Related papers: Sparse domination for rough multilinear singular i…

We investigate new pointwise bounds for a class of rough integral operators, $T_{\Omega,\alpha}$, for a parameter $0<\alpha <n$ that includes classical rough singular integrals of Calder\'on and Zygmund, rough hypersingular integrals, and…

Let $\T (0\leq \alpha <n)$ be the singular and fractional integrals with variable kernel $\Omega(x,z)$, and $[b,\T]$ be the commutator generated by $\T$ and a Lipschitz function $b$. In this paper, the authors study the boundedness of…

In this paper we prove the sharp boundedness for a fractional type operator given by a kernel that satisfy a $L^{\alpha,r'}$-H\"ormander conditions and a fractional size condition, where $0<\alpha<n$ and $1< r'\leq \infty$. To prove this…

Given $\Omega(\subseteq\;R^{1+m})$, a smooth bounded domain and a nonnegative measurable function $f$ defined on $\Omega$ with suitable summability. In this paper, we will study the existence and regularity of solutions to the quasilinear…

We present a general approach to sparse domination based on single-scale $L^p$-improving as a key property. The results are formulated in the setting of metric spaces of homogeneous type and avoid completely the use of dyadic-probabilistic…

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 $\mathcal T:=\{T_\varepsilon\}_{\varepsilon>0}$…

We show some simple sufficient conditions for which the multilinear embedding theorem holds for fractional sparse operators. By verifying these conditions, we establish the theorem for power weights. We also provide Morrey-type sufficient…

In this work we extend Lacey's domination theorem to prove the pointwise control of bilinear Calder\'on--Zygmund operators with Dini--continuous kernel by sparse operators. The precise bounds are carefully tracked following the spirit in a…

In this article, we conduct a study of integral operators defined in terms of non-convolution type kernels with singularities of various degrees. The operators that fall within our scope of research include fractional integrals, fractional…

Let $\Omega\in L^q(S^{n-1})$ 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…

We study the two-weighted off-diagonal compactness of commutators of rough singular integral operators $T_\Omega$ that are associated with a kernel $\Omega\in L^q(\mathbb{S}^{d-1})$. We establish a characterisation of compactness of the…

We study weighted norm inequalities of $(p,r)$-type, $ \Vert \mathbf{G} (f \, d \sigma) \Vert_{L^r(\Omega, d\sigma)} \le C \Vert f \Vert_{L^p(\Omega, \sigma)}, \quad \forall \, f \in L^p(\sigma),$ for $0 < r < p$ and $p>1$, where…

For polynomial $ P (x,y)$, and any Calder\'{o}n-Zygmund kernel, $K$, the operator below satisfies a $ (1,r)$ sparse bound, for $ 1< r \leq 2$. $$ \sup _{\epsilon >0} \Bigl\lvert \int_{|y| > \epsilon} f (x-y) e ^{2 \pi i P (x,y) } K(y) \; dy…

The purpose of this paper is to describe the smooth homogeneous Calderon-Zygmund operators for which the maximal singular integral T*f may be controlled by the singular integral Tf. We consider two types of control. The first is the L2…

In this paper it is shown that for $\Omega\in L\log L(\mathbb{S}^{d-1})$, the rough maximal singular integral operator $T_\Omega^*$ is of weak type $L\log\log L(\mathbb{R}^d)$. Furthermore, for $w\in A_1$ and $\Omega\in…

On a Riemannian manifold with a smooth function $f: M\to \mathbb{R}$, we consider the linearization of the Perelman scalar curvature $\mathcal{R}$ and its $L^2$-formal adjoint operator $\delta\mathcal{R}^*$. A manifold endowed with a metric…

In this paper, we systematically study the Fefferman-Stein inequality and Coifman-Fefferman inequality for the general commutators of singular integral operators that satisfy H\"{o}rmander conditions of Young type. Specifically, we first…

In this paper, we give a sharp sparse domination of pseudodifferential operators associated with symbols belonging to the H\"{o}rmander class, and fundamental solutions of dispersive equations. Furthermore, we give boundedness results of…

Using the Calder\'on-Zygmund decomposition, we give a novel and simple proof that $L^2$ bounded dyadic shifts admit a domination by positive sparse forms with linear growth in the complexity of the shift. Our estimate, coupled with…

We consider the diffusion equation in the setting of operator theory. In particular, we study the characterization of the limit of the diffusion operator for diffusivities approaching zero on a subdomain $\Omega_1$ of the domain of…