Related papers: Weighted norm inequalities, off-diagonal estimates…

We give an overview of the generalized Calder\'on-Zygmund theory for "non-integral" singular operators, that is, operators without kernels bounds but appropriate off-diagonal estimates. This theory is powerful enough to obtain weighted…

This is the second part of a series of four articles on weighted norm inequalities, off-diagonal estimates and elliptic operators. We consider a substitute to the notion of pointwise bounds for kernels of operators which usually is a…

This is the first part of a series of four articles. In this work, we are interested in weighted norm estimates. We put the emphasis on two results of different nature: one is based on a good-$\lambda$ inequality with two-parameters and the…

This article focuses on $L^p$ estimates for objects associated to elliptic operators in divergence form: its semigroup, the gradient of the semigroup, functional calculus, square functions and Riesz transforms. We introduce four critical…

We prove weighted norm inequalities for fractional powers of elliptic operators together with their commutators with BMO functions, encompassing what is known for the classical Riesz potentials and elliptic operators with Gaussian…

This is the fourth article of our series. Here, we study weighted norm inequalities for the Riesz transform of the Laplace-Beltrami operator on Riemannian manifolds and of subelliptic sum of squares on Lie groups, under the doubling volume…

This is the final part of a series of papers where we study perturbations of divergence form second order elliptic operators $-\operatorname{div} A \nabla$ by first and zero order terms, whose complex coefficients lie in critical spaces,…

Let $L$ be a second order divergence form elliptic operator with complex bounded measurable coefficients. The operators arising in connection with $L$, such as the heat semigroup and Riesz transform, are not, in general, of…

In this paper, we study both elliptic and parabolic equations in non-divergence form with singular degenerate coefficients. Weighted and mixed-norm $L_p$-estimates and solvability are established under some suitable partially weighted BMO…

We consider Muckenhoupt weights $w$, and define weighted Hardy spaces $H^p_{\mathcal{T}}(w)$, where $\mathcal{T}$ denotes a conical square function or a non-tangential maximal function defined via the heat or the Poisson semigroup generated…

We dominate non-integral singular operators by adapted sparse operators and derive optimal norm estimates in weighted spaces. Our assumptions on the operators are minimal and our result applies to an array of situations, whose prototype are…

We consider the operator $L=-{\rm div}(A\nabla)$, where the $n\times n$ matrix $A$ is real-valued, elliptic, with the symmetric part of $A$ in $L^\infty(\mathbb{R}^n)$, and the anti-symmetric part of $A$ only belongs to the space…

We give a new proof of the sharp one weight $L^p$ inequality for any operator $T$ that can be approximated by Haar shift operators such as the Hilbert transform, any Riesz transform, the Beurling-Ahlfors operator. Our proof avoids the…

We study singular integral operators with variable Calder\'on--Zygmund kernels and their commutators with $VMO$ functions in the framework of Orlicz spaces. After revisiting the classical $L^p$ theory, we establish boundedness results in…

This is the first part of a series of three articles. In this paper, we obtain weighted norm inequalities for different conical square functions associated with the Heat and the Poisson semigroups generated by a second order divergence form…

Let $L=-\sum_{i,j=1}^n a_{ij}D_iD_j$ be the elliptic operator in non-divergence form with smooth real coefficients satisfying uniformly elliptic condition. Let $W$ be the global nonnegative adjoint solution. If $W\in A_2$, we prove that the…

In this article, we will consider second order uniformly elliptic operators of divergence form defined on R^n with measurable coefficients. Mainly, we will give estimates on the dimension of space of solutions that grow at most polynomially…

This paper is a contribution to the study of regularity theory for nonlinear elliptic equations. The aim of this paper is to establish some global estimates for non-uniformly elliptic in divergence form as follows \begin{align*}…

Consider the second order divergence form elliptic operator $L$ with complex bounded coefficients. In general, the operators related to it (such as Riesz transform or square function) lie beyond the scope of the Calder\'{o}n-Zygmund theory.…

We establish optimal L^p bounds for the nontangential maximal function of the gradient of the solution to a second order elliptic operator in divergence form, possibly non-symmetric, with bounded measurable coefficients independent of the…