes of symbols and improve estimates for the corresponding operators. A key idea is to consider the operators in Wiener amalgam spaces.
Motivated by the study of kernels of bilinear pseudodifferential operators with symbols in a Hörmander class of critical order, we investigate boundedness properties of strongly singular Calderón--Zygmund operators in the bilinear setting. For such operators, whose kernels satisfy integral-type conditions, we establish boundedness properties in the setting of Lebesgue spaces as well as endpoint mappings involving the space of functions of bounded mean oscillations and the Hardy space. Assuming pointwise-type conditions on the kernels, we show that strongly singular bilinear Calderón--Zygmund o…
In this paper we show that the solution of the supercrti- cal surface quasi-geostrophic (SQG) equation, starting from initial data in homogeneous critical Besov spaces belong to a subanalytic Gevrey class. In particular, we improve upon the result of Dong and Li in [26], where they showed that the solutions of Chen-Miao-Zhang (cf. [11]) are classical solutions. We extend the approach of Biswas (cf. [7]) to critical, L^p -based Besov spaces, and adapt the point of view of Lemarie- Rieusset (cf. [36]), who treated the operator arising from applying the analytic Gevrey operator to a product of an…
We consider the Kato-Ponce type inequality for bilinear pseudo-differential operators with $S_{0,0}$-type symbols in the scale of Besov spaces. In particular, the borderline whether the boundedness of those operators holds or not is discussed.
This paper mainly dedicates to prove a plethora of weighted estimates on Morrey spaces for bilinear fractional integral operators and their general commutators with BMO functions of the form $$B_α(f,g)(x)=\int_{\mathbb{R}^{n}}\frac{f(x-y)g(x+y)}{|y|^{n-α}}dy,\qquad 0<α<n.$$ We also prove some maximal function control theorems for these operators, that is, the weighted Morrey norm is bounded by the weighted Morrey norm of a natural maximal operator when the weight belongs to $A_{\infty}$. As a corollary, some new weighted estimates for the bilinear maximal function associated to the bilinear Hi…
Boundedness of bilinear pseudo-differential operators with $BS^m_{0,0}$ symbols on Sobolev spaces
本文研究双线性Hörmander类$BS^m_{0,0}$象征对应的伪微分算子，借助象征演算和$S_{0,0}$型算子的Lebesgue空间有界性，证明其在Sobolev空间上的有界性。
Commutators of bilinear Calderón-Zygmund operators and multiplication by functions in a certain subspace of the space of functions of bounded mean oscillations are shown to be compact on appropriate products of weighted Lebesgue spaces.
In this paper, we are interested in the following bilinear fractional integral operator $B\mathcal{I}_α$ defined by \[ B\mathcal{I}_α({f,g})(x)=\int_{% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{n}}\frac{f(x-y)g(x+y)}{|y|^{n-α}}dy, \] with $0< α<n$. We prove the weighted boundedness of $B\mathcal{I}_α$ on the Morrey type spaces. Moreover, an Olsen type inequality for $B\mathcal{I}_α$ is also given.
In this article we extend recent results by the first author on the necessity of $BMO$ for the boundedness of commutators on the classical Lebesgue spaces. We generalize these results to a large class of Banach function spaces. We show that with modest assumptions on the underlying spaces and on the operator $T$, if the commutator $[b,T]$ is bounded, then the function $b$ is in $BMO$.
Weak type estimates for spherical multipliers on noncompact symmetric spaces
本文证明了非紧型对称空间上球面Fourier乘子的强弱1型估计。这补充了J. - Ph. Anker和A.D. Ionescu。
By a new method derived from Nicola--Primo--Tabacco[24], we study the boundedness on $α$-modulation spaces of unimodular multipliers with symbol $e^{iμ(ξ)}$. Comparing with the previous results, the boundedness result is established for a larger family of unimodular multipliers under weaker assumptions.
We prove that the class of trilinear multiplier forms with singularity over a one dimensional subspace, including the bilinear Hilbert transform, admit bounded $L^p$-extension to triples of intermediate $\mathrm{UMD}$ spaces. No other assumption, for instance of Rademacher maximal function type, is made on the triple of $\mathrm{UMD}$ spaces. Among the novelties in our analysis is an extension of the phase-space projection technique to the $\mathrm{UMD}$-valued setting. This is then employed to obtain appropriate single tree estimates by appealing to the $\mathrm{UMD}$-valued bound for bilinea…
Local estimates and global continuities in Lebesgue spaces for bilinear operators
本文首先证明了具有截断符号的双线性算子（与双线性希尔伯特变换及类似的奇异算子密切相关）的一些局部估计。根据海森堡不确定性原理，这种估计对应于“非对角”衰变的描述。此外，它们还允许我们证明具有空间依赖符号的双线性算子在Lebesgue空间中的整体连续性。
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