arXiv subjects

Moyan Qin

Publications and source records attributed to Moyan Qin.

5 recordsLinked to original sources

arXiv2026

Bilinear rough singular integrals under a fractional geometric condition

We establish the Banach-range boundedness of bilinear rough singular integral operators, together with their maximal and maximally truncated forms, under the fractional geometric condition on the mean-zero angular kernel \[ \sup_{\xi \in \mathbb{S}^{1}}\int_{\mathbb{S}^{1}} \frac{|\Omega(\theta)|}{|\theta \cdot \xi|^{a}} \, d\sigma(\theta) < \infty, \qquad \frac12 < a < 1. \] This condition imposes integrability strictly weaker than the $L^q(\mathbb{S}^1) (q>1)$ constraints considered by Grafakos, He, Honz\'ik (Adv. Math., 2018), Dosidis and Slav\'ikov\'a (Math. Ann., 2024), while defining a class of functions that is neither contained in nor contains the classical Orlicz space $L(\log L)^\alpha(\mathbb{S}^1) $ ($\alpha>1$). Our proof avoids traditional wavelet decompositions of the multiplier, instead using local Fourier series expansions of the input functions.

math.CA↗

arXiv2021

On the Bounds of Weak $(1,1)$ Norm of Hardy-Littlewood Maximal Operator with $L\log L({\mathbb S^{n-1}})$ Kernels

Let $Ω\in L^1{({\mathbb S^{n-1}})}$, be a function of homogeneous of degree zero, and $M_Ω$ be the Hardy-Littlewood maximal operator associated with $Ω$ defined by $M_Ω(f)(x) = \sup_{r>0}\frac1{r^n}\int_{|x-y| λ\}| = n^{-1}\|Ω\|_{L^1({\mathbb S^{n-1}})}\|f\|_{L^1({\mathbb R^n})}.$$ This removes the smoothness restrictions on the kernel $Ω$, such as Dini-type conditions, in previous results. To prove our result, we present a new upper bound of $\|M_Ω\|_{L^1\to L^{1,\infty}}$, which essentially improves the upper bound $C(\|Ω\|_{L\log L({\mathbb S^{n-1}})}+1)$ given by Christ and Rubio de Francia. As a consequence, the upper and lower bounds of $\|M_Ω\|_{L^1\to L^{1,\infty}}$ are obtained for $Ω\in L\log L {({\mathbb S^{n-1}})}$.

math.CA↗

arXiv2021

Limiting weak-type behaviors for singular integrals with rough $L\log L(\mathbb{S}^n)$ kernels

Let $Ω$ be a function of homogeneous of degree zero and vanish on the unit sphere $\mathbb {S}^n$. In this paper, we investigate the limiting weak-type behavior for singular integral operator $T_Ω$ associated with rough kernel $Ω$. We show that, if $Ω\in L\log L(\mathbb S^{n})$, then $\lim_{λ\to0^+}λ|\{x\in\mathbb{R}^n:|T_Ω(f)(x)|>λ\}| = n^{-1}\|Ω\|_{L^1(\mathbb {S}^n)}\|f\|_{L^1(\mathbb{R}^n)},\quad0\le f\in L^1(\mathbb{R}^n).$ Moreover,$(n^{-1}\|Ω\|_{L^1(\mathbb{S}^{n-1})}$ is a lower bound of weak-type norm of $T_Ω$ when $Ω\in L\log L(\mathbb{S}^{n-1})$. Corresponding results for rough bilinear singular integral operators defined in the form $T_{\vecΩ}(f_1,f_2) = T_{Ω_1}(f_1)\cdot T_{Ω_2}(f_2)$ have also been established.

math.CA↗

arXiv2021

The limiting weak type behaviors and The lower bound for a new weak $L\log L$ type norm of strong maximal operators

It is well known that the weak ($1,1$) bounds doesn't hold for the strong maximal operators, but it still enjoys certain weak $L\log L$ type norm inequality. Let $Φ_n(t)=t(1+(\log^+t)^{n-1})$ and the space $L_{Φ_n}({\mathbb R^{n}})$ be the set of all measurable functions on ${\mathbb R^{n}}$ such that $\|f\|_{L_{Φ_n}({\mathbb R^{n}})} :=\|Φ_n(|f|)\|_{L^1({\mathbb R^{n}})}<\infty$. In this paper, we introduce a new weak norm space $L_{Φ_n}^{1,\infty}({\mathbb R^{n}})$, which is more larger than $L^{1,\infty}({\mathbb R^{n}})$ space, and establish the correspondng limiting weak type behaviors of the strong maximal operators. As a corollary, we show that $ \max\{{2^n}{((n-1)!)^{-1}},1\}$ is a lower bound for the best constant of the $L_{Φ_n}\to L_{Φ_n}^{1,\infty}$ norm of the strong maximal operators. Similar results have been extended to the multilinear strong maximal operators.

math.CA↗

arXiv2020

Limiting weak-type behavior for rough bilinear operators

Let $Ω_1,Ω_2$ be functions of homogeneous of degree $0$ and $\vecΩ=(Ω_1,Ω_2)\in L\log L(\mathbb{S}^{n-1})\times L\log L(\mathbb{S}^{n-1})$. In this paper, we investigate the limiting weak-type behavior for bilinear maximal function $M_{\vecΩ}$ and bilinear singular integral $T_{\vecΩ}$ associated with rough kernel $\vecΩ$. For all $f,g\in L^1(\mathbb{R}^n)$, we show that $$\lim_{λ\to 0^+}λ|\big\{ x\in\mathbb{R}^n:M_{\vecΩ}(f_1,f_2)(x)>λ\big\}|^2 = \frac{\|Ω_1Ω_2\|_{L^{1/2}(\mathbb{S}^{n-1})}}{ω_{n-1}^2}\prod\limits_{i=1}^2\| f_i\|_{L^1}$$ and $$\lim_{λ\to 0^+}λ|\big\{ x\in\mathbb{R}^n:| T_{\vecΩ}(f_1,f_2)(x)|>λ\big\}|^{2} = \frac{\|Ω_1Ω_2\|_{L^{1/2}(\mathbb{S}^{n-1})}}{n^2}\prod\limits_{i=1}^2\| f_i\|_{L^1}.$$ As consequences, the lower bounds of weak-type norms of $M_{\vecΩ}$ and $T_{\vecΩ}$ are obtained. These results are new even in the linear case. The corresponding results for rough bilinear fractional maximal function and fractional integral operator are also discussed.

math.CA↗