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 via sparse domination, we obtain some quantitative estimates in terms of the $A_{p}$ characteristics of the weights. The abstract theorem is also shown to apply to multilinear Calderón-Zygmund operators with a standard smoothness assumption. Due to the generality of the sparse domination theorem, future applications not considered in this paper are expected.
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 establish a uniform domination of the family of trilinear multiplier forms with singularity over a one-dimensional subspace by positive sparse forms involving $L^p$-averages. This class includes the adjoint forms to the bilinear Hilbert transforms. Our result strengthens the $L^p$-boundedness proved in \cite{MTT} and entails as a corollary a rich multilinear weighted theory. In particular, we obtain $L^{q_1}(v_1) \times L^{q_2}(v_2)$-boundedness of the bilinear Hilbert transform when the weights $v_j$ belong to the class $A_{\frac{q+1}{2}}\cap RH_2$. Our proof relies on a stopping time cons…
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 establish a quantitative weighted inequality for the bilinear rough singular integral, where the bound is controlled by the cube of the weight constant.
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…
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 $$\lim_{λ\to 0^+}λ|\big\{ x\in\mathbb{R}^n:| T_{\ve…
In this work we extend Lacey's domination theorem to prove the pointwise control of bilinear Calderón--Zygmund operators with Dini--continuous kernel by sparse operators. The precise bounds are carefully tracked following the spirit in a recent work of Hytönen, Roncal and Tapiola. We also derive new mixed weighted estimates for a general class of bilinear dyadic positive operators using multiple $A_{\infty}$ constants inspired in the Fujii-Wilson and Hrusčěv classical constants. These estimates have many new applications including mixed bounds for multilinear Calderón--Zygmund operators and th…
We study the rough bilinear singular integral, introduced by Coifman and Meyer , $$ 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, $$ when $Ω$ is a function in $L^q(\mathbb S^{2n-1})$ with vanishing integral and $2\le q\le \infty$. When $q=\infty$ we obtain boundedness for $T_Ω$ from $L^{p_1}(\mathbb R^n)\times L^{p_2}(\mathbb R^n)$ to $ L^p(\mathbb R^n) $ when $1<p_1, p_2<\infty$ and $1/p=1/p_1+1/p_2$. For $q=2$ we obtain that $T_Ω$ is bounded from $L^{2}(\mathbb R^n)\times L^{ 2}(\mathbb R^n)$ to $ L^1(\mathbb R^n) $. For $q…
In this paper, we establish some bilinear endpoint estimates of Calderón commutator $\mathcal{C}[\nabla A,f](x)$ with a homogeneous kernel when $Ω\in L\log^+L(\mathbf{S}^{d-1})$. More precisely, we prove that $\mathcal{C}[\nabla A,f]$ maps $L^q(\mathbb{R}^d)\times L^1(\mathbb{R}^d)$ to $L^{r,\infty}(\mathbb{R}^d)$ if $q>d$ which improves previous result essentially. If $q=d$, we show that Calderón commutator maps $L^{d,1}(\mathbb{R}^d)\times L^1(\mathbb{R}^d)$ to $L^{r,\infty}(\mathbb{R}^d)$ which is new even if the kernel is smooth. The novelty in the paper is that we prove a new endpoint est…
We obtain a weak type $(1,1)$ estimate for a maximal operator associated with the classical rough homogeneous singular integrals $T_Ω$. In particular, this provides a different approach to a sparse domination for $T_Ω$ obtained recently by Conde-Alonso, Culiuc, Di Plinio and Ou.
We study bilinear rough singular integral operators $\mathcal{L}_Ω$ associated with a function $Ω$ on the sphere $\mathbb{S}^{2n-1}$. In the recent work of Grafakos, He, and Slavíková (Math. Ann. 376: 431-455, 2020), they showed that $\mathcal{L}_Ω$ is bounded from $L^2\times L^2$ to $L^1$, provided that $Ω\in L^q(\mathbb{S}^{2n-1})$ for $4/3<q\le \infty$ with mean value zero. In this paper, we provide a generalization of their result. We actually prove $L^{p_1}\times L^{p_2}\to L^p$ estimates for $\mathcal{L}_Ω$ under the assumption $$Ω\in L^q(\mathbb{S}^{2n-1}) \quad \text{ for }~\max{\Big(\…
We prove pointwise variational Lp bounds for a bilinear Fourier integral operator in a large but not necessarily sharp range of exponents. This result is a joint strengthening of the corresponding bounds for the classical Carleson operator, the bilinear Hilbert transform, the variation norm Carleson operator, and the bi-Carleson operator. Terry Lyon's rough path theory allows for extension of our result to multilinear estimates. We consider our result a proof of concept for a wider array of similar estimates with possible applications to ordinary differential equations.
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 $q\in (1,\,\infty]$.
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 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 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…
We prove a sparse bound for the $m$-sublinear form associated to vector-valued maximal functions of Fefferman-Stein type. As a consequence, we show that the sparse bounds of multisublinear operators are preserved via $\ell^r$-valued extension. This observation is in turn used to deduce vector-valued, multilinear weighted norm inequalities for multisublinear operators obeying sparse bounds, which are out of reach for the extrapolation theory recently developed by Cruz-Uribe and Martell. As an example, vector-valued multilinear weighted inequalities for bilinear Hilbert transforms are deduced fr…
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.
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…
Given $1\leq q<p<\infty$ quantitative weighted L^p estimates, in terms of Aq weights, for vector valued maximal functions, Calderón-Zygmund operators, commutators and maximal rough singular integrals are obtained. The results for singular operators will rely upon suitable convex body domination results, which in the case of commutators will be provided in this work, obtaining as a byproduct a new proof for the scalar case as well.
We consider bilinear restriction estimates for wave-Schrödinger interactions and provided a sharp condition to ensure that the product belongs to $L^q_t L^r_x$ in the full bilinear range $\frac{2}{q} + \frac{d+1}{r} < d+1$, $1 \leqslant q, r \leqslant 2$. Moreover, we give a counter-example which shows that the bilinear restriction estimate can fail, even in the transverse setting. This failure is closely related to the lack of curvature of the cone. Finally we mention extensions of these estimates to adapted function spaces. In particular we give a general transference type principle for $U^2…
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…
In this paper we provide some quantitative mixed-type estimates assuming conditions that imply that $uv\in A_{\infty}$ for Calderón-Zygmund operators, rough singular integrals and commutators. The main novelty of this paper lies in the fact that we rely upon sparse domination results, pushing an approach to endpoint estimates that was introduced by Domingo-Salazar, Lacey and Rey and extended in works by Lerner, Ombrosi and the second author and Li, Perez, the second author and Roncal.
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 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 show that bilinear variational estimates of Do, Muscalu, and Thiele (arXiv:1009.5187) remain valid for a pair of general martingales with respect to the same filtration. Our result can also be viewed as an off-diagonal generalization of the Burkholder--Davis--Gundy inequality for martingale rough paths by Chevyrev and Friz (arXiv:1704.08053).
