Searcharxiv⌕ Search

arXiv subjects

Xudong Lai

Publications and source records attributed to Xudong Lai.

23 records · Page 2Linked to original sources
arXiv2017

Bilinear endpoint estimates for Calderón commutator with rough kernel

In this paper, we establish some bilinear endpoint estimates of Calderón commutator $\mathcal{C}[\nabla A,f](x)$ with a homogeneous kernel when $Ω\in L\log^+L(\mathbf{S}^{d-1})$. More precisely, we prove that $\mathcal{C}[\nabla A,f]$ maps $L^q(\mathbb{R}^d)\times L^1(\mathbb{R}^d)$ to $L^{r,\infty}(\mathbb{R}^d)$ if $q>d$ which improves previous result essentially. If $q=d$, we show that Calderón commutator maps $L^{d,1}(\mathbb{R}^d)\times L^1(\mathbb{R}^d)$ to $L^{r,\infty}(\mathbb{R}^d)$ which is new even if the kernel is smooth. The novelty in the paper is that we prove a new endpoint estimate of the Mary Weiss maximal function which may have its own interest in the theory of singular integral.

math.CA↗
arXiv2017

Weak type (1,1) bound criterion for singular integral with rough kernel and its applications

In this paper, a weak type (1,1) bound criterion is established for singular integral operator with rough kernel. As some applications of this criterion, we prove some important operators with rough kernel in harmonic analysis, such as Calderón commutator, higher order Calderón commutator, general Calderón commutator, Calderón commutator of Bajsanski-Coifman type and general singular integral of Muckenhoupt type, are all of weak type (1,1).

math.CA↗
arXiv2016

$L^1$-Dini conditions and limiting behavior of weak type estimates for singular integrals

In 2006, Janakiraman [10] showed that if $Ω$ with mean value zero on $S^{n-1}$ satisfies the condition \[ \sup_{|ξ|=1}\int_{S^{n-1}}|Ω(θ)-Ω(θ+δξ)|dσ(θ)\leq Cnδ\int_{S^{n-1}}|Ω(θ)|dσ(θ),\quad 0<δ<\frac{1}{n},\ (\ast) \] then for the singular integral operator $T_Ω$ with homogeneous kernel, the following limiting behavior holds: \[\lim\limits_{λ\rightarrow 0}λm(\{x\in\mathbb{R}^n:|T_Ωf(x)|>λ\})= \frac{1}{n}\|Ω\|_{1}\|f\|_{1},\quad \text{for}\ f\in L^1(\mathbb{R}^n)\ \text{with}\ f\geq 0.\ (\ast\ast)\] In the present paper, we prove that if replacing the condition $(\ast)$ by more general condition, the $L^1$-Dini condition, then the limiting behavior $(\ast\ast)$ still holds for the singular integral $T_Ω$. In particular, we give an example which satisfies the $L^1$-Dini condition, but does not satisfy $(\ast)$. Hence, we improve essentially the above result given in [10]. To prove our conclusion, we show that the $L^1$-Dini conditions defined respectively via the rotation and translation on $\mathbb{R}^n$ are equivalent (see Theorem 2.5 below), which has its own interest in the theory of singular integrals. Moreover, similar limiting behavior for the fractional integral operator $T_{Ω,α}$ with homogeneous kernel is also established in this paper.

math.AP↗
arXiv2015

Weighted bound for commutators

Let $K$ be the Calderón-Zygmund convolution kernel on $\mathbb{R}^d (d\geq2)$. Define the commutator associated with $K$ and $a\in L^\infty(\mathbb{R}^d)$ by \[ T_af(x)=p.v. \int K(x-y)m_{x,y}a\cdot f(y)dy. \] Recently, Grafakos and Honz\'ık [5] proved that $T_a$ is of weak type (1,1) for $d=2$. In this paper, we show that $T_a$ is also weighted weak type (1,1) with the weight $|x|^α\,(-2<α<0)$ for $d=2$. Moreover, we prove that $T_a$ is bounded on weighted $L^p(\mathbb{R}^d)\,(1<p<\infty)$ for all $d\ge2$.

math.CA↗