Michael T Lacey
Rubio de Francia's Littlewood Paley inequality is an extension of the classical Littlewood Paley inequality to one that holds for a decomposition of frequency space into arbitrary disjoint intervals. We survey this inequality, its higher dimensional analogs, and the implications for multipliers.
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The one-sided Littlewood--Paley inequality for arbitrary intervals was proved by Rubio de Francia. Later, N. Osipov proved its analogue for the system of Walsh functions. In this paper, this inequality is proved for more general Vilenkin systems.
Littlewood--Paley--Rubio de Francia inequality for unbounded Vilenkin systems
本文将 Rubio de Francia 的单侧 Littlewood--Paley 不等式推广到任意 Vilenkin 系统，且不要求循环群的阶数满足任何条件。
Let $Ω$ be a collection of disjoint dyadic squares $ω$, let $π_ω$ denote the non-smooth bilinear projection onto $ω$ \[ π_ω(f,g)(x):=\int\int \mathbf{1}_ω(ξ,η) \widehat{f}(ξ) \widehat{g}(η) e^{2πi (ξ+ η) x} d ξdη\] and let $r>2$. We show that the bilinear Rubio de Francia operator \[ \Big(\sum_{ω\inΩ} |π_ω (f,g)|^r \Big)^{1/r} \] is $L^p \times L^q \rightarrow L^s$ bounded with constant independent of $Ω$ whenever $1/p + 1/q = 1/s$, $r'<p,q<r$, $r'/2 < s < r/2$.
We consider the Rubio de Francia's Littlewood--Paley square function associated with an arbitrary family of intervals in $\mathbb{R}$ with finite overlapping. Quantitative weighted estimates are obtained for this operator. The linear dependence on the characteristic of the weight $[w]_{A_{p/2}}$ turns out to be sharp for $3\le p<\infty$, whereas the sharpness in the range $2<p<3$ remains as an open question. Weighted weak-type estimates in the endpoint $p=2$ are also provided. The results arise as a consequence of a sparse domination shown for these operators, obtained by suitably adapting the…
On Rubio de Francia's maximal theorem
本文补充了鲁比奥-德-弗兰西亚最大定理的研究，探讨了最大函数的受限弱型端点界和$L^p$--$L^q$界，以及弗罗斯曼条件对最大界的影响。
Let $\mathscr{R}$ be a collection of disjoint dyadic rectangles $R$ with sides parallel to the axes, let $π_R$ denote the non-smooth bilinear projection onto $R$ \[ π_R (f,g)(x):=\iint \mathbf{1}_{R}(ξ,η) \widehat{f}(ξ) \widehat{g}(η) e^{2πi (ξ+ η) x} dξdη\] and let $r>2$. We show that the bilinear Rubio de Francia operator associated to $\mathscr{R}$ given by \[ f,g \mapsto \Big(\sum_{R\in\mathscr{R}} |π_{R} (f,g)|^r \Big)^{1/r} \] is $L^p \times L^q \rightarrow L^s$ bounded whenever $1/p + 1/q = 1/s$, $r'<p,q<r$. This extends from squares to rectangles a previous result by the same authors, …
Rubio de Francia proved the one-sided Littlewood--Paley inequality for arbitrary intervals in $L^p$, $2 \le p < \infty$. In this article, his methods are developed and employed to prove an analogue of such an inequality "beyond the index $p=\infty$", i.e., for spaces of Hölder functions and BMO.
We prove the boundedness of a smooth bilinear Rubio de Francia operator associated with an arbitrary collection of squares (with sides parallel to the axes) in the frequency plane\[\left(f, g \right)\mapsto \left( \sum\_{ω\in Ω}\left| \int\_{\mathbb{R}^2} \hat{f}(ξ) \hat{g}(η) Φ\_ω(ξ, η) e^{2 πi x\left(ξ+η\right)} d ξd η\right|^r \right)^{1/r},\] provided $r\textgreater{}2$. More exactly, we show that the above operator maps $L^p \times L^q \to L^s$ whenever $p, q, s'$ are in the "local $L^{r'}$" range, i.e. $\displaystyle \frac{1}{p}+\frac{1}{q}+\frac{1}{s'}=1$, $\displaystyle0 \leq \frac{1}{…
Rubio de Francia proved the one-sided Littlewood--Paley inequality for arbitrary intervals in $L^p$, $2 \le p < \infty$. In this article, such an inequality is proved for the Walsh system.
J. L. Rubio de Francia proved the one-sided Littlewood--Paley inequality for arbitrary intervals in $L^p$, $2\le p<\infty$ and later N. N. Osipov proved the similar inequality for Walsh functions. In this paper we investigate some properties of Banach spaces $X$ such that the latter inequality holds for $X$-valued functions.
Matrix weights, singular integrals, Jones factorization and Rubio de Francia extrapolation
本文概述了奇异积分矩阵加权范数不等式中寻找精确常数的矩阵A2猜想，回顾了标量情形历史与定性结果，并介绍了作者与Bownik发展的凸集值函数调和分析理论，证明了Jones因子分解与Rubio de Francia外推定理，最后讨论了矩阵A2猜想被证伪的问题。
We obtain one variant of the extrapolation theorem of Rubio de Fracia for variable exponent Lebesgue spaces. As a consequence we obtain conditions guarantee boundedness of strongly singular integral operators, singular integral operators with rough kernels, fractional maximal operators related to spherical means, Bochner-Riesz operators in variable Lebesgue spaces.
Rubio de Francia Extrapolation Theorem for Quasi non-increasing Sequences
本文证明了准非递增序列的离散Rubio de Francia插值定理，并给出了广义离散Hardy平均算子有界性的权重刻画。
In $R^d$, define a maximal function in the directions $v\in \directions\subset\{x \mid \abs x=1\}$ by $$ M^\directions f(x)=\sup_{v\in\directions} \sup_{\zve} \int_{-\ze}^\ze \abs{f(x-vy)} dy. $$ For a function $f$ on $\ZR^d$, let $S_\zw f$ denote the Fourier restriction of $f$ to a region $\zw$. We are especially interested taking \zw to be a sector of $R^d$ with base points at the origin. A sector is a product of the interval $(0,\infty)$ with respect to a choice of (non orthogonal) basis. What is most important is that the basis is a subset of $\directions$. Consider a collection $\zW$ of p…
A note on Rubio de Francia's extrapolation in tent spaces and applications
本文推广 Rubio de Francia 外插定理至帐篷空间，从而自动获得 Calderón-Zygmund 算子、Kato 猜想相关算子及分数阶算子的新加权估计。
Let $T$ be an arbitrary operator bounded from $L^{p_0}(w)$ into $L^{p_0, \infty}(w)$ for every weight $w$ in the Muckenhoupt class $A_{p_0}$. It is proved in this article that the distribution function of $Tf$ with respect to any weight $u$ can be essentially majorized by the distribution function of $Mf$ with respect to $u$ (plus an integral term easy to control). As a consequence, well-known extrapolation results, including results in a multilinear setting, can be obtained with very simple proofs. New applications in extrapolation for two-weight problems and estimates on rearrangement invari…
Rubio de Francia's Extrapolation Theorem and Sparse Bounds
本文针对次可加算子证明了加权有界性蕴含稀疏形式界的部分逆命题，扩展了Rubio de Francia外推定理并得到粗糙奇异积分算子的新稀疏界。
Pointwise localization and sharp weighted bounds for Rubio de Francia square functions
本文证明 Rubio de Francia 平方函数可由稀疏算子逐点控制，从而得到尖锐的加权范数不等式，并部分验证了关于 $A_1$ 权的猜想。
We extend Rubio de Francia's extrapolation theorem for functions valued in UMD Banach function spaces, leading to short proofs of some new and known results. In particular we prove Littlewood-Paley-Rubio de Francia-type estimates and boundedness of variational Carleson operators for Banach function spaces with UMD concavifications.
The purpose of this note is to correct an error in a paper of M. Cowling, G. Fendler and J.J.F. Fournier, and to give a counterexample to a conjecture of J.-L. Rubio de Francia.
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.
Extrapolation and Factorization of matrix weights
针对矩阵 $\mathcal A_p$ 权，本文通过建立凸集值函数、可测半范数函数及相应极大算子与迭代算法，证明了 Jones 分解和 Rubio de Francia 外推定理。
We extend the theory of Rubio de Francia extrapolation, including off-diagonal, limited range, and $A_{\infty}$ extrapolation, to the weighted variable Lebesgue spaces. As a consequence we are able to show that a number of different operators from harmonic analysis are bounded on these spaces. The proofs of our extrapolation results are developed in a way that outlines a general approach to proving extrapolation theorems on other Banach function spaces.
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 $n\ge 4$.
Let $Ω\in L^1{({\mathbb S^{n-1}})}$, be a function of homogeneous of degree zero, and $M_Ω$ be the Hardy-Littlewood maximal operator associated with $Ω$ defined by $M_Ω(f)(x) = \sup_{r>0}\frac1{r^n}\int_{|x-y|<r}|Ω(x-y)f(y)|dy.$ It was shown by Christ and Rubio de Francia that $\|M_Ω(f)\|_{L^{1,\infty}({\mathbb R^n})} \le C(\|Ω\|_{L\log L({\mathbb S^{n-1}})}+1)\|f\|_{L^1({\mathbb R^n})}$ provided $Ω\in L\log L {({\mathbb S^{n-1}})}$. In this paper, we show that, if $Ω\in L\log L({\mathbb S^{n-1}})$, then for all $f\in L^1({\mathbb R^n})$, $M_Ω$ enjoys the limiting weak-type behaviors that $$\l…
Parabolic Extrapolation and Its Applications to Characterizing Parabolic BMO Spaces via Parabolic Fractional Commutators
本文建立抛物版Rubio de Francia外推定理，利用带时间滞后的抛物分数次算子交换子，首次在高维情形下刻画抛物BMO型空间。
An Extrapolation Theorem in the setting of Hausdorff capacities
该研究在Hausdorff容量框架下，借助经典外推技术与最新的容量型Muckenhoupt权理论，证明了Rubio de Francia外推定理的类似结果，拓展了相关理论的适用范围。
We generalize the extrapolation theory of Rubio de Francia to the context of Banach function spaces and modular spaces. Our results are formulated in terms of some natural weighted estimates for the Hardy-Littlewood maximal function and are stated in measure spaces and for general Muckenhoupt bases. Finally, we give several applications in analysis and partial differential equations.
Off-diagonal matrix extrapolation for Muckenhoupt bases
本文将矩阵权Rubio de Francia外推理论扩展至非对角情形，推广矩阵加权外推理论到一般基的矩阵$\mathcal{A}_p$类（需Christ-Goldberg极大算子有界），并证明所有多参数基满足该性质。
Extrapolation via Sawyer-type inequalities
本研究提出不依赖Rubio de Francia迭代算法的多变量受限弱型外推理论，依托加权Hardy-Littlewood极大算子的Sawyer型不等式实现，可恢复含端点结果的弱型外推格式，还拓展了特征函数弱型等价范围、填补了多变量算子权估计的文献空白。
In this paper we show weighted estimates in mixed norm spaces for the Riesz transform associated with the harmonic oscillator in spherical coordinates. In order to prove the result we need a weighted inequality for a vector-valued extension of the Riesz transform related to the Laguerre expansions which is of independent interest. The main tools to obtain such extension are a weighted inequality for the Riesz transform independent of the order of the involved Laguerre functions and an appropriate adaptation of Rubio de Francia's extrapolation theorem.
In this paper we prove mixed norm estimates for Riesz transforms related to Laplace--Beltrami operators on compact Riemannian symmetric spaces of rank one. These operators are closely related to the Riesz transforms for Jacobi polynomials expansions. The key point is to obtain sharp estimates for the kernel of the Jacobi--Riesz transforms with uniform control on the parameters, together with an adaptation of Rubio de Francia's extrapolation theorem. The latter results are of independent interest.
Properties and applications of partial multiple weights for fractional integrals
本文引入部分多重权，在部分Muckenhoupt类中建立Rubio de Francia外推定理及分数积分加权估计，推导多个基本不等式，并刻画交换子，部分解答了Cruz-Uribe的开放问题。
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