Local estimates and global continuities in Lebesgue spaces for bilinear operators

The bilinear Hilbert transform in UMD spaces

We prove $L^p$-bounds for the bilinear Hilbert transform acting on functions valued in intermediate UMD spaces. Such bounds were previously unknown for UMD spaces that are not Banach lattices. Our proof relies on bounds on embeddings from Bochner spaces $L^p(\mathbb{R};X)$ into outer Lebesgue spaces on the time-frequency-scale space $\mathbb{R}^3_+$.

Lp estimates for non smooth bilinear Littlewood-Paley square functions on R

In this work, some non smooth bilinear analogues of linear Littlewood-Paley square functions on the real line are studied. These bilinear operators are closely related to the bilinear Hilbert transforms and vector valued version of these ones. Mainly we prove boundedness-properties in Lebesgue spaces for them.

Extrapolation of compactness on weighted spaces: Bilinear operators

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…

The Marcinkiewicz multiplier condition for bilinear operators

This article is concerned with the question of whether Marcinkiewicz multipliers on $\mathbb R^{2n}$ give rise to bilinear multipliers on $\mathbb R^n\times \mathbb R^n$. We show that this is not always the case. Moreover we find necessary and sufficient conditions for such bilinear multipliers to be bounded. These conditions in particular imply that a slight logarithmic modification of the Marcinkiewicz condition gives multipliers for which the corresponding bilinear operators are bounded on products of Lebesgue and Hardy spaces.

加权Riesz-Kolmogorov准则与变指数Lebesgue空间上紧性的多线性外推

Compact sets in quasi-Banach function spaces and multilinear extrapolation of compactness on $L^{p(\cdot)}(w)$

本文通过建立加权Riesz-Kolmogorov定理，在变指数Lebesgue空间上实现了多线性紧算子的外推，并得到了交换子、分数次积分和Fourier乘子的新加权紧性估计。

Boundedness and Compactness of commutator of Hardy-Littlewood maximal operator

We study the mapping property of the commutator of bilinear Hardy-Littlewood maximal operator in homogeneous Triebel-Lizorkin space. We also show that the commutator of bilinear Hardy-Littlewood maximal operator is a compact operator acting on product of Lebesgue spaces. The results are new even in the linear case.

Notes on bilinear multipliers on Orlicz spaces

Let $Φ_1 , Φ_2 $ and $ Φ_3$ be Young functions and let $L^{Φ_1}(\mathbb{R})$, $L^{Φ_2}(\mathbb{R})$ and $L^{Φ_3}(\mathbb{R})$ be the corresponding Orlicz spaces. We say that a function $m(ξ,η)$ defined on $\mathbb{R}\times \mathbb{R}$ is a bilinear multiplier of type $(Φ_1,Φ_2,Φ_3)$ if \[ B_m(f,g)(x)=\int_\mathbb{R} \int_\mathbb{R} \hat{f}(ξ) \hat{g}(η)m(ξ,η)e^{2πi (ξ+η) x}dξdη\] defines a bounded bilinear operator from $L^{Φ_1}(\mathbb{R}) \times L^{Φ_2}(\mathbb{R})$ to $L^{Φ_3}(\mathbb{R})$. We denote by $BM_{(Φ_1,Φ_2,Φ_3)}(\mathbb{R})$ the space of all bilinear multipliers of type $(Φ_1,Φ_2…

The multilinear strong maximal function

A multivariable version of the strong maximal function is introduced and a sharp distributional estimate for this operator in the spirit of the Jessen, Marcinkiewicz, and Zygmund theorem is obtained. Conditions that characterize the boundedness of this multivariable operator on products of weighted Lebesgue spaces equipped with multiple weights are obtained. Results for other multi(sub)linear maximal functions associated with bases of open sets are studied too. Certain bilinear interpolation results between distributional estimates, such as that obtained for the multivariable strong maximal fu…

一般拟巴拿赫函数空间中的外推法

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。

稀疏控制蕴含向量值稀疏控制

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。

The weak-type Carleson theorem via wave packet estimates

We prove that the weak-$L^{p}$ norms, and in fact the sparse $(p,1)$-norms, of the Carleson maximal partial Fourier sum operator are $\lesssim (p-1)^{-1}$ as $p\to 1^+$. This is an improvement on the Carleson-Hunt theorem, where the same upper bound on the growth order is obtained for the restricted weak-$L^p$ type norm, and which was the strongest quantitative bound prior to our result. Furthermore, our sparse $(p,1)$-norms bound imply new and stronger results at the endpoint $p=1$. In particular, we obtain that the Fourier series of functions from the weighted Arias de Reyna space $ QA_{\inf…

Interpolation for analytic families of multilinear operators on metric measure spaces

Let (X j , d j , $μ$ j), j = 0, 1,. .. , m be metric measure spaces. Given 0 < p $κ$ $\le$ $\infty$ for $κ$ = 1,. .. , m and an analytic family of multilinear operators T z : L p 1 (X 1) x $\bullet$ $\bullet$ $\bullet$ L p m (X m) $\rightarrow$ L 1 loc (X 0), for z in the complex unit strip, we prove a theorem in the spirit of Stein's complex interpolation for analytic families. Analyticity and our admissibility condition are defined in the weak (integral) sense and relax the pointwise definitions given in [9]. Continuous functions with compact support are natural dense subspaces of Lebesgue s…