Riesz Transforms Associated with Schrdinger Operators Acting on Weighted Hardy Spaces

Let L =-? + V be a Schrdinger operator acting on L2(Rn), n ≥ 1, where V ≡ 0 is a nonnegative locally integrable function on Rn. In this article, we will intropduce weighted Hardy spaces H L(w) associated with L by means of the square function and then study their atomic decomposition theory. We will also show that the Riesz transform ?L-1/2associated with L is bounded from our new space Hp L(w) to the classical weighted Hardy space Hp(w) when n/(n +1) p 1 and w ∈ A1∩ RH(2/p)′.     

Riesz Transforms Associated with Higher-Order Schrödinger Type Operators

Let L = L ₀ + V be a Schrödinger type operator, where L ₀ is a higher order elliptic operator with bounded complex coefficients in divergence form and V is a signed measurable function. Under the strongly subcritical assumption on V, we study the L ^q boundedness of Riesz transform ? ^m L ^?1/2 for q ≤ 2 based on the off-diagonal estimates of semigroup e ^{?t L} . Furthermore, the authors impose extra regularity assumptions on V to obtain the L ^q boundedness of Riesz transform ? ^m L ^?1/2 for some q > 2. In particular, these results are applied to the more interesting Schrödinger operators L = P(D) + V, where P(D) is any homogeneous positive elliptic operator with constant coefficients.     

Besov and Hardy Spaces Associated with the Schrödinger Operator on the Heisenberg Group

We introduce the Besov space $\dot{B}^{0,L}_{1,1}$ associated with the Schrödinger operator L with a nonnegative potential satisfying a reverse Hölder inequality on the Heisenberg group, and obtain the molecular decomposition. We also develop the Hardy space $H_{L}^{1}$ associated with the Schrödinger operator via the Littlewood–Paley area function and give equivalent characterizations via atoms, molecules, and the maximal function. Moreover, using the molecular decomposition, we prove that $\dot{B}^{0,L}_{1,1}$ is a subspace of $H_{L}^{1}$ .     

Boundedness of Commutators Generated by Campanato-type Functions and Riesz Transforms Associated with Schrdinger Operators

Let■=-△+V be a Schrdinger operator on R~n,n3,where △is the Laplacian on R~n and V≠0 is a nonnegative function satisfying the reverse Holder's inequality.Let[b,T]be the commutator generated by the Campanatotype function b∈■ and the Riesz transform associated with Schrdinger operator T=▽(-△+V)~(-1/2).In the paper,we establish the boundedness of[b,T]on Lebesgue spaces and Campanato-type spaces.     

Hardy Spaces H_L~p(R~n) Associated with Higher-Order Schrdinger Type Operators

Let L = L0+V be the higher order Schrdiger type operator where L0 is a homogeneous elliptic operator of order 2m in divergence form with bounded coefficients and V is a real measurable function as multiplication operator(e.g., including(-?)m+V(m∈N) as special examples). In this paper, assume that V satisfies a strongly subcritical form condition associated with L0, the authors attempt to establish a theory of Hardy space Hp L(Rn)(0 p ≤ 1) associated with the higher order Schrdinger type operator L. Specifically, we first define the molecular Hardy space Hp L(Rn) by the so-called( p, q, ε, M) molecule associated to L and then establish its characterizations by the area integral defined by the heat semigroup e-t L.     

Riesz Transforms of Schr?dinger Operators on Manifolds

We consider Schr?dinger operators A=???+V on L ^p (M) where M is a complete Riemannian manifold of homogeneous type and V=V ⁺?V ^? is a signed potential. We study boundedness of Riesz transform type operators $\nabla A^{-\frac{1}{2}}$ and $|V|^{\frac{1}{2}}A^{-\frac{1}{2}}$ on L ^p (M). When V ^? is strongly subcritical with constant ????(0,1) we prove that such operators are bounded on L ^p (M) for $p\in(p_{0}', 2]$ where $p_{0}'=1$ if N??2, and $p_{0}'=(\frac{2N}{(N-2)(1-\sqrt{1-\alpha })})' \in (1, 2)$ if N>2. We also study the case p>2. With additional conditions on V and M we obtain boundedness of ?A ^?1/2 and |V|^1/2 A ^?1/2 on L ^p (M) for p??(1,inf?(q ₁,N)) where q ₁ is such that $\nabla(-\Delta)^{-\frac{1}{2}}$ is bounded on L ^r (M) for r??[2,q ₁).     

The Weighted Estimates of the Schrdinger Operators on the Nilpotent Lie Group

In this paper we consider the Schrdinger operator-△G + W on the nilpotent Lie group G where the nonnegative potential W belongs to the reverse Hlder class Bq1 for some q1 ≥ D2 and D is the dimension at infinity of G.The weighted Lp-Lq estimates for the operators Wα(-△G + W)-β and Wα G(-△G + W)-β are obtained.     

Variation Operators for Semigroups and Riesz Transforms on BMO in the Schrödinger Setting

In this paper we prove that the variation operators of the heat semigroup and the truncations of Riesz transforms associated to the Schrödinger operator are bounded on a suitable BMO type space.     

Local Dispersive and Strichartz Estimates for the Schrödinger Operator on the Heisenberg Group

It was proved by Bahouri et al. [9] that the Schrödinger equation on the Heisenberg group $\mathbb{H}^d,$ involving the sublaplacian, is an example of a totally non-dispersive evolution equation: for this reason global dispersive estimates cannot hold. This paper aims at establishing local dispersive estimates on $\mathbb{H}^d$ for the linear Schrödinger equation, by a refined study of the Schrödinger kernel $S_t$ on $\mathbb{H}^d.$ The sharpness of these estimates is discussed through several examples. Our approach, based on the explicit formula of the heat kernel on $\mathbb{H}^d$ derived by Gaveau [19], is achieved by combining complex analysis and Fourier-Heisenberg tools. As a by-product of our results we establish local Strichartz estimates and prove that the kernel $S_t$ concentrates on quantized horizontal hyperplanes of $\mathbb{H}^d.$     

Morrey Estimates for Schrdinger Type Elliptic Equations

In this paper, by means of Olsen type inequalities related to the fractional integral operator, the authors establish the interior estimates in Morrey spaces for Schrdinger type elliptic equations with potentials satisfying a reverse Hlder condition.     

Boundedness of Generalized Riesz Transforms on Orlicz–Hardy Spaces Associated to Operators

Let ${\Phi}$ be a continuous, strictly increasing and concave function on (0, ∞) of critical lower type index ${p_\Phi^- \in(0,\,1]}$ . Let L be an injective operator of type ω having a bounded H _∞ functional calculus and satisfying the k-Davies–Gaffney estimates with ${k \in {\mathbb Z}_+}$ . In this paper, the authors first introduce an Orlicz–Hardy space ${H^{\Phi}_{L}(\mathbb{R}^n)}$ in terms of the non-tangential L-adapted square function and then establish its molecular characterization. As applications, the authors prove that the generalized Riesz transform ${D_{\gamma}L^{-\delta/(2k)}}$ is bounded from the Orlicz–Hardy space ${H^{\Phi}_{L}(\mathbb{R}^n)}$ to the Orlicz space ${L^{\widetilde{\Phi}}(\mathbb{R}^n)}$ when ${p_\Phi^- \in (0, \frac{n}{n+ \delta - \gamma}]}$ , ${0 < \gamma \le \delta < \infty}$ and ${\delta- \gamma < n (\frac{1}{p_-(L)}-\frac{1}{p_+(L)})}$ , or from ${H^{\Phi}_{L}(\mathbb{R}^n)}$ to the Orlicz–Hardy space ${H^{\widetilde \Phi}(\mathbb{R}^n)}$ when ${p_\Phi^-\in (\frac{n}{n + \delta+ \lfloor \gamma \rfloor- \gamma},\,\frac{n}{n+ \delta- \gamma}]}$ , ${1\le \gamma \le \delta < \infty}$ and ${\delta- \gamma < n (\frac{1}{p_-(L)}-\frac{1}{p_+(L)})}$ , or from ${H^{\Phi}_{L}(\mathbb{R}^n)}$ to the weak Orlicz–Hardy space ${WH^\Phi(\mathbb{R}^n)}$ when ${\gamma = \delta}$ and ${p_\Phi=n/(n + \lfloor \gamma \rfloor)}$ or ${p_\Phi^-=n/(n + \lfloor \gamma \rfloor)}$ with ${p_\Phi^-}$ attainable, where ${\widetilde{\Phi}}$ is an Orlicz function whose inverse function ${\widetilde{\Phi}^{-1}}$ is defined by ${\widetilde{\Phi}^{-1}(t):=\Phi^{-1}(t)t^{\frac{1}{n}(\gamma- \delta)}}$ for all ${t \in (0,\,\infty)}$ , ${p_\Phi}$ denotes the strictly critical lower type index of ${\Phi}$ , ${\lfloor \gamma \rfloor}$ the maximal integer not more than ${\gamma}$ and ${(p_-(L),\,p_+(L))}$ the range of exponents ${p \in[1,\, \infty]}$ for which the semigroup ${\{e^{-tL}\}_{t >0 }}$ is bounded on ${L^p(\mathbb{R}^n)}$ .     

Hardy Spaces HpL(Rn) Associated with Higher-Order Schr?dinger Type Operators

Let L=L0+V be the higher order Schr?diger type operator where L0 is a homogeneous elliptic operator of order 2m in divergence form with bounded coeffi-cients and V is a real measurable function as mult...     

On Isomorphisms of Hardy Spaces Associated with Schrödinger Operators

Let L=?Δ+V is a Schrödinger operator on $\mathbb{R}^{d}$ , d≥3, V≥0. Let $H^{1}_{L}$ denote the Hardy space associated with L. We shall prove that there is an L-harmonic function w, 0<δ≤w(x)≤C, such that the mapping $$H_L^1 \ni f\mapsto wf\in H^1\bigl(\mathbb{R}^d\bigr) $$ is an isomorphism from the Hardy space $H_{L}^{1}$ onto the classical Hardy space $H^{1}(\mathbb{R}^{d})$ if and only if $\Delta^{-1}V(x)=-c_{d}\int_{\mathbb{R}^{d}} |x-y|^{2-d} V(y) dy$ belongs to $L^{\infty}(\mathbb{R}^{d})$ .     

Endpoint Properties of Localized Riesz Transforms and Fractional Integrals Associated to Schrödinger Operators

Let \({\mathcal L}\equiv-\Delta+V\) be the Schrödinger operator in \({{\mathbb R}^n}\), where V is a nonnegative function satisfying the reverse Hölder inequality. Let ρ be an admissible function modeled on the known auxiliary function determined by V. In this paper, the authors characterize the localized Hardy spaces \(H^1_\rho({{\mathbb R}^n})\) in terms of localized Riesz transforms and establish the boundedness on the BMO-type space \({\mathop\mathrm{BMO_\rho({\mathbb R}^n)}}\) of these operators as well as the boundedness from \({\mathop\mathrm{BMO_\rho({\mathbb R}^n)}}\) to \({\mathop\mathrm{BLO_\rho({\mathbb R}^n)}}\) of their corresponding maximal operators, and as a consequence, the authors obtain the Fefferman–Stein decomposition of \({\mathop\mathrm{BMO_\rho({\mathbb R}^n)}}\) via localized Riesz transforms. When ρ is the known auxiliary function determined by V, \({\mathop\mathrm{BMO_\rho({\mathbb R}^n)}}\) is just the known space \(\mathop\mathrm{BMO}_{\mathcal L}({{\mathbb R}^n})\), and \({\mathop\mathrm{BLO_\rho({\mathbb R}^n)}}\) in this case is correspondingly denoted by \(\mathop\mathrm{BLO}_{\mathcal L}({{\mathbb R}^n})\). As applications, when n?≥?3, the authors further obtain the boundedness on \(\mathop\mathrm{BMO}_{\mathcal L}({{\mathbb R}^n})\) of Riesz transforms \(\nabla{\mathcal L}^{-1/2}\) and their adjoint operators, as well as the boundedness from \(\mathop\mathrm{BMO}_{\mathcal L}({{\mathbb R}^n})\) to \(\mathop\mathrm{BLO}_{\mathcal L}({{\mathbb R}^n})\) of their maximal operators. Also, some endpoint estimates of fractional integrals associated to \({\mathcal L}\) are presented.     

Commutators of Riesz Transforms Related to?Schr?dinger Operators

In this work we obtain boundedness on L ^p , for 1<p<??, of commutators T _b f=bTf?T(bf) where T is any of the Riesz transforms or their conjugates associated to the Schr?dinger operator ???+V with V satisfying an appropriate reverse H?lder inequality. The class where b belongs is larger than the usual BMO. We also obtain a substitute result for p=??, under a slightly stronger condition on?b.     

Some Estimates of Higher Order Riesz Transform Related to Schrödinger Type Operators

In this paper we consider the Schrödinger type operators \(H_2=(-\Delta)^2 +V^2\), where the nonnegative potential V belongs to the reverse Hölder class \(B_{q_{_1}}\) for \(q_{_1}\geq \frac{n}{2}, n\geq 5\). The L ^p and weak type (1, 1) estimates of higher order Riesz transform \(\nabla^2H^{-\frac{1}{2}}_2 \) related to Schrödinger type operators H ₂ are obtained. In particular, \(\nabla^2H^{-\frac{1}{2}}_2 \) is a Calderón-Zygmund operator if V?∈?B _2n or \(V\in B_\frac{n}{2}\) and there exists a constant C such that V(x)?≤?Cm(x,V)².     

A Characterization of Hardy Spaces Associated with Certain Schrödinger Operators

Let {K _t } _t>0 be the semigroup of linear operators generated by a Schrödinger operator ?L = Δ ? V (x) on ? ^d , d ≥ 3, where V (x) ≥ 0 satisfies Δ ^?1 V ∈ L ^∞. We say that an L ¹-function f belongs to the Hardy space \({H^{1}_{L}}\) if the maximal function ? _L f (x) = sup _t>0 |K _t f (x)| belongs to L ¹ (? ^d ). We prove that the operator (?Δ)^1/2 L ^?1/2 is an isomorphism of the space \({H^{1}_{L}}\) with the classical Hardy space H ¹(? ^d ) whose inverse is L ^1/2(?Δ)^?1/2. As a corollary we obtain that the space \({H^{1}_{L}}\) is characterized by the Riesz transforms \(R_{j}=\frac {\partial }{\partial x_{j}}L^{-1\slash 2}\) .     

Heat Kernel Estimates of Fractional Schrödinger Operators with Negative Hardy Potential

Potential Analysis - We obtain two-sided estimates for the heat kernel (or the fundamental function) associated with the following fractional Schrödinger operator with negative Hardy potential...     

Sharp Estimates for Schrödinger Groups on Hardy Spaces for $$0

Journal of Fourier Analysis and Applications - In this paper, we discuss optimal constants and extremisers of Kato-smoothing estimates for the 2D Dirac equation. Smoothing estimates are...     

Heat Kernel Estimates for Schrödinger Operators on Exterior Domains with Robin Boundary Conditions

We study Schrödinger operators with Robin boundary conditions on exterior domains in ? ^d . We prove sharp point-wise estimates for the associated semigroups which show, in particular, how the boundary conditions affect the time decay of the heat kernel in dimensions one and two. Applications to spectral estimates are discussed as well.