Related papers: The Kato Square Root Problem on Submanifolds

We consider smooth, complete Riemannian manifolds which are exponentially locally doubling. Under a uniform Ricci curvature bound and a uniform lower bound on injectivity radius, we prove a Kato square root estimate for certain coercive…

We solve the Kato square root problem for general elliptic operators and systems with measurable and complex coefficients on any domain of the Euclidean space. The operators are subject to Dirichlet boundary conditions. We also allow…

We solve the Kato square root problem for second order elliptic systems in divergence form under mixed boundary conditions on Lipschitz domains. This answers a question posed by J.-L. Lions in 1962. To do this we develop a general theory of…

We obtain the Kato square root estimate for second order elliptic operators in divergence form with mixed boundary conditions on an open and possibly unbounded set in $\mathbb{R}^d$ under two simple geometric conditions: The Dirichlet…

The Kato square root problem for divergence form elliptic operators with potential $V : \mathbb{R}^{n} \rightarrow \mathbb{C}$ is the equivalence statement $\left\Vert (L + V)^{\frac{1}{2}} u\right\Vert_{2} \simeq \left\Vert \nabla u…

On a domain $\Omega \subseteq \mathbb{R}^d$ we consider second order elliptic systems in divergence form with bounded complex coefficients, realized via a sesquilinear form with domain $V \subseteq H^1(\Omega)$. Under very mild assumptions…

We study the persistence of quadratic estimates related to the Kato square root problem across a change of metric on smooth manifolds by defining a class of Riemannian-like metrics that are permitted to be of low regularity and degenerate…

We solve the Kato square root problem for bounded measurable perturbations of subelliptic operators on connected Lie groups. The subelliptic operators are divergence form operators with complex bounded coefficients, which may have lower…

We consider the negative Laplacian subject to mixed boundary conditions on a bounded domain. We prove under very general geometric assumptions that slightly above the critical exponent $\frac{1}{2}$ its fractional power domains still…

We consider the Kato square root problem for non-divergence second order elliptic operators $L =- a_{ij} D_iD_j$, and, especially, the normalized adjoints of such operators. In particular, our results are applicable to the case of real…

We obtain the Kato square root property for coupled second-order elliptic systems in divergence form subject to mixed boundary conditions on an open and possibly unbounded set in $\mathbb{R}^n$ under two simple geometric conditions: The…

We prove the Kato conjecture for degenerate elliptic operators in R^n. More precisely, we consider the divergence form operator L_w = -1/w div (wA) grad, where w is a Muckenhoupt A_2 weight and A is a complex valued n x n matrix which is…

We provide a direct proof of a quadratic estimate that plays a central role in the determination of domains of square roots of elliptic operators and, as shown more recently, in some boundary value problems with $L^2$ boundary data. We…

We give a simplified and direct proof of the Kato square root estimate for parabolic operators with elliptic part in divergence form and coefficients possibly depending on space and time in a merely measurable way. The argument relies on…

We consider perturbations of Dirac type operators on complete, connected metric spaces equipped with a doubling measure. Under a suitable set of assumptions, we prove quadratic estimates for such operators and hence deduce that these…

We consider the Kato problem and extensions for degenerate elliptic operators of arbitrary order $2m$ ($m\geq 1$), whose coefficients are measurable, complex-valued and satisfy the G$\mathring{a}$rding inequality with respect to a…

In this paper, we introduce several new secondary invariants for Dirac operators on a complete Riemannian manifold with a uniform positive scalar curvature metric outside a compact set and use these secondary invariants to establish a…

We solve the Kato square root problem for parabolic operators of arbitrary order $2m$ whose coefficients are allowed to depend on both space and time in a merely measurable way and possess boundedness and ellipticity controlled by a…

We prove a Kato square root estimate with anisotropically degenerate matrix coefficients. We do so by doing the harmonic analysis using an auxiliary Riemannian metric adapted to the operator. We also derive $L^2$-solvability estimates for…

We establish refinements of the classical Kato inequality for sections of a vector bundle which lie in the kernel of a natural injectively elliptic first-order linear differential operator. Our main result is a general expression which…