Denis Potapov, Fedor Sukochev, Quanhua Xu
The paper studies Banach spaces satisfying the Littlewood-Paley-Rubio de Francia property LPR_p, 2 \leq p < \infty. The paper shows that every Banach lattice whose 2-concavification is a UMD Banach lattice has this property. The paper also shows that every space having LPR_q also has LPR_p with q \leq p < \infty.
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Discrete Rubio de Francia extrapolation theorem via factorization of weights and iterated algorithms
该研究利用离散 Muckenhoupt 权分解、迭代 Rubio de Francia 算法及对偶方法，建立了离散形式的 Rubio de Francia 外推定理。
We discuss generalizations of Rubio de Francia's inequality for Triebel--Lizorkin and Besov spaces, continuing the research from [5]. Two versions of Rubio de Francia's operator are discussed: it is shown that a rotation factor is needed for the boundedness of the operator in some smooth spaces while it is not essential in other spaces. We study the operators on some "end" spaces of the Triebel--Lizorkin scale and then use usual interpolation methods.
We prove Rubio de Francia extrapolation results in Lebesgue and grand Lebesgue spaces for quasi monotone functions with $QB_{β,p}$ weights. The extrapolation in Lebesgue spaces with the weight class $QB_{β,\infty}$ has also been investigated. As an application, we characterize the boundedness of the Hardy averaging operator for quasi monotone functions in the grand Lebesgue spaces.
Let $X$ be a Banach space. It is proved that an analogue of the Rubio de Francia square function estimate for partial sums of the Fourier series of $X$-valued functions holds true for all disjoint collections of subintervals of the set of integers of equal length and for all exponents $p$ greater or equal than 2 if and only if the space $X$ is a UMD space of type 2. The same criterion is obtained for the case of subintervals of the real line and Fourier integrals instead of Fourier series.
We prove the following correction theorem: every function $f$ on the circumference $\mathbb{T}$ that is bounded by the $α_1$-weight $w$ (this means that $Mw^2 \leq C w^2$) can be modified on a set $e$ with $\int\limits_{e} w \leq \eps$ so that its quadratic function built up from arbitary sequence of nonintersecting intervals in $\mathbb{Z}$ will not exceed $C \log \frac{1}{\eps} w$.
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.
A version of Littlewood-Paley-Rubio de Francia inequality for the two-parameter Walsh system is proved: for any family of disjoint rectangles $I_k = I_k^1 \times I_k^2$ in ${\mathbb{Z}_+ \times \mathbb{Z}_+}$ and a family of functions $f_k$ with Walsh spectrum inside $I_k$ the following is true \[\left\lVert\sum\limits_k f_k\right\rVert_{L^p} \leq C_p \left\lVert\left(\sum\limits_k \left|f_k\right|^2\right)^{1/2}\right\rVert_{L^p}, \qquad 1 < p \leq 2,\] where $C_p$ does not depend on the choice of rectangles $\left\{I_k\right\}$ or functions $\left\{f_k\right\}$. The arguments are based on th…
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.
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。
Vector-valued Littlewood--Paley--Rubio de Francia Inequalities on Vilenkin Systems
本文研究Vilenkin系统上等长Littlewood--Paley--Rubio de Francia不等式的两种表述，发现典范表述在Schatten类上失效，而Vilenkin表述等价于UMD加type 2性质。
Littlewood-Paley-Rubio de Francia inequality for multi-parameter Vilenkin systems
本文证明了有界多参数Vilenkin系统下的Littlewood-Paley-Rubio de Francia不等式，基于多参数鞅Hardy空间原子理论论证，得到多参数Gundy定理及若干推广推论。
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…
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.
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.
The Maurey-Rosenthal theorem states that each bounded and linear operator T from a quasi normed space E into some L_p(ν) which satisfies a certain vector-valued inequality even allows a weighted norm inequality. Continuing the work of Garcia Cuerva and Rubio de Francia we give several scalar and vector-valued variants of this fundamental result within the framework of quasi K"othe function spaces over measure spaces.
In this paper, we generalize the $A_\fz$ extrapolation theorem in \cite{cmp} and the $A_p$ extrapolation theorem of Rubio de Francia to Schrödinger settings. In addition, we also establish the weighted vector-valued inequalities for Schrödinger type maximal operators by using weights belonging to $ A_p^{ρ,\tz}$ which includes $A_p$. As their applications, we establish the weighted vector-valued inequalities for some Schödinger type operators and pseudo-differential operators.
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。
Operator-valued Fourier multipliers of bounded s-variation
本文研究了具有有限s-变差的算子值傅里叶乘子在加权空间中的有界性，通过加权向量值变分Carleson不等式和Littlewood-Paley-Rubio de Francia类型的估计，建立了相关定理。
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。
We prove the $\ell^s$-boundedness of a family of integral operators with an operator-valued kernel on UMD Banach function spaces. This generalizes and simplifies earlier work by Gallarati, Veraar and the author, where the $\ell^s$-boundedness of this family of integral operators was shown on Lebesgue spaces. The proof is based on a characterization of $\ell^s$-boundedness as weighted boundedness by Rubio de Francia.
We prove a vector-valued version of Carleson's theorem: Let Y=[X,H]_t be a complex interpolation space between a UMD space X and a Hilbert space H. For p\in(1,\infty) and f\in L^p(T;Y), the partial sums of the Fourier series of f converge to f pointwise almost everywhere. Apparently, all known examples of UMD spaces are of this intermediate form Y=[X,H]_t. In particular, we answer affirmatively a question of Rubio de Francia on the pointwise convergence of Fourier series of Schatten class valued functions.
This paper is devoted to studying the extrapolation theory of Rubio de Francia on general function spaces. We present endpoint extrapolation results including $A_1$, $A_p$, and $A_\infty$ extrapolation in the context of Banach function spaces, and also on modular spaces. We also include several applications that can be easily obtained using extrapolation: local decay estimates for various operators, Coifman--Fefferman inequalities that can be used to show some known sharp $A_1$ inequalities, Muckenhoupt--Wheeden and Sawyer's conjectures are also presented for many operators, which go beyond Ca…
Fractional type operators on Hardy spaces associated with ball quasi-Banach function spaces
本文研究了由m-正交矩阵生成的分数型算子T_{α,m}，证明了在特定条件下，T_{α,m}可扩展为有界算子，应用于多种函数空间，其中某些是新结果。
Extrapolation in general quasi-Banach function spaces
该研究在一般拟Banach函数空间中建立多类Rubio de Francia外推定理，证明Hardy-Littlewood极大算子推广的映射性质与相关空间有界性，推广已有结果并给出算子应用。
Sparse domination implies vector-valued sparse domination
本文证明多线性算子的标量值稀疏控制蕴含向量值稀疏控制，引入UMD条件的多线性类比，从而直接获得精确的向量值加权界，并应用于改进Calderón-Zygmund算子与双线性Hilbert变换的界。
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。
Product Hardy spaces meet ball quasi-Banach function spaces
本文通过 Littlewood-Paley-Stein 理论引入了与球拟 Banach 函数空间关联的乘积 Hardy 空间，建立了基于离散 Calderón 恒等式的分解定理，并利用 Rubio de Francia 外推定理证明了双参数奇异积分算子的有界性。
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