Kato Square Research Articles

The weighted Kato square root problem of elliptic operators having a BMO anti-symmetric part

The weighted Kato square root problem of elliptic operators having a BMO anti-symmetric part

On Kato square root problem for a parabolic operator and applications

On Kato square root problem for a parabolic operator and applications

Kato square root problem for degenerate elliptic operators on bounded Lipschitz domains

Let n≥2, w be a Muckenhoupt A2(Rn) weight, Ω a bounded Lipschitz domain of Rn, and L:=−w−1div(A∇⋅) the degenerate elliptic operator on Ω with the Dirichlet or the Neumann boundary condition. In this article, the authors establish the following weighted Lp estimate for the Kato square root of L:‖L1/2(f)‖Lp(Ω,vw)∼‖∇f‖Lp(Ω,vw) for any f∈W01,p(Ω,vw) when L satisfies the Dirichlet boundary condition, or, for any f∈W1,p(Ω,vw) with ∫Ωf(x)dx=0 when L satisfies the Neumann boundary condition, where p is in an interval including 2, v belongs to both some Muckenhoupt weight class and the reverse Hölder class with respect to w, W01,p(Ω,vw) and W1,p(Ω,vw) denote the weighted Sobolev spaces on Ω, and the positive equivalence constants are independent of f. As a corollary, under some additional assumptions on w, via letting v:=w−1, the unweighted L2 estimate for the Kato square root of L that ‖L1/2(f)‖L2(Ω)∼‖∇f‖L2(Ω) for any f∈W01,2(Ω) when L satisfies the Dirichlet boundary condition, or, for any f∈W1,2(Ω) with ∫Ωf(x)dx=0 when L satisfies the Neumann boundary condition, are obtained. Moreover, as applications of these unweighted L2 estimates, the unweighted L2 regularity estimates for the weak solutions of the corresponding degenerate parabolic equations in Ω with the Dirichlet or the Neumann boundary condition are also established.

Weighted Norm Inequalities for Commutators of the Kato Square Root of Second Order Elliptic Operators on ℝn

Weighted Norm Inequalities for Commutators of the Kato Square Root of Second Order Elliptic Operators on ℝn

On the Kato square root problem

In the infinite-dimensional separable complex Hilbert space we construct new abstract examples of unbounded maximal accretive and maximal sectorial operators B for which domB12≠domB⁎12. New criterions for the equality are established.

The Kato square root problem on locally uniform domains

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 Rd under two simple geometric conditions: The Dirichlet boundary part is Ahlfors–David regular and a quantitative connectivity property in the spirit of locally uniform domains holds near the Neumann boundary part. This improves upon all existing results even in the case of pure Dirichlet or Neumann boundary conditions. We also treat elliptic systems with lower order terms. As a side product we establish new regularity results for the fractional powers of the Laplacian with boundary conditions in our geometric setup.

L$^2$ well-posedness of boundary value problems for parabolic systems with measurable coefficients

We prove the first positive results concerning boundary value problems in the upper half-space for second order parabolic systems only assuming measurability and some transversal regularity in the coefficients of the elliptic part. To do so, we introduce and develop a first order strategy by means of a parabolic Dirac operator at the boundary to obtain, in particular, Green’s representation for solutions in natural classes involving square functions and non-tangential maximal functions, well-posedness results with data in L ^2 -Sobolev spaces together with invertibility of layer potentials, and perturbation results. On the way, we solve the Kato square root problem for parabolic operators with coefficients of the elliptic part depending measurably on all variables. The major new challenge, compared to the earlier results by one of us under time and transversal independence of the coefficients, is to handle non-local half-order derivatives in time which are unavoidable in our situation.

The Kato Square Root Problem for Divergence Form Operators with Potential

The Kato square root problem for divergence form elliptic operators with potential V : {mathbb {R}}^{n} rightarrow {mathbb {C}} is the equivalence statement left| left( L + V right) ^{frac{1}{2}} u right| _{2} simeq left| nabla u right| _{2} + left| V^{frac{1}{2}} u right| _{2}, where L + V := - mathrm {div} (A nabla ) + V and the perturbation A is an L^{infty } complex matrix-valued function satisfying an accretivity condition. This relation is proved for any potential with range contained in some positive sector and satisfying left| left| V right| ^{frac{alpha }{2}} u right| _{2} + left| (-Delta )^{frac{alpha }{2}}u right| _{2} lesssim left| left( left| V right| - Delta right) ^{frac{alpha }{2}}u right| _{2} for all u in D(left| V right| - Delta ) and some alpha in (1,2]. The class of potentials that will satisfy such a condition is known to contain the reverse Hölder class RH_{2} and L^{frac{n}{2}}left( {mathbb {R}}^{n} right) in dimension n > 4. To prove the Kato estimate with potential, a non-homogeneous version of the framework introduced by Axelsson, Keith and McIntosh for proving quadratic estimates is developed. In addition to applying this non-homogeneous framework to the scalar Kato problem with zero-order potential, it will also be applied to the Kato problem for systems of equations with zero-order potential.

Lp-gradient estimates for the commutators of the Kato square roots of second-order elliptic operators on ℝn

Let $L=-{\rm~div}(A\nabla)$ be a second-order divergent-form elliptic operator, where $A$ is an accretive $n\times~n$ matrix with bounded and measurable complex coefficients on ${\Bbb~R}^n.$ Herein, we prove that the commutator $[b,\sqrt{L}]$ of the Kato square root $\sqrt{L}$ and $b$ with $\nabla~b\in~L^n({\Bbb~R}^n)~(n>2)$, is bounded from the homogenous Sobolev space $\dot{L}^{p}_{1}(\mathbb{R}^{n})$ to $L^{p}(\mathbb{R}^{n})$ $(p_{-}(L)<p<p_{+}(L))$

Nonlocal problems and functional-differential equations: theoretical aspects and applications to mathematical modelling

This paper presents a review of results on nonlocal problems, functional-differential equations, and their applications, obtained during several last years. The following research areas are covered: the Kato square root problem for functional-differential operators, Vlasov equations and their applications to the modelling of high-temperature plasma, specific properties of differential-difference equations with incommensurable translations, degenerate functional-differential equations and their applications, functional-differential equations with contractions and extensions of independent variables, and operational methods for parabolic and elliptic functional-differential equations.

Kato square root problem with unbounded leading coefficients

We prove the Kato conjecture for elliptic operators, L = − ∇ ⋅ ( ( A + D ) ∇ ) L=-\nabla \cdot \left ((\mathbf A+\mathbf D)\nabla \ \right ) , with A \mathbf A a complex measurable bounded coercive matrix and D \mathbf D a measurable real-valued skew-symmetric matrix in R n \mathbb {R}^n with entries in B M O ( R n ) BMO(\mathbb {R}^n) ; i.e., the domain of L \sqrt {L}\, is the Sobolev space H ˙ 1 ( R n ) \dot H^1(\mathbb {R}^n) in any dimension, with the estimate ‖ L f ‖ 2 ≲ ‖ ∇ f ‖ 2 \|\sqrt {L}\, f\|_2 \lesssim \| \nabla f\|_2 .

Elliptic differential‐difference operators with degeneration and the Kato square root problem

AbstractWe prove correctness of the Kato square root conjecture for elliptic differential‐difference operators with degeneration.

Lp-estimates for the square root of elliptic systems with mixed boundary conditions

This article focuses on Lp-estimates for the square root of elliptic systems of second order in divergence form on a bounded domain. We treat complex bounded measurable coefficients and allow for mixed Dirichlet/Neumann boundary conditions on domains beyond the Lipschitz class. If there is an associated bounded semigroup on Lp0, then we prove that the square root extends for all p∈(p0,2) to an isomorphism between a closed subspace of W1,p carrying the boundary conditions and Lp. This result is sharp and extrapolates to exponents slightly above 2. As a byproduct, we obtain an optimal p-interval for the bounded H∞-calculus on Lp. Estimates depend holomorphically on the coefficients, thereby making them applicable to questions of (non-autonomous) maximal regularity and optimal control. For completeness we also provide a short summary on the Kato square root problem in L2 for systems with lower order terms in our setting.

Об одном свойстве регулярно аккретивных дифференциально-разностных операторов с вырождением

We consider elliptic differential-difference operators with degeneration in a bounded domain with piecewise smooth boundary. It is proved that these operators are regular accretive and satisfy the Kato square root conjecture.

On the Kato problem and extensions for degenerate elliptic operators

We study the Kato problem for degenerate divergence form operators. This was begun by Cruz-Uribe and Rios who proved that given an operator $L_w=-w^{-1}{\rm div}(A\nabla)$, where $w\in A_2$ and $A$ is a $w$-degenerate elliptic measure (i.e, $A=w\,B$ with $B$ an $n\times n$ bounded, complex-valued, uniformly elliptic matrix), then $L_w$ satisfies the weighted estimate $\|\sqrt{L_w}f\|_{L^2(w)}\approx\|\nabla f\|_{L^2(w)}$. Here we solve the $L^2$-Kato problem: under some additional conditions on the weight $w$, the following unweighted $L^2$-Kato estimates hold $$ \|L_w^{1/2}f\|_{L^2(\mathbb{R}^n)}\approx\|\nabla f\|_{L^2(\mathbb{R}^n)}. $$ This extends the celebrated solution to the Kato conjecture by Auscher, Hofmann, Lacey, McIntosh, and Tchamitchian, allowing the differential operator to have some degeneracy in its ellipticity. For example, we consider the family of operators $L_\gamma=-|x|^{\gamma}{\rm div}(|x|^{-\gamma}B(x)\nabla)$, where $B$ is any bounded, complex-valued, uniformly elliptic matrix. We prove that there exists $\epsilon>0$, depending only on dimension and the ellipticity constants, such that $$ \|L_\gamma^{1/2}f\|_{L^2(\mathbb{R}^n)}\approx\|\nabla f\|_{L^2(\mathbb{R}^n)}, \qquad -\epsilon<\gamma<\frac{2\,n}{n+2}. $$ This gives a range of $\gamma$'s for which the classical Kato square root $\gamma=0$ is an interior point. Our main results are obtained as a consequence of a rich Calder\'on-Zygmund theory developed for some operators associated with $L_w$. These results, which are of independent interest, establish estimates on $L^p(w)$, and also on $L^p(v\,dw)$ with $v\in A_\infty(w)$, for the associated semigroup, its gradient, the functional calculus, the Riesz transform, and square functions. As an application, we solve some unweighted $L^2$-Dirichlet, Regularity and Neumann boundary value problems for degenerate elliptic operators.

Conical square functions for degenerate elliptic operators

Abstract The aim of the present paper is to study the boundedness of different conical square functions that arise naturally from second-order divergence form degenerate elliptic operators. More precisely, let L w = - w - 1 ⁢ div ( w ⁢ A ⁢ ∇ ) {L_{w}=-w^{-1}\mathop{\rm div}(wA\nabla)} , where w ∈ A 2 {w\in A_{2}} and A is an n × n {n\times n} bounded, complex-valued, uniformly elliptic matrix. Cruz-Uribe and Rios solved the L 2 ⁢ ( w ) {L^{2}(w)} -Kato square root problem obtaining that L w {\sqrt{L_{w}}} is equivalent to the gradient on L 2 ⁢ ( w ) {L^{2}(w)} . The same authors in collaboration with the second named author of this paper studied the L p ⁢ ( w ) {L^{p}(w)} -boundedness of operators that are naturally associated with L w {L_{w}} , such as the functional calculus, Riesz transforms, and vertical square functions. The theory developed admitted also weighted estimates (i.e., estimates in L p ⁢ ( v ⁢ d ⁢ w ) {L^{p}(v\,dw)} for v ∈ A ∞ ⁢ ( w ) {v\in A_{\infty}(w)} ), and in particular a class of “degeneracy” weights w was found in such a way that the classical L 2 {L^{2}} -Kato problem can be solved. In this paper, continuing this line of research, and also that originated in some recent results by the second and third named authors of the current paper, we study the boundedness on L p ⁢ ( w ) {L^{p}(w)} and on L p ⁢ ( v ⁢ d ⁢ w ) {L^{p}(v\,dw)} , with v ∈ A ∞ ⁢ ( w ) {v\in A_{\infty}(w)} , of the conical square functions that one can construct using the heat or Poisson semigroup associated with L w {L_{w}} . As a consequence of our methods, we find a class of degeneracy weights w for which L 2 {L^{2}} -estimates for these conical square functions hold. This opens the door to the study of weighted and unweighted Hardy spaces and of boundary value problems associated with L w {L_{w}} .

Riesz continuity of the Atiyah–Singer Dirac operator under perturbations of the metric

We prove that the Atiyah-Singer Dirac operator ${\mathrm D}_{\mathrm g}$ in ${\mathrm L}^2$ depends Riesz continuously on ${\mathrm L}^{\infty}$ perturbations of complete metrics ${\mathrm g}$ on a smooth manifold. The Lipschitz bound for the map ${\mathrm g} \to {\mathrm D}_{\mathrm g}(1 + {\mathrm D}_{\mathrm g}^2)^{-\frac{1}{2}}$ depends on bounds on Ricci curvature and its first derivatives as well as a lower bound on injectivity radius. Our proof uses harmonic analysis techniques related to Calder\'on's first commutator and the Kato square root problem. We also show perturbation results for more general functions of general Dirac-type operators on vector bundles.

Weighted $L^p$ estimates of Kato square roots associated to degenerate elliptic operators

Let w be a Muckenhoupt A2(Rn) weight and Lw := −w−1 div(A∇) the degenerate elliptic operator on the Euclidean space Rn, n ≥ 2. In this article, the authors establish some weighted Lp estimates of Kato square roots associated to the degenerate elliptic operators Lw. More precisely, the authors prove that, for w ∈ Ap(Rn), p ∈ (2n n+1 , 2] and any f ∈ C∞c (Rn), kL 1/2 w (f)kLp(w,Rn) ∼ k∇fkLp(w,Rn), where C∞c (Rn) denotes the set of all infinitely differential functions with compact supports and the implicit equivalent positive constants are independent of f.

Continuity of solutions to space-varying pointwise linear elliptic equations

We consider pointwise linear elliptic equations of the form $\mathrm{L}_x u_x = \eta_x$ on a smooth compact manifold where the operators $\mathrm{L}_x$ are in divergence form with real, bounded, measurable coefficients that vary in the space variable $x$. We establish $\mathrm{L}^{2}$-continuity of the solutions at $x$ whenever the coefficients of $\mathrm{L}_x$ are $\mathrm{L}^{\infty}$-continuous at $x$ and the initial datum is $\mathrm{L}^{2}$-continuous at $x$. This is obtained by reducing the continuity of solutions to a homogeneous Kato square root problem. As an application, we consider a time evolving family of metrics $\mathrm{g}_t$ that is tangential to the Ricci flow almost-everywhere along geodesics when starting with a smooth initial metric. Under the assumption that our initial metric is a rough metric on $\mathcal{M}$ with a $\mathrm{C}^{1}$ heat kernel on a "non-singular" nonempty open subset $\mathcal{N}$, we show that $x \mapsto \mathrm{g}_t(x)$ is continuous whenever $x \in \mathcal{N}$.

Mixed boundary value problems on cylindrical domains

We study second-order divergence-form systems on half-infinite cylindrical domains with a bounded and possibly rough base, subject to homogeneous mixed boundary conditions on the lateral boundary and square integrable Dirichlet, Neumann, or regularity data on the cylinder base. Assuming that the coefficients $A$ are close to coefficients $A_0$ that are independent of the unbounded direction with respect to the modified Carleson norm of Dahlberg, we prove a priori estimates and establish well-posedness if $A_0$ has a special structure. We obtain a complete characterization of weak solutions whose gradient either has an $L^2$-bounded non-tangential maximal function or satisfies a Lusin area bound. To this end, we combine the first-order approach to elliptic systems with the Kato square root estimate for operators with mixed boundary conditions.