Weighted norm estimates and representation formulas are proved for non-homogeneous singular integrals with no regularity condition on the kernel and only an L log L integrability condition. The representation formulas involve averages over a star-shaped set naturally associated with the kernel. The proof of the norm estimates is based on the representation formulas, some new variations of the Hardy-Littlewood maximal function, and weighted Littlewood-Paley theory.
We establish a quantitative weighted inequality for the bilinear rough singular integral, where the bound is controlled by the cube of the weight constant.
In this paper, we investigate the behavior of the bounds of the composition for rough singular integral operators on the weighted space. More precisely, we obtain the quantitative weighted bounds of the composite operator for two singular integral operators with rough homogeneous kernels on $L^p(\mathbb{R}^d,\,w)$, $p\in (1,\,\infty)$, which is smaller than the product of the quantitative weighted bounds for these two rough singular integral operators. Moreover, at the endpoint $p=1$, the $L\log L$ weighted weak type bound is also obtained, which has interests of its own in the theory of rough…
We obtain a weak type $(1,1)$ estimate for a maximal operator associated with the classical rough homogeneous singular integrals $T_Ω$. In particular, this provides a different approach to a sparse domination for $T_Ω$ obtained recently by Conde-Alonso, Culiuc, Di Plinio and Ou.
We show that the $L^2(w)$ operator norm of the composition $M\!\circ T_Ω$, where $M$ is the maximal operator and $T_Ω$ is a rough homogeneous singular integral with angular part $Ω\in L^{\infty}(S^{n-1})$, depends quadratically on $[w]_{A_2}$, and this dependence is sharp.
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).
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.
In this paper it is shown that for $Ω\in L\log L(\mathbb{S}^{d-1})$, the rough maximal singular integral operator $T_Ω^*$ is of weak type $L\log\log L(\mathbb{R}^d)$. Furthermore, for $w\in A_1$ and $Ω\in L^\infty(\mathbb{S}^{d-1})$, it is shown that $T_Ω^*$ is of weak type $L\log\log L(w)$ with weight dependence $[w]_{A_1}[w]_{A_{\infty}}\log([w]_{A_{\infty}}+1),$ which is same as the best known constant for the singular integral $T_Ω$.
We study $m$-linear homogeneous rough singular integral operators $\mathcal{L}_Ω$ associated with integrable functions $Ω$ on $\mathbb{S}^{mn-1}$ with mean value zero. We prove boundedness for $\mathcal{L}_Ω$ from $L^{p_1}\times \cdots \times L^{p_m}$ to $L^p$ when $1<p_1,\dots, p_m<\infty$ and $1/p=1/p_1+\cdots +1/p_m$ in the largest possible open set of exponents when $Ω\in L^q(\mathbb S^{mn-1})$ and $q\ge 2$. This set can be described by a convex polyhedron in $\mathbb R^m$.
Quantitative $A_1-A_\infty$ estimates for rough homogeneous singular integrals $T_Ω$ and commutators of $BMO$ symbols and $T_Ω$ are obtained. In particular the following estimates are proved: % \[ \|T_Ω\|_{L^p(w)}\le c_{n,p}\|Ω\|_{L^\infty} [w]_{A_1}^{\frac{1}{p}}\,[w]_{A_{\infty}}^{1+\frac{1}{p'}}\|f\|_{L^p(w)} \] % and % \[ \| [b,T_Ω]f\|_{L^{p}(w)}\leq c_{n,p}\|b\|_{BMO}\|Ω\|_{L^{\infty}} [w]_{A_1}^{\frac{1}{p}}[w]_{A_{\infty}}^{2+\frac{1}{p'}}\|f\|_{L^{p}\left(w\right)}, \] % for $1<p<\infty$ and $1/p+1/p'=1$.
Let $Ω$ be homogeneous of degree zero and have mean value zero on the unit sphere ${S}^{d-1}$, $T_Ω$ be the homogeneous singular integral operator with kernel $\frac{Ω(x)}{|x|^d}$ and $T_{Ω,\,b}$ be the commutator of $T_Ω$ with symbol $b$. In this paper, we prove that if $Ω\in L(\log L)^2(S^{d-1})$, then for $b\in {\rm BMO}(\mathbb{R}^d)$, $T_{Ω,\,b}$ satisfies an endpoint estimate of $L\log L$ type.
Let $Ω$ be homogeneous of degree zero and have mean value zero on the unit sphere ${S}^{n-1}$, $T_Ω$ be the convolution singular integral operator with kernel $\frac{Ω(x)}{|x|^n}$. For $b\in{\rm BMO}(\mathbb{R}^n)$, let $T_{Ω,\,b}$ be the commutator of $T_Ω$. In this paper, by establishing suitable sparse dominations, the authors establish some weak type endpoint estimates of $L\log L$ type for $T_{Ω,\,b}$ when $Ω\in L^q(S^{n-1})$ for some $q\in (1,\,\infty]$.
We consider homogeneous singular kernels, whose angular part is bounded, but need not have any continuity. For the norm of the corresponding singular integral operators on the weighted space $L^2(w)$, we obtain a bound that is quadratic in the $A_2$ constant $[w]_{A_2}$. We do not know if this is sharp, but it is the best known quantitative result for this class of operators. The proof relies on a classical decomposition of these operators into smooth pieces, for which we use a quantitative elaboration of Lacey's dyadic decomposition of Dini-continuous operators: the dependence of constants on…
We prove weighted estimates for rough bilinear singular integral operators with kernel $$K(y_1, y_2) = \frac{Ω((y_1,y_2)/|(y_1,y_2)|)}{|(y_1, y_2)|^{2d}},$$ where $y_i \in \mathbb{R}^{d}$ and $Ω\in L^{\infty}(S^{2d-1})$ with $\int_{S^{2d-1}}Ωdσ= 0.$ The argument is by sparse domination of rough bilinear operators, via an abstract theorem that is a multilinear generalization of recent work by Conde-Alonso, Culiuc, Di Plinio and Ou. We also use recent results due to Grafakos, He, and Honz\'ık for the application to rough bilinear operators. In particular, since the weighted estimates are proved …
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…
We prove that bilinear forms associated to the rough homogeneous singular integrals $T_Ω$ on $\mathbb R^d$, where the angular part $Ω\in L^q (S^{d-1})$ has vanishing average and $1<q\leq \infty$, and to Bochner-Riesz means at the critical index in $\mathbb R^d$ are dominated by sparse forms involving $(1,p)$ averages. This domination is stronger than the weak-$L^1$ estimates for $T_Ω$ and for Bochner-Riesz means, respectively due to Seeger and Christ. Furthermore, our domination theorems entail as a corollary new sharp quantitative $A_p$-weighted estimates for Bochner-Riesz means and for homog…
In this paper we provide weighted estimates for rough operators, including rough homogeneous singular integrals $T_Ω$ with $Ω\in L^\infty(\mathbb{S}^{n-1})$ and the Bochner-Riesz multiplier at the critical index $B_{(n-1)/2}$. More precisely, we prove qualitative and quantitative versions of Coifman-Fefferman type inequalities and their vector-valued extensions, weighted $A_p-A_\infty$ strong and weak type inequalities for $1<p<\infty$, and $A_1-A_\infty$ type weak $(1,1)$ estimates. Moreover, Fefferman-Stein type inequalities are obtained, proving in this way a conjecture raised by the second…
An $A_1-A_\infty$ estimate improving a previous result in arXiv:1607.06432 is obtained. Also new a result in terms of the ${A_\infty}$ constant and the one supremum $A_q-A_\infty^{\exp}$ constant, is proved, providing a counterpart for the result obained in arXiv:1705.08364. Both of the preceding results rely upon a sparse domination in terms of bilinear forms for $[b,T_Ω]$ with $Ω\in L^\infty(\mathbb{S}^{n-1})$ and $b\in BMO$ which is established relying upon techniques from arXiv:1705.07397.
We study the rough bilinear singular integral, introduced by Coifman and Meyer , $$ T_Ω(f,g)(x)=p.v. \! \int_{\mathbb R^{n}}\! \int_{\mathbb R^{n}}\! |(y,z)|^{-2n} Ω((y,z)/|(y,z)|)f(x-y)g(x-z) dydz, $$ when $Ω$ is a function in $L^q(\mathbb S^{2n-1})$ with vanishing integral and $2\le q\le \infty$. When $q=\infty$ we obtain boundedness for $T_Ω$ from $L^{p_1}(\mathbb R^n)\times L^{p_2}(\mathbb R^n)$ to $ L^p(\mathbb R^n) $ when $1<p_1, p_2<\infty$ and $1/p=1/p_1+1/p_2$. For $q=2$ we obtain that $T_Ω$ is bounded from $L^{2}(\mathbb R^n)\times L^{ 2}(\mathbb R^n)$ to $ L^1(\mathbb R^n) $. For $q…
Weak type $(1,1)$ boundedness of noncommutative singular integral operators with rough kernels
Institute for Advanced Study in Mathematics Harbin Institute of Technology（哈尔滨工业大学数学高等研究院）；Harbin Institute of Technology（哈尔滨工业大学）；Zhengzhou Research Institute Harbin Institute of Technology（哈尔滨工业大学郑州研究院）
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 $$Ω\in L^q(\mathbb{S}^{2n-1}) \quad \text{ for }~\max{\Big(\…
Let $Ω$ be homogeneous of degree zero, have vanishing moment of order one on the unit sphere $\mathbb {S}^{d-1}$($d\ge 2$). In this paper, our object of investigation is the following rough non-standard singular integral operator $$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,$$ where $A$ is a function defined on $\mathbb{R}^d$ with derivatives of order one in ${\rm BMO}(\mathbb{R}^d)$. We show that $T_{Ω,\,A}$ enjoys the endpoint $L\log L$ type estimate and is $L^p$ bounded if $Ω\in L(\log L)^{2}(\mathbb{S}^{d-1})$. Th…
We prove sparse bounds for maximal oscillatory rough singular integral operator $$T^{P}_{Ω,*}f(x):=\sup_{ε>0} \left|\int_{|x-y|>ε}e^{ιP(x,y)}\frac{Ω\big((x-y)/|x-y|\big)}{|x-y|^{n}}f(y)dy\right|,$$ where $P(x,y)$ is a real-valued polynomial on $\mathbb{R}^{n}\times \mathbb{R}^{n}$ and $Ω\in L^{\infty}(\mathbb{S}^{n-1})$ is a homogeneous function of degree zero with $\int_{\mathbb{S}^{n-1}}Ω(θ)~dθ=0$. This allows us to conclude weighted $L^p-$estimates for the operator $T^{P}_{Ω,*}$. Moreover, the norm $\|T^P_{Ω,*}\|_{L^p\rightarrow L^p}$ depends only on the total degree of the polynomial $P(x,…
In this paper, we study the quantitative weighted bounds for the $q$-variational singular integral operators with rough kernels. The main result is for the sharp truncated singular integrals itself $$ \|V_q\{T_{Ω,\varepsilon}\}_{\varepsilon>0}\|_{L^p(w)\rightarrow L^p(w)}\leq c_{p,q,n} \|Ω\|_{ L^\infty}(w)_{A_p}^{1+1/q}\{w\}_{A_p},$$ where the quantity $(w)_{A_p}$, $\{w\}_{A_p}$ will be recalled in the introduction; we do not know whether this is sharp, but it is the best known quantitative result for this class of operators, since when $q=\infty$, it coincides with the best known quantitative…
We consider Coifman--Fefferman inequalities for rough homogeneous singular integrals $T_Ω$ and $C_p$ weights. It was recently shown by Li-Pérez-Rivera-Ríos-Roncal that $$ \|T_Ω\|_{L^p(w)} \le C_{p,T,w} \|Mf\|_{L^p(w)} $$ for every $0< p < \infty$ and every $w \in A_\infty$. Our first goal is to generalize this result for every $w \in C_q$ where $q > \max\{1,p\}$ without using extrapolation theory. Although the bounds we prove are new even in a qualitative sense, we also give the quantitative bound with respect to the $C_q$ characteristic. Our techniques rely on recent advances in sparse domina…
We prove that the class of convolution-type kernels satisfying suitable decay conditions of the Fourier transform, appearing in the works of Christ, Christ-Rubio de Francia, and Duoandikoetxea-Rubio de Francia gives rise to maximally truncated singular integrals satisfying a sparse bound by $(1+\varepsilon,1+\varepsilon)$-averages for all $\varepsilon>0$, with linear growth in $\varepsilon^{-1}$. This is an extension of the sparse domination principle by Conde-Alonso, Culiuc, Ou and the first author to maximally truncated singular integrals. Our results cover the rough homogeneous singular int…
We consider singular integral operators and maximal singular integral operators with rough kernels on homogeneous groups. We prove certain estimates for the operators that imply $L^p$ boundedness of them by an extrapolation argument under a sharp condition for the kernels. Also, we prove some weighted $L^p$ inequalities for the operators.
In this paper, starting with a relatively simple observation that the variational estimates of the commutators of the standard Calderón-Zygmund operators with the BMO functions can be deduced from the weighted variational estimates of the standard Calderón-Zygmund operators themselves, we establish similar variational estimates for the commutators of the BMO functions with rough singular integrals which do not admit any weighted variational estimates. The proof involves many Littlewood-Paley type inequalities with commutators as well as Bony decomposition and related para-product estimates.
In this paper, we systematically study weighted jump and variational inequalities for rough operators. More precisely, we show some weighted jump and variational inequalities for the families $\mathcal T:=\{T_\varepsilon\}_{\varepsilon>0}$ of truncated singular integrals and $\mathcal M_Ω:=\{M_{Ω,t}\}_{t>0}$ of averaging operators with rough kernels, which are defined respectively by $$ T_\varepsilon f(x)=\int_{|y|>\varepsilon}\frac{Ω(y')}{|y|^n}f(x-y)dy$$ and $$M_{Ω,t} f(x)=\frac1{t^n}\int_{|y|<t}Ω(y')f(x-y)dy, $$ where the kernel $Ω$ belongs to $L^q(\mathbf S^{n-1})$ for $q>1$.
Let $Ω\in L^q(S^{n-1})$ with $1<q\le\infty$ be homogeneous of degree zero and has mean value zero on $S^{n-1}$. In this paper, we will study the boundedness of homogeneous singular integrals and Marcinkiewicz integrals with rough kernel on the weighted Morrey spaces $L^{p,κ}(w)$ for $q'\le p<\infty$(or $q'<p<\infty$) and $0<κ<1$. We will also prove that the commutator operators formed by a $BMO(\mathbb R^n)$ function $b(x)$ and these rough operators are bounded on the weighted Morrey spaces $L^{p,κ}(w)$ for $q'<p<\infty$ and $0<κ<1$.
In this paper quantitative weighted matrix estimates for vector valued extensions of $L^{r'}$-Hörmander operators and rough singular integrals are studied. Strong type $(p,p)$ estimates, endpoint estimates, and some new results on Coifman-Fefferman estimates assuming $A_\infty$ and $C_p$ condition counterparts are provided. To prove the aforementioned estimates we rely upon some suitable convex body domination results that we settle as well in this paper.
This article is devoted to the extension of the theory of rough paths in the context of Volterra equations with possibly singular kernels. We begin to describe a class of two parameter functions defined on the simplex called Volterra paths. These paths are used to construct a so-called Volterra-signature, analogously to the signature used in Lyon's theory of rough paths. We provide a detailed algebraic and analytic description of this object. Interestingly, the Volterra signature does not have a multiplicative property similar to the classical signature, and we introduce an integral product be…
A criterion on weak type $(1,1)$ bound of rough singular integrals
Feng Liu: College of Mathematics；System Science Shandong University of Science；Simin Liu: College of Mathematics；Minglei Shi: Department of Mathematics Zhejiang Shuren University Shaoxing, Zhejiang 312028 People's Republic of China
