Related papers: Elliptic operators with unbounded diffusion coeffi…

We study the boundary regularity of solutions of elliptic operators in divergence form with $C^{0,\alpha}$ coefficients or operators which are small perturbations of the Laplacian in non-smooth domains. We show that, as in the case of the…

Let $L$ be the divergence form elliptic operator with complex bounded measurable coefficients, $\omega$ the positive concave function on $(0,\infty)$ of strictly critical lower type $p_\oz\in (0, 1]$ and…

We consider fractional Schr\"odinger operators $H=(-\Delta)^\alpha+V(x)$ in $n$ dimensions with real-valued potential $V$ when $n>2\alpha$, $\alpha>1$. We show that the wave operators extend to bounded operators on $L^p(\mathbb R^n)$ for…

Let $f:[0,+\infty) \to \mathbb{R}$ be a (locally) Lipschitz function and $\Omega \subset \mathbb{R}^2$ a $C^{1,\alpha}$ domain whose boundary is unbounded and connected. If there exists a positive bounded solution to the overdetermined…

We study boundedness on $L^p(R^d)$ of vertical Littlewood-Paley-Stein functions for Schr\"odinger operators $-\Delta + V$ with nonnegative potential $V$. These functions are proved to be bounded on $L^p$ for all $p \in (1, 2]$. The…

We generalize to the setting of 1-sided chord-arc domains, that is, to domains satisfying the interior Corkscrew and Harnack Chain conditions (these are respectively scale-invariant/quantitative versions of the openness and…

We study symmetric diffusion operators on metric measure spaces. Our main question is whether or not the restriction of the operator to a suitable core continues to be essentially self-adjoint or $L^p$-unique if a small closed set is…

Weighted inequality on the Hardy-Littlewood maximal function is completely understood while it is not well understood for the spherical maximal function. For the power weight $|x|^{\alpha}$, it is known that the spherical maximal operator…

We consider second order divergence form elliptic operators with $W^{1,1}$ coefficients, in a uniform domain $\Omega$ with Ahlfors regular boundary. We show that the $A_\infty$ property of the elliptic measure associated to any such…

Let $\Omega \subset \mathbb{R}^N$ ($N \geq 3$) be a $C^2$ bounded domain and $\Sigma \subset \Omega$ be a compact, $C^2$ submanifold without boundary, of dimension $k$ with $0\leq k < N-2$. Put $L_\mu = \Delta + \mu d_\Sigma^{-2}$ in…

Building on arXiv:1902.03807, this paper develops a unifying study on the boundedness properties of several representative classes of hybrid operators, i.e. operators that enjoy both zero and non-zero curvature features. Specifically, via…

This article studies the Fourier spectrum characterization of functions in the Clifford algebra-valued Hardy spaces $H^p(\mathbf R^{n+1}_+), 1\leq p\leq \infty.$ Namely, for $f\in L^p(\mathbf R^n)$, Clifford algebra-valued, $f$ is further…

We consider elliptic operators with measurable coefficients and Robin boundary conditions on a bounded domain $\Omega \subset \mathbb{R}^d$ and show that the first eigenfunction $v$ satisfies $v(x) \ge \delta > 0$ for all $x \in…

In this article we investigate $L^p$ boundedness of the spherical maximal operator $\mathfrak{m}^\alpha$ of (complex) order $\alpha$ on the $n$-dimensional hyperbolic space $\mathbb{H}^n$, which was introduced and studied by El Kohen. We…

We introduce a new condition on elliptic operators $L= {1/2}\triangle + b \cdot \nabla $ which ensures the validity of the Liouville property for bounded solutions to $Lu=0$ on $\R^d$. Such condition is sharp when $d=1$. We extend our…

We prove that for operators satistying weighted inequalities with $A_p$ weights the boundedness on a certain class of Morrey spaces holds with weights of the form $|x|^\alpha w(x)$ for $w\in A_p$. In the case of power weights the shift with…

For a given second-order linear elliptic operator $L$ which admits a positive minimal Green function, and a given positive weight function $W$, we introduce a family of weighted Lebesgue spaces $L^p(\phi_p)$ with their dual spaces, where…

We prove an analogue of Wolff's inequality for the so-called intrinsic nonlinear potentials associated with the quasilinear elliptic equation \[ -\Delta_{p} u = \sigma u^{q} \quad \text{in} \;\; \mathbb{R}^n, \] in the sub-natural growth…

We investigate selfadjoint $C_0$-semigroups on Euclidean domains satisfying Gaussian upper bounds. Major examples are semigroups generated by second order uniformly elliptic operators with Kato potentials and magnetic fields. We study the…

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…