Non-negative Self-adjoint Operator Research Articles

Maximal function characterizations of Hardy spaces associated with both non-negative self-adjoint operators satisfying Gaussian estimates and ball quasi-Banach function spaces

Maximal function characterizations of Hardy spaces associated with both non-negative self-adjoint operators satisfying Gaussian estimates and ball quasi-Banach function spaces

Hardy Spaces Associated with Non-negative Self-adjoint Operators and Ball Quasi-Banach Function Spaces on Doubling Metric Measure Spaces and Their Applications

Hardy Spaces Associated with Non-negative Self-adjoint Operators and Ball Quasi-Banach Function Spaces on Doubling Metric Measure Spaces and Their Applications

On sharp estimates for Schrödinger groups of fractional powers of nonnegative self-adjoint operators

Let L be a non negative, selfadjoint operator on L2(X), where X is a metric space endowed with a doubling measure. Consider the Schrödinger group for fractional powers of L. If the heat flow e−tL satisfies suitable conditions of Davies–Gaffney type, we obtain the following estimate in Hardy spaces associated to L:‖(I+L)−β/2eiτLγ/2f‖HLp(X)≤C(1+|τ|)nsp‖f‖HLp(X) where p∈(0,1], γ∈(0,1], β/γ=n|12−1p|=nsp and τ∈R.If in addition e−tL satisfies a localized Lp0→L2 polynomial estimate for some p0∈[1,2), we obtain‖(I+L)−β/2eiτLγ/2f‖p0,∞≤C(1+|τ|)nsp0‖f‖p0,∀τ∈R, provided 0<γ≠1, β/γ=n|12−1p|=nsp and τ∈R. By interpolation, the second estimate implies also, for all p∈(p0,p0′), the strong (p,p) type estimate‖(I+L)−β/2eiτLγ/2f‖p≤C(1+|τ|)nsp‖f‖p. The applications of our theory span a diverse spectrum, ranging from the Schrödinger operator with an inverse square potential to the Dirichlet Laplacian on open domains. It showcases the effectiveness of our theory across various settings.

New Atomic Decomposition for Besov Type Space dot{B}^0_{1,1} Associated with Schrödinger Type Operators

Let (X, d, mu ) be a space of homogeneous type. Let L be a nonnegative self-adjoint operator on L^2(X) satisfying certain conditions on the heat kernel estimates which are motivated from the heat kernel of the Schrödinger operator on mathbb {R}^n. The main aim of this paper is to prove a new atomic decomposition for the Besov space dot{B}^{0, L}_{1,1}(X) associated with the operator L. As a consequence, we prove the boundedness of the Riesz transform associated with L on the Besov space dot{B}^{0, L}_{1,1}(X).

Sharp endpoint estimates for Schrödinger groups on Hardy spaces

Let L be a non-negative self-adjoint operator acting on L2(X) where X is a space of homogeneous type with a dimension n. Suppose that the heat kernel of L satisfies the Davies-Gaffney estimates of order m≥2. Let HL1(X) be the Hardy space associated with L. In this paper we obtain the sharp endpoint estimate for the Schrödinger group eitL associated with L such that‖(I+L)−n/2eitLf‖L1(X)+‖(I+L)−n/2eitLf‖HL1(X)≤C(1+|t|)n/2‖f‖HL1(X),t∈R for some constant C=C(n,m)>0 independent of t. We further apply our result to provide the sharp estimate for Schrödinger group of the Kohn Laplacian □b on polynomial model domains treated by Nagel–Stein [41], where e−t□b satisfies only the second order Davies-Gaffney estimates.Moreover, when the heat kernel of L satisfies a Gaussian upper bound, by a duality and interpolation argument, it gives a new proof of a recent result of [13] for sharp endpoint Lp-Sobolev bound for eitL:‖(I+L)−seitLf‖Lp(X)≤C(1+|t|)s‖f‖Lp(X),t∈R,s≥n|12−1p| for every 1<p<∞, which extends the classical results due to Miyachi ([39,40]) for the Laplacian on the Euclidean space Rn.

Boundedness of Oscillation and Variation of Semigroups in Musielak–Orlicz–Hardy Spaces

In this paper, let $$({\cal X},d,\mu)$$ be a space of homogeneous type, L be a non-negative self-adjoint operator on $${L^2}({\cal X})$$ , and the kernels of the semigroup {(tL)me−tL}t>0 satisfy the Gaussian upper bound estimates for m ∈ ℤ+ ≔ {0} ∪ ℤ. The author shows that the oscillation operator and the variation operator of the semigroup {(tL)me−tL}t>0 are bounded from Musielak–Orlicz–Hardy space $${H_{\varphi,L}}({\cal X})$$ to Musielak–Orlicz space $${L^\varphi}({\cal X})$$ , where φ is a growth function.

Variable Besov–Morrey Spaces Associated with Operators

Let (X,d,μ) be a space of homogenous type and L be a non-negative self-adjoint operator on L2(X) with heat kernels satisfying Gaussian upper bounds. In this paper, we introduce the variable Besov–Morrey space associated with the operator L and prove that this space can be characterized via the Peetre maximal functions. Then, we establish its atomic decomposition.

Weak Type (1,1) Bounds for Schrödinger Groups

Let L be a nonnegative self-adjoint operator acting on L2(X), where X is a space of homogeneous type of dimension n. Suppose that the heat kernel of L satisfies a Gaussian upper bound. It is known that the operator (I+L)−seitL is bounded on Lp(X) for s>n|1/2−1/p| and p∈(1,∞) (see, e.g., [7, 22, 33]). The index s=n|1/2−1/p| was only obtained recently in [9, 10], and this range of s is sharp since it is precisely the range known in the case where L is the Laplace operator Δ on X=Rn [30]. In this paper, we establish that for p=1, the operator (1+L)−n/2eitL is of weak type (1,1), that is, there is a constant C, independent of t and f, such that μ({x:|(I+L)−n/2eitLf(x)|>λ})≤Cλ−1(1+|t|)n/2‖f‖ L1(X),t∈R (for λ>0 when μ(X)=∞ and λ>μ(X)−1‖f‖L1(X) when μ(X)<∞). Moreover, we also show that the index n/2 is sharp when L is the Laplacian on Rn by providing an example. Our results are applicable to Schrödinger groups for large classes of operators including elliptic operators on compact manifolds, Schrödinger operators with nonnegative potentials and Laplace operators acting on Lie groups of polynomial growth or irregular nondoubling domains of Euclidean spaces.

On a nonlocal problem for the first-order differential-operator equations

In this work, we study the spaces of generalised elements identified with formal Fourier series and constructed via a non-negative self-adjoint operator in Hilbert space. The spectrum of this operator is purely discrete. For a differential-operator equation of the first order, we formulate a nonlocal multipoint by time problem if the corresponding condition is satisfied in a positive or negative space that is constructed via such operator; such problem can be treated as a generalisation of an abstract Cauchy problem for the specified differential-operator equation. The correct solvability of the aforementioned problem is proven, a fundamental solution is constructed, and its structure and properties are studied. The solution is represented as an abstract convolution of a fundamental solution with a boundary element. This boundary element is used to formulate a multipoint condition, and it is a linear continuous functional defined in the space of main elements. Furthermore, this solution satisfies multipoint condition in a negative space that is adjoint with a corresponding positive space of elements.

Sharp Estimates for Schrödinger Groups on Hardy Spaces for 0

Let X be a space of homogeneous type with the doubling order n. Let L be a nonnegative self-adjoint operator on L^2(X) and suppose that the kernel of e^{-tL} satisfies a Gaussian upper bound. This paper shows that for 0<ple 1 and s=n(1/p-1/2), ‖(I+L)-seitLf‖HLp(X)≲(1+|t|)s‖f‖HLp(X)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned}\\Vert (I+L)^{-s}e^{itL}f\\Vert _{H^p_L(X)} \\lesssim (1+|t|)^{s}\\Vert f\\Vert _{H^p_L(X)} \\end{aligned}$$\\end{document}for all tin {mathbb {R}}, where H^p_L(X) is the Hardy space associated to L. This recovers the classical results in the particular case when L=-Delta and extends a number of known results.

Imaginary power operators on Hardy spaces

Let ( X , d , μ ) (X,d, \mu ) be an Ahlfors n n -regular metric measure space. Let L \mathcal {L} be a non-negative self-adjoint operator on L 2 ( X ) L^2(X) with heat kernel satisfying Gaussian estimate. Assume that the kernels of the spectral multiplier operators F ( L ) F(\mathcal {L}) satisfy an appropriate weighted L 2 L^2 estimate. By the spectral theory, we can define the imaginary power operator L i s , s ∈ R \mathcal {L}^{is}, s\in \mathbb R , which is bounded on L 2 ( X ) L^2(X) . The main aim of this paper is to prove that for any p ∈ ( 0 , ∞ ) p \in (0,\infty ) , ‖ L i s f ‖ H L p ( X ) ≤ C ( 1 + | s | ) n | 1 / p − 1 / 2 | ‖ f ‖ H L p ( X ) , s ∈ R , \begin{equation*} \big \|\mathcal {L}^{is} f\big \|_{H^p_{\mathcal {L}}(X)} \leq C (1+|s|)^{n|1/p-1/2|} \|f\|_{H^p_{\mathcal {L}}(X)}, \quad s \in \mathbb {R}, \end{equation*} where H L p ( X ) H^p_\mathcal {L}(X) is the Hardy space associated to L \mathcal {L} , and C C is a constant independent of s s . Our result applies to sub-Laplaicans on stratified Lie groups and Hermite operators on R n \mathbb {R}^n with n ≥ 2 n\ge 2 .

Almost everywhere convergence of spectral sums for self-adjoint operators

Let L L be a non-negative self-adjoint operator acting on the space L 2 ( X ) L^2(X) , where X X is a positive measure space. Let L = ∫ 0 ∞ λ d E L ( λ ) { L}=\int _0^{\infty } \lambda dE_{ L}({\lambda }) be the spectral resolution of L L and S R ( L ) f = ∫ 0 R d E L ( λ ) f S_R({ L})f=\int _0^R dE_{ L}(\lambda ) f denote the spherical partial sums in terms of the resolution of L { L} . In this article we give a sufficient condition on L L such that lim R → ∞ S R ( L ) f ( x ) = f ( x ) ,     a.e. \begin{equation*} \lim _{R\rightarrow \infty } S_R({ L})f(x) =f(x),\ \ \text {a.e.} \end{equation*} for any f f such that log ⁡ ( 2 + L ) f ∈ L 2 ( X ) \operatorname {log}(2+L) f\in L^2(X) . These results are applicable to large classes of operators including Dirichlet operators on smooth bounded domains, the Hermite operator and Schrödinger operators with inverse square potentials.

Oracle inequalities and upper bounds for kernel density estimators on manifolds and more general metric spaces

We prove oracle inequalities and upper bounds for kernel density estimators on a very broad class of metric spaces. Precisely we consider the setting of a doubling measure metric space in the presence of a non-negative self-adjoint operator whose heat kernel enjoys Gaussian regularity. Many classical settings like Euclidean spaces, spheres, balls, cubes as well as general Riemannian manifolds, are contained in our framework. Moreover the rate of convergence we achieve is the optimal one in these special cases. Finally we provide the general methodology of constructing the proper kernels when the manifold under study is given and we give precise examples for the case of the sphere.

Higher-order asymptotic expansion for abstract linear second-order differential equations with time-dependent coefficients

This paper is concerned with the asymptotic expansion of solutions to the initial-value problem of u″(t)+Au(t)+b(t)u′(t)=0 in a Hilbert space with a nonnegative selfadjoint operator A and a coefficient b(t)∼(1+t)−β(−1<β<1). In the case b(t)≡1, it is known that the higher-order asymptotic profiles are determined via a family of first-order differential equations of the form v′(t)+Av(t)=Fn(t) (Sobajima (2021) [10]). For the time-dependent case, it is only known that the asymptotic behavior of such a solution is given by the one of b(t)v′(t)+Av(t)=0. The result of this paper is to find the equations for all higher-order asymptotic profiles. It is worth noticing that the equation for n-th order profile u˜n is given via v′(t)+mn(t)Av(t)=Fn(t) which coefficient mn (time-scale) differs each other.

Littlewood–Paley Characterization for Musielak–Orlicz–Hardy Spaces Associated with Self-Adjoint Operators

Let X , d , μ be a metric measure space endowed with a metric d and a non-negative Borel doubling measure μ . Let L be a non-negative self-adjoint operator on L 2 X . Assume that the (heat) kernel associated to the semigroup e − t L satisfies a Gaussian upper bound. In this paper, we prove that the Musielak–Orlicz–Hardy space H φ , L X associated with L in terms of the Lusin-area function and the Musielak–Orlicz–Hardy space H L , G , φ X associated with L in terms of the Littlewood–Paley function coincide and their norms are equivalent. To do this, we first establish the discrete characterization of these two spaces. It improves the known results in the literature.

Continuous characterizations of inhomogeneous Besov and Triebel-Lizorkin spaces associated to non-negative self-adjoint operators

Let $$(M, \rho , \mu )$$ be a metric measure space satisfying the doubling, reverse doubling and noncollapsing conditions, and let $$\mathscr {L}$$ be a nonnegative self-adjoint operator acting on $$L^2 (M, d\mu )$$ whose heat kernel satisfies the small-time Gaussian upper bound, Hölder continuity and Markov property. In this paper, we establish new characterizations of the “classical” and “nonclassical” Besov and Triebel-Lizorkin spaces associated to $$\mathscr {L}$$ introduced by Kerkyacharian and Petrushev. More precisely, we obtain characterizations of these spaces in terms of continuous Littlewood-Paley and Lusin functions associated to the heat semigroup generated by $$\mathscr {L}$$ , for complete range of indices. This extends related known results in the classical Euclidean setting to our general setting, and extends corresponding results in (Trans Am Math Soc, 367:121–189, 2015) to complete range of indices.

Spectral multiplier theorem and sub-Gaussian heat kernel estimates

We study general spectral multiplier theorems for nonnegative self-adjoint operators on spaces of homogeneous type. We show that a Hörmander type spectral multiplier theorem follows from sub-Gaussian upper bounds for the corresponding heat kernel. Our res

Functional calculus of Laplace transform type on non-doubling parabolic manifolds with ends

Let $M$ be a non-doubling parabolic manifold with ends and $L$ a non-negative self-adjoint operator on $L^{2}(M)$ which satisfies a suitable heat kernel upper bound named the upper bound of Gaussian type. These operators include the Schrödinger operators $L = \Delta + V$ where $\Delta$ is the Laplace–Beltrami operator and $V$ is an arbitrary non-negative potential. This paper will investigate the behaviour of the Poisson semi-group kernels of $L$ together with its time derivatives and then apply them to obtain the weak type $(1, 1)$ estimate of the functional calculus of Laplace transform type of $\sqrt{L}$ which is defined by $\mathfrak{M}(\sqrt{L}) f(x) := \int_{0}^{\infty} \bigl[\sqrt{L} e^{-t \sqrt{L}} f(x)\bigr] m(t) dt$ where $m(t)$ is a bounded function on $[0, \infty)$. In the setting of our study, both doubling condition of the measure on $M$ and the smoothness of the operators' kernels are missing. The purely imaginary power $L^{is}$, $s \in \mathbb{R}$, is a special case of our result and an example of weak type $(1, 1)$ estimates of a singular integral with non-smooth kernels on non-doubling spaces.

Regularity estimates for Green operators of Dirichlet and Neumann problems on weighted Hardy spaces

In this paper we first study the generalized weighted Hardy spaces $H^{p}_{L,w}(X)$ for $0 < p \le 1$ associated to nonnegative self-adjoint operators $L$ satisfying Gaussian upper bounds on the space of homogeneous type $X$ in both cases of finite and infinite measure. We show that the weighted Hardy spaces defined via maximal functions and atomic decompositions coincide. Then we prove weighted regularity estimates for the Green operators of the inhomogeneous Dirichlet and Neumann problems in suitable bounded or unbounded domains including bounded semiconvex domains, convex regions above a Lipschitz graph and upper half-spaces. Our estimates are in terms of weighted $L^{p}$ spaces for the range $1 < p <\infty$ and in terms of the new weighted Hardy spaces for the range $0 < p \le 1$. Our regularity estimates for the Green operators under the weak smoothness assumptions on the boundaries of the domains are new, especially the estimates on Hardy spaces for the full range $0 < p \le 1$ and the case of unbounded domains.

Approximate Solutions of One Abstract Cauchy Problem

We find approximate solutions of the Cauchy problem for a differential-operator equation of hyperbolic type with degeneration in a Hilbert space. In terms of these approximations, we give a characteristic of the Gevrey classes for a nonnegative self-adjoint operator.