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.
Let $r>\frac{4}{3}$ and let $Ω\in L^{r}(\mathbb{S}^{2n-1})$ have vanishing integral. We show that the bilinear rough singular integral $$T_Ω(f,g)(x)= \textrm{p.v.} \int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\frac{Ω((y,z)/|(y,z)|)}{|(y,z)|^{2n}}f(x-y)g(x-z)\,dydz,$$ satisfies a sparse bound by $(p,p,p)$-averages, where $p$ is bigger than a certain number explicitly related to $r$ and $n$. As a consequence we deduce certain quantitative weighted estimates for bilinear homogeneous singular integrals associated with rough homogeneous kernels.
We study bilinear rough singular integral operators $\mathcal{L}_Ω$ associated with a function $Ω$ on the sphere $\mathbb{S}^{2n-1}$. In the recent work of Grafakos, He, and Slavíková (Math. Ann. 376: 431-455, 2020), they showed that $\mathcal{L}_Ω$ is bounded from $L^2\times L^2$ to $L^1$, provided that $Ω\in L^q(\mathbb{S}^{2n-1})$ for $4/3<q\le \infty$ with mean value zero. In this paper, we provide a generalization of their result. We actually prove $L^{p_1}\times L^{p_2}\to L^p$ estimates for $\mathcal{L}_Ω$ under the assumption $$Ω\in L^q(\mathbb{S}^{2n-1}) \quad \text{ for }~\max{\Big(\…
By a reduction method, the limiting weak-type behaviors of factional maximal operators and fractional integrals are established without any smoothness assumption on the kernel, which essentially improve and extend previous results. As a byproduct, we characterize the boundedness of several operators by the membership of their kernel in Lebesgue space on sphere.
We investigate two types of boundedness criteria for bilinear Fourier multiplier operators with symbols with bounded partial derivatives of all (or sufficiently many) orders. Theorems of the first type explicitly prescribe only a certain rate of decay of the symbol itself while theorems of the second type require, in addition, the same rate of decay of all derivatives of the symbol. We show that even though these two types of bilinear multiplier theorems are closely related, there are some fundamental differences between them which arise in limiting cases. Also, since theorems of the latter ty…
Let $Ω$ be a function of homogeneous of degree zero and vanish on the unit sphere $\mathbb {S}^n$. In this paper, we investigate the limiting weak-type behavior for singular integral operator $T_Ω$ associated with rough kernel $Ω$. We show that, if $Ω\in L\log L(\mathbb S^{n})$, then $\lim_{λ\to0^+}λ|\{x\in\mathbb{R}^n:|T_Ω(f)(x)|>λ\}| = n^{-1}\|Ω\|_{L^1(\mathbb {S}^n)}\|f\|_{L^1(\mathbb{R}^n)},\quad0\le f\in L^1(\mathbb{R}^n).$ Moreover,$(n^{-1}\|Ω\|_{L^1(\mathbb{S}^{n-1})}$ is a lower bound of weak-type norm of $T_Ω$ when $Ω\in L\log L(\mathbb{S}^{n-1})$. Corresponding results for rough bili…
Let $Ω$ be homogeneous of degree zero, have mean value zero and integrable on the unit sphere, and $μ_Ω$ be the higher-dimensional Marcinkiewicz integral defined by $$μ_Ω(f)(x)= \Big(\int_0^\infty\Big|\int_{|x-y|\leq t}\frac{Ω(x-y)}{|x-y|^{n-1}}f(y)dy\Big|^2\frac{dt}{t^3}\Big)^{1/2}. $$ In this paper, the authors establish a bilinear sparse domination for $μ_Ω$ under the assumption $Ω\in L^{\infty}(S^{n-1})$. As applications, some quantitative weighted bounds for $μ_Ω$ are obtained.
We obtain a sharp $L^2\times L^2 \to L^1$ boundedness criterion for a class of bilinear operators associated with a multiplier given by a signed sum of dyadic dilations of a given function, in terms of the $L^q$ integrability of this function; precisely we show that boundedness holds if and only if $q<4$. We discuss applications of this result concerning bilinear rough singular integrals and bilinear dyadic spherical maximal functions. Our second result is an optimal $L^2\times L^2\to L^1$ boundedness criterion for bilinear operators associated with multipliers with $L^\infty$ derivatives. Thi…
Let $Ω$ be homogeneous of degree zero, have vanishing moment of order one on the unit sphere $\mathbb {S}^{d-1}$($d\ge 2$). In this paper, our object of investigation is the following rough non-standard singular integral operator $$T_{Ω,\,A}f(x)={\rm p.\,v.}\int_{\mathbb{R}^d}\frac{Ω(x-y)}{|x-y|^{d+1}}\big(A(x)-A(y)-\nabla A(y)(x-y)\big)f(y){\rm d}y,$$ where $A$ is a function defined on $\mathbb{R}^d$ with derivatives of order one in ${\rm BMO}(\mathbb{R}^d)$. We show that $T_{Ω,\,A}$ enjoys the endpoint $L\log L$ type estimate and is $L^p$ bounded if $Ω\in L(\log L)^{2}(\mathbb{S}^{d-1})$. Th…
