Remarks on Product VMO

Compactness of commutator of Riesz transforms in the two weight setting

We characterize the compactness of commutators in the Bloom setting. Namely, for a suitably non-degenerate Calderón--Zygmund operator $T$, and a pair of weights $ σ, ω\in A_p$, the commutator $ [T, b]$ is compact from $ L ^{p} (σ) \to L ^{p} (ω)$ if and only if $ b \in VMO _{ν}$, where $ ν= (σ/ ω) ^{1/p}$. This extends the work of the first author, Holmes and Wick. The weighted $VMO$ spaces are different from the classical $ VMO$ space. In dimension $ d =1$, compactly supported and smooth functions are dense in $ VMO _{ν}$, but this need not hold in dimensions $ d \geq 2$. Moreover, the commut…

Remarks on the operator-norm convergence of the Trotter product formula

We revise the operator-norm convergence of the Trotter product formula for a pair {A,B} of generators of semigroups on a Banach space. Operator-norm convergence holds true if the dominating operator A generates a holomorphic contraction semigroup and B is a A-infinitesimally small generator of a contraction semigroup, in particular, if B is a bounded operator. Inspired by studies of evolution semigroups it is shown in the present paper that the operator-norm convergence generally fails even for bounded operators B if A is not a holomorphic generator. Moreover, it is shown that operator norm co…

关于外部Lipschitz区域上Riesz变换的若干注记

Some remarks on Riesz transform on exterior Lipschitz domains

本文研究外部Lipschitz区域上Dirichlet椭圆算子的Riesz变换，在CMO或VMO系数条件下证明梯度与算子半幂的Lp范数等价，并给出热半群梯度一致有界性的替代结果。

A remark on bilinear pseudo-differential operators with symbols in the Sjöstrand class

In this short note, we consider bilinear pseudo-differential operators with symbols belonging to the Sjöstrand class. We show that those operators are bounded from the product of the $L^2$-based Sobolev spaces $H^{s_1} \times H^{s_2}$ to $L^r$ for $s_1,s_2 > 0$, $s_1+s_2=n/2$, and $1 \leq r \leq 2$.

Some remarks on the $n$-linear Hilbert transform for $n\geq 4$

We prove that for every integer $n\geq 4$, the $n$-linear operator whose symbol is given by a product of two generic symbols of $n$-linear Hilbert transform type, does not satisfy any $L^p$ estimates similar to those in Hölder inequality. Then, we extend this result to multi-linear operators whose symbols are given by a product of an arbitrary number of generic symbols of $n$-linear Hilbert transform kind. As a consequence, under the same assumption $n\geq 4$,these immediately imply that for any $1< p_1, ..., p_n \leq \infty$ and $0<p<\infty$ with $1/p_1 + ... + 1/p_n = 1/p$, there exist non-d…