Related papers: Commutators of bilinear bi-parameter singular inte…

In this paper we extend the bump conjecture and a particular case of the separated bump conjecture with logarithmic bumps to iterated commutators $T_b^m$. Our results are new even for the first order commutator $T_b^1$. A new bump type…

Let $L=-\Delta +V$ with non-negative potential $V$ satisfying some appropriate reverse H\"older inequality. In this paper, we study the boundedness of the commutators of some singular integrals associated to $L$ such as Riesz transforms and…

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^{2\pi ix\cdot…

Let $1<p<\infty$. We show the boundedness of operator-valued commutators $[\pi_a,M_b]$ on the noncommutative $L_p(L_\infty(\mathbb{R})\otimes \mathcal{M})$ for any von Neumann algebra $\mathcal{M}$, where $\pi_a$ is the $d$-adic martingale…

We introduce multilinear analogues of dyadic paraproduct operators and Haar Multipliers, and study boundedness properties of these operators and their commutators. We also characterize dyadic BMO functions via the boundedness of certain…

In this paper we develop the theory of Fourier multiplier operators $T_{m}:L^{p}(\mathbb{R}^{d};X)\to L^{q}(\mathbb{R}^{d};Y)$, for Banach spaces $X$ and $Y$, $1\leq p\leq q\leq \infty$ and $m:\mathbb{R}^d\to \mathcal{L}(X,Y)$ an…

We study a multilinear singular integral obtained by taking averages of simplex Hilbert transforms. This multilinear form is also closely related to Calder\'on commutators and the twisted paraproduct. We prove $L^p$ bounds in dimensions two…

In this paper, we prove that the weighted BMO space as follows $${\rm BMO}^{p}(\omega)=\Big\{f\in L^{1}_{\rm loc}:\sup_{Q}\|\chi_{Q}\|^{-1}_{L^{p}(\omega)}\big\|(f-f_{Q})\omega^{-1}\chi_{Q}\big\|_{L^{p}(\omega)}<\infty\Big\}$$ is…

Denote by $T$ and $I_{\alpha}$ the bilinear Calder\'{o}n-Zygmund operators and bilinear fractional integrals, respectively. In this paper, it is proved that if $b_{1},b_{2}\in {\rm CMO}$ (the {\rm BMO}-closure of…

We show that a bilinear radial Fourier multiplier operator with symbol $\sigma$ is $L^2(\R^n)\times L^2(\R^n) \to L^1(\R^n)$ bounded, $n\in \mathbb N,$ if the function $\sigma$ satisfies the smoothness condition $\sigma(2^j\cdot)\Phi\in…

Let $\alpha\in (0, 1]$, $\beta\in [0, n)$ and $T_{\Omega,\beta}$ be a singular or fractional integral operator with homogeneous kernel $\Omega$. In this article, a CMO type space ${\rm CMO}_\alpha(\mathbb R^n)$ is introduced and studied. In…

Let $I_{\alpha}$ be the bilinear fractional integral operator, $B_{\alpha}$ be a more singular family of bilinear fractional integral operators and $\vec{b}=(b,b)$. B\'{e}nyi et al. in \cite{B1} showed that if $b\in {\rm CMO}$, the {\rm…

We prove uniform uniform $L^{p}$ bounds for the family of bilinear Hilbert transforms $\mathrm{BHT}_{\beta} [f_1, f_2] (x) := \mathrm{p.v.} \int_{\mathbb{R}} f_1 (x - t) f_2 (x + \beta t) \frac{\mathrm{d} t}{t}$. We show that the operator…

In this paper, the behavior for commutators of a class of bilinear singular integral operator associated with non-smooth kernels on the products of weighted Lebesgue spaces is considered. By some new maximal functions to control the…

We present a new proof of the compactness of bilinear paraproducts with CMO symbols. By drawing an analogy to compact linear operators, we first explore further properties of compact bilinear operators on Banach spaces and present examples.…

We develop a general framework for the analysis of operator-valued multilinear multipliers acting on Banach-valued functions. Our main result is a Coifman-Meyer type theorem for operator-valued multilinear multipliers acting on suitable…

Let $T$ be an operator on Banach space $X$ that is similar to $- T$ via an involution $U$. Then $U$ decomposes the Banach space $X$ as $X = X_1 \oplus X_2$ with respect to which decomposition we have $U = \left(\begin{matrix} I_1 & 0 \\ 0 &…

We prove a compact version of the $T1$ theorem for bi-parameter singular integrals. That is, if a bi-parameter singular integral operator $T$ admits the compact full and partial kernel representations, and satisfies the weak compactness…

In this paper we prove several weighted estimates for bilinear fractional integral operators and their commutators with BMO functions. We also prove maximal function control theorem for these operators, that is, we prove the weighted $L^p$…

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 $\chi_D$ outside the local $L^2$ setting,…