Non-negative Self-adjoint Operator Research Articles

Endpoint boundedness of Riesz transforms on Hardy spaces associated with operators

Let L 1 be a nonnegative self-adjoint operator in L 2(ℝ n ) satisfying the Davies-Gaffney estimates and L 2 a second order divergence form elliptic operator with complex bounded measurable coefficients. A typical example of L 1 is the Schrödinger operator −Δ+V, where Δ is the Laplace operator on ℝ n and $0\le V\in L^{1}_{\mathop{\mathrm{loc}}} ({\mathbb{R}}^{n})$ . Let $H^{p}_{L_{i}}(\mathbb{R}^{n})$ be the Hardy space associated to L i for i∈{1, 2}. In this paper, the authors prove that the Riesz transform $D (L_{i}^{-1/2})$ is bounded from $H^{p}_{L_{i}}(\mathbb{R}^{n})$ to the classical weak Hardy space WH p (ℝ n ) in the critical case that p=n/(n+1). Recall that it is known that $D(L_{i}^{-1/2})$ is bounded from $H^{p}_{L_{i}}(\mathbb{R}^{n})$ to the classical Hardy space H p (ℝ n ) when p∈(n/(n+1), 1].

Sharp spectral stability estimates via the Lebesgue measure of domains for higher order elliptic operators

We prove sharp stability estimates for the variation of the eigenvalues of non-negative self-adjoint elliptic operators of arbitrary even order upon variation of the open sets on which they are defined. These estimates are expressed in terms of the Lebesgue measure of the symmetric difference of the open sets. Both Dirichlet and Neumann boundary conditions are considered.

Notes on boundedness of spectral multipliers on Hardy spaces associated to operators

AbstractLet L be a nonnegative self-adjoint operator on L2 (X), where X is a space of homogeneous type. Assume that L generates an analytic semigroup e–tl whose kernel satisfies the standard Gaussian upper bounds. We prove that the spectral multiplier F(L) is bounded on for 0 < p < 1, the Hardy space associated to operator L, when F is a suitable function.

Notes on boundedness of spectral multipliers on Hardy spaces associated to operators

AbstractLetLbe a nonnegative self-adjoint operator onL2(X), whereXis a space of homogeneous type. Assume thatLgenerates an analytic semigroupe–tlwhose kernel satisfies the standard Gaussian upper bounds. We prove that the spectral multiplierF(L) is bounded onfor 0< p< 1, the Hardy space associated to operatorL, whenFis a suitable function.

Spectral multipliers from [formula omitted] to [formula omitted

Let L be a non-negative self-adjoint operator acting on L 2 ( X ) , where X is a space of homogeneous type. Assume that the heat kernels p t ( x , y ) corresponding to the semigroup e − t L satisfy Gaussian upper bounds but possess no regularity in variables x and y. In this article, we prove a spectral multiplier theorem for F ( L ) from H L 1 ( X ) to L q ( X ) for some 1 ⩽ q ⩽ 2 , if the function F possesses the Sobolev norm of order s with suitable bounds and s > n ( 1 q − 1 2 ) where n is a measure of the dimension of the space. We also study the weighted L p – L q estimates for spectral multiplier theorem.

Square Laplacian perturbed by inverse fourth-power potential. I Self-adjointness (real case)

The self-adjointness of Δ2 + κ|x|−4 (κ>κ0 = κ0(N)) in L2(ℝN) is established as an application of the perturbation theorem in terms of Re(Au, Bεu), u ∈ D(A), for two non-negative self-adjoint operators A, B in a Hilbert space, where the family {Bε}ε>0 is the Yosida approximation of B. A key to the proof lies in a new inequality for the functions ν ∈ L2(ℝN) with |x|2Δν ∈ L2(ℝN) derived by using two real parameters.

ORLICZ–HARDY SPACES ASSOCIATED WITH OPERATORS SATISFYING DAVIES–GAFFNEY ESTIMATES

Let [Formula: see text] be a metric space with doubling measure, L a nonnegative self-adjoint operator in [Formula: see text] satisfying the Davies–Gaffney estimate, ω a concave function on (0, ∞) of strictly lower type pω∈(0, 1] and ρ(t) = t-1/ω-1(t-1) for all t∈(0, ∞). The authors introduce the Orlicz–Hardy space [Formula: see text] via the Lusin area function associated to the heat semigroup, and the BMO-type space [Formula: see text]. The authors then establish the duality between [Formula: see text] and [Formula: see text]; as a corollary, the authors obtain the ρ-Carleson measure characterization of the space [Formula: see text]. Characterizations of [Formula: see text], including the atomic and molecular characterizations and the Lusin area function characterization associated to the Poisson semigroup, are also presented. Let [Formula: see text] and L = -Δ+V be a Schrödinger operator, where [Formula: see text] is a nonnegative potential. As applications, the authors show that the Riesz transform ∇L-1/2is bounded from Hω, L(ℝn) to L(ω). Moreover, if there exist q1, q2∈(0, ∞) such that q1<1<q2and [ω(tq2)]q1is a convex function on (0, ∞), then several characterizations of the Orlicz–Hardy space Hω, L(ℝn), in terms of the Lusin-area functions, the non-tangential maximal functions, the radial maximal functions, the atoms and the molecules, are obtained. All these results are new even when ω(t) = tpfor all t ∈ (0, ∞) and p ∈ (0, 1).

Remarks on the Trotter–Kato Product Formula for Unitary Groups

Let A and B be non-negative self-adjoint operators in a separable Hilbert space such that their form sum C is densely defined. It is shown that the Trotter product formula holds for imaginary parameter values in the L 2-norm, that is, one has $$ \lim_{n\to+\infty} \int\limits^T_{-T} \left\|\left(e^{-itA/n}e^{-itB/n} \right)^nh - e^{-itC}h\right\|^2dt = 0 $$ for each element h of the Hilbert space and any T > 0. This result is extended to the class of holomorphic Kato functions, to which the exponential function belongs. Moreover, for a class of admissible functions: $${\phi(\cdot),\psi(\cdot):{\mathbb R}_+ \longrightarrow {\mathbb C}}$$ , where $${{\mathbb R}_+ := [0,\infty)}$$ , satisfying in addition $${{\Re{\rm e}}\,(\phi(y))\ge 0, {\Im{\rm m}}\,(\phi(y) \le 0}$$ and $${{\Im{\rm m}}\,(\psi(y)) \le 0}$$ for $${y \in {\mathbb R}_+}$$ , we prove that $$ \,\mbox{\rm s-}\hspace{-2pt} \lim_{n\to\infty}(\phi(tA/n)\psi(tB/n))^n = e^{-itC} $$ holds true uniformly on $${[0,T]\ni t}$$ for any T > 0.

Spectral multipliers for Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates

Let $(X,d,\mu)$ be a metric measure space endowed with a distance $d$ and a nonnegative Borel doubling measure $\mu$. Let $L$ be a non-negative self-adjoint operator on $L^2(X)$. Assume that the semigroup $e^{-tL}$ generated by $L$ satisfies the Davies-Gaffney estimates. Let $H_L^p(X)$ be the Hardy space associated with $L$. We prove a Hormander-type spectral multiplier theorem for $L$ on $H_L^p(X)$ for $0 n(1/p - 1/2)$ where $n$ is the dimension of $X$. By interpolation, $m(L)$ is bounded on $H_L^p(X)$ for all $0 < p < \infty$ if $m$ is infinitely differentiable with suitable bounds on its derivatives. We also obtain a spectral multiplier theorem on $L^p$ spaces with appropriate weights in the reverse Holder class.

A Substructuring Domain Decomposition Scheme for Unsteady Problems

AbstractDomain decomposition methods are used for the approximate solution of boundary-value problems for partial differential equations on parallel computing systems. Specific features of unsteady problems are fully taken into account in iteration-free domain decomposition schemes. Regionally-additive schemes are based on various classes of splitting schemes. In this paper we highlight a class of domain decomposition schemes which are based on the partition of the initial domain into subdomains with common boundary nodes. Using a partition of unity we construct and analyze unconditionally stable schemes for domain decomposition based on a two-component splitting: the problem within each subdomain and the problem at their boundaries. As an example we consider a Cauchy problem of first or second order in time with a non-negative self-adjoint second order operator in space. The theoretical discussion is supplemented with the numerical solution of a model problem for a two-dimensional parabolic equation.

Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates

Let $X$ be a metric space with doubling measure, and $L$ be a non-negative, self-adjoint operator satisfying Davies-Gaffney bounds on $L^2(X)$. In this article the authors present a theory of Hardy and BMO spaces associated to $L$, including an atomic (or molecular) decomposition, square function characterization, and duality of Hardy and BMO spaces. Further specializing to the case that $L$ is a Schrodinger operator on $\mathbb{R}^n$ with a non-negative, locally integrable potential, the authors establish additional characterizations of such Hardy spaces in terms of maximal functions. Finally, they define Hardy spaces $H^p_L(X)$ for $p>1$, which may or may not coincide with the space $L^p(X)$, and show that they interpolate with $H^1_L(X)$ spaces by the complex method.

Weighted norm inequalities, Gaussian bounds and sharp spectral multipliers

Let L be a non-negative self-adjoint operator acting on L 2 ( X ) where X is a space of homogeneous type. Assume that L generates a holomorphic semigroup e − t L whose kernels p t ( x , y ) have Gaussian upper bounds but there is no assumption on the regularity in variables x and y. In this article, we study weighted L p -norm inequalities for spectral multipliers of L. We show that sharp weighted Hörmander-type spectral multiplier theorems follow from Gaussian heat kernel bounds and appropriate L 2 estimates of the kernels of the spectral multipliers. These results are applicable to spectral multipliers for large classes of operators including Laplace operators acting on Lie groups of polynomial growth or irregular non-doubling domains of Euclidean spaces, elliptic operators on compact manifolds and Schrödinger operators with non-negative potentials.

Monotone convergence theorems for semi-bounded operators and forms with applications

AbstractLet Hn be a monotone sequence of non-negative self-adjoint operators or relations in a Hilbert space. Then there exists a self-adjoint relation H∞ such that Hn converges to H∞ in the strong resolvent sense. This result and related limit results are explored in detail and new simple proofs are presented. The corresponding statements for monotone sequences of semi-bounded closed forms are established as immediate consequences. Applications and examples, illustrating the general results, include sequences of multiplication operators, Sturm–Liouville operators with increasing potentials, forms associated with Kreĭn–Feller differential operators, singular perturbations of non-negative self-adjoint operators and the characterization of the Friedrichs and Kreĭn–von Neumann extensions of a non-negative operator or relation.

A uniqueness result for Kirchhoff equations with non-Lipschitz nonlinear term

We consider the second order Cauchy problem u ″ + m ( | A 1 / 2 u | 2 ) A u = 0 , u ( 0 ) = u 0 , u ′ ( 0 ) = u 1 , where m : [ 0 , + ∞ ) → [ 0 , + ∞ ) is a continuous function, and A is a self-adjoint nonnegative operator with dense domain on a Hilbert space. It is well known that this problem admits local-in-time solutions provided that u 0 and u 1 are regular enough, depending on the continuity modulus of m. It is also well known that the solution is unique when m is locally Lipschitz continuous. In this paper we prove that if either 〈 A u 0 , u 1 〉 ≠ 0 , or | A 1 / 2 u 1 | 2 ≠ m ( | A 1 / 2 u 0 | 2 ) | A u 0 | 2 , then the local solution is unique even if m is not Lipschitz continuous.

THE REPRESENTATION OF STRONGLY CONTINUOUS OPERATOR SEMIGROUPS ON VECTOR BUNDLES AS HIDA DISTRIBUTIONS

Let E → M be a vector bundle over a compact manifold. In this short note we prove that the Schrödinger semigroup (complex heat semigroup) associated to a self-adjoint (non-negative self-adjoint) operator in ΓL2(M, E) corresponds to a ΓL2(M, E)*-valued Hida distribution, which depends continuously (holomorphically) on the time parameter.

Spectral gap global solutions for degenerate Kirchhoff equations

We consider the second order Cauchy problem u ″ + m ( | A 1 / 2 u | 2 ) A u = 0 , u ( 0 ) = u 0 , u ′ ( 0 ) = u 1 , where m : [ 0 , + ∞ ) → [ 0 , + ∞ ) is a continuous function, and A is a self-adjoint nonnegative operator with dense domain on a Hilbert space. It is well known that this problem admits local-in-time solutions provided that u 0 and u 1 are regular enough, depending on the continuity modulus of m , and on the strict/weak hyperbolicity of the equation. We prove that for such initial data ( u 0 , u 1 ) there exist two pairs of initial data ( u ¯ 0 , u ¯ 1 ) , ( u ̂ 0 , u ̂ 1 ) for which the solution is global, and such that u 0 = u ¯ 0 + u ̂ 0 , u 1 = u ¯ 1 + u ̂ 1 . This is a byproduct of a global existence result for initial data with a suitable spectral gap, which extends previous results obtained in the strictly hyperbolic case with a smooth nonlinearity m .

On symmetries in the theory of finite rank singular perturbations

For a nonnegative self-adjoint operator A 0 acting on a Hilbert space H singular perturbations of the form A 0 + V , V = ∑ 1 n b i j 〈 ψ j , ⋅ 〉 ψ i are studied under some additional requirements of symmetry imposed on the initial operator A 0 and the singular elements ψ j . A concept of symmetry is defined by means of a one-parameter family of unitary operators U that is motivated by results due to R.S. Phillips. The abstract framework to study singular perturbations with symmetries developed in the paper allows one to incorporate physically meaningful connections between singular potentials V and the corresponding self-adjoint realizations of A 0 + V . The results are applied for the investigation of singular perturbations of the Schrödinger operator in L 2 ( R 3 ) and for the study of a (fractional) p-adic Schrödinger type operator with point interactions.

Spectral stability of nonnegative self-adjoint operators

The survey is devoted to spectral stability problems for uniformly elliptic differential operators under the variation of the domain and to the accompanying estimates for the difference of the eigenvalues. Two approaches to the problem are discussed in detail. In the first one it is assumed that the domain is transformed by means of a transformation of a certain class, and the spectral stability with respect to this transformation is investigated. The second approach is based on the notion of a transition operator and allows direct comparison of the eigenvalues on domains which are close in that or another sense.

Convergence Rates For Lavrentiev-Type Regularization In Hilbert Scales

AbstractFor solving linear ill-posed problems with noisy data regularization methods are required. We analyze a simplified regularization scheme in Hilbert scales for operator equations with nonnegative self-adjoint operators. By exploiting the op-erator monotonicity of certain functions, order-optimal error bounds are derived that characterize the accuracy of the regularized approximations. These error bounds have been obtained under general smoothness conditions.

Sharp estimates for large coupling convergence with applications to Dirichlet operators

Let H be a nonnegative selfadjoint operator, E the closed quadratic form associated with H, and P a nonnegative quadratic form such that E + P is closed and D ( P ) ⊃ D ( H ) . For every β > 0 let H β be the selfadjoint operator associated with E + β P . The pairs ( H , P ) satisfying L ( H , P ) : = lim inf β → ∞ β ‖ ( H β + 1 ) −1 − lim β ′ → ∞ ( H β ′ + 1 ) −1 ‖ < ∞ are characterized. A sufficient condition for convergence of the operators ( H β + 1 ) −1 within a Schatten–von Neumann class of finite order is derived. It is shown that L ( H , P ) = 1 , if E is a regular conservative Dirichlet form with the strong local property and P the killing form corresponding to the equilibrium measure of a closed set with finite capacity and nonempty interior. An example is given where L ( H , P ) is finite, H is a regular Dirichlet operator and P the killing form corresponding to a measure which has infinite mass and a support with infinite capacity.