End Point Estimate of Littlewood-Paley Operator Associated to the Generalized Schrödinger Operator

Let L = − Δ + μ be the generalized Schrödinger operator on ℝ d , d ≥ 3 , where μ ≠ 0 is a nonnegative Radon measure satisfying certain scale-invariant Kato conditions and doubling conditions. In this work, we give a new BMO space associated to the generalized Schrödinger operator L , BM O θ , L , which is bigger than the BMO spaces related to the classical Schrödinger operators A = − Δ + V , with V a potential satisfying a reverse Hölder inequality introduced by Dziubański et al. in 2005. Besides, the boundedness of the Littlewood-Paley operators associated to L in BM O θ , L also be proved.

BMO-type estimates of Riesz transforms associated with generalized Schrödinger operators

In this paper we show that the dual Riesz transform associated with the generalized Schrödinger operator {\mathcal{L}} is bounded from {\mathrm{BMO}} into {\mathrm{BMO}_{\mathcal{L}}} and give the Fefferman–Stein-type decomposition of {\mathrm{BMO}_{\mathcal{L}}} functions in terms of Riesz transforms.

𝐻𝑝 spaces for generalized Schrödinger operators and applications

In this paper, we study the Hardy type space {H_{\mathcal{L}}^{p}({\mathbb{R}^{n}})} by means of local maximal functions associated with the heat semigroup {e^{-t\mathcal{L}}} generated by {-\mathcal{L}}, where {\mathcal{L}=-\Delta+\mu} is the generalized Schrödinger operator in {{\mathbb{R}^{n}}} ({n\geq 3}) and {\mu\not\equiv 0} is a nonnegative Radon measure satisfying certain scale-invariant Kato conditions and doubling conditions. Via the equivalence of the norms between various local maximal functions, we show that the norms {\lVert f\rVert_{H_{\mathcal{L}}^{p}({\mathbb{R}^{n}})}^{p}} and {\lVert f\rVert_{H_{m}^{p,q}({\mathbb{R}^{n}})}^{p}} are equivalent for {\frac{n}{n+\delta^{\prime}}<p\leq 1\leq q\leq\infty} ({p\neq q}) with some {\delta^{\prime}>0}. As applications, we prove that Calderón–Zygmund operators related to the auxiliary function {m(x,\mu)} are bounded from {H_{\mathcal{L}}^{p}({\mathbb{R}^{n}})} into {L^{p}({\mathbb{R}^{n}})} for {\frac{n}{n+\gamma_{1}}<p\leq 1} with {\gamma_{1}>0}. In particular, we show that the Riesz transform {\nabla(-\Delta+\mu)^{-\frac{1}{2}}}, which is a special example of the above Calderón–Zygmund operators, is bounded from {H_{\mathcal{L}}^{p}({\mathbb{R}^{n}})} into {{H}^{p}({\mathbb{R}^{n}})} for {\frac{n}{n+\gamma_{1}}<p\leq 1} with {0<\gamma_{1}<1}.

Endpoint Estimates of Riesz Transforms Associated with Generalized Schrödinger Operators

In this paper we establish the endpoint estimates and Hardy type estimates for the Riesz transform associated with the generalized Schrödinger operator.

WAVEGUIDES COUPLED THROUGH A SEMITRANSPARENT BARRIER: A BIRMAN–SCHWINGER ANALYSIS

The paper is devoted to a model of a mesoscopic system consisting of a pair of parallel planar waveguides separated by an infinitely thin semitransparent boundary modeled by a transverse δ interaction. We develop the Birman–Schwinger theory for the corresponding generalized Schrödinger operator. The spectral properties become nontrivial if the barrier coupling is not invariant with respect to longitudinal translations, in particular, there are bound states if the barrier is locally more transparent in the mean and the coupling parameter reaches the same asymptotic value in both directions along the guide axis. We derive the weak-coupling expansion of the ground-state eigenvalue for the cases when the perturbation is small in the supremum and the L 1-norms. The last named result applies to the situation when the support of the leaky part shrinks: the obtained asymptotics differs from that of a double guide divided by a pierced Dirichlet barrier. We also derive an upper bound on the number of bound states.