L^p BOUNDEDNESS OF COMMUTATOR OPERATOR ASSOCIATED WITH SCHRDINGER OPERATORS ON HEISENBERG GROUP

Let L = -△Hn ＋ V be a SchrSdinger operator on Heisenberg group Hn, where AHn is the sublaplacian and the nonnegative potential V belongs to the reverse HSlder class BQ/2 where Q is the homogeneous dimension of Hn. Let T1 = （--△Hn ＋V）-1V, T2 = （-△Hn ＋V）-1/2V1/2, and T3 = （--AHn ＋V）-I/2△Hn, then we verify that [b, Ti], i = 1, 2, 3 are bounded on some LP（Hn）, where b ∈ BMO（Hn）. Note that the kernel of Ti, i = 1, 2, 3 has no smoothness.

LITTLEWOOD-PALEY THEOREM FOR SCHR DINGER OPERATORS

Let H be a Schroedinger operator on R^n. Under a polynomial decay condition for the kernel of its spectral operator, we show that the Besov spaces and Triebel-Lizorkin spaces associated with H are well defined. We further give a Littlewood-Paley characterization of Lp spaces in terms of dyadic functions of H. This generalizes and strengthens the previous result when the heat kernel of H satisfies certain upper Gaussian bound.

On Characterization of Poisson Integrals of Schr?dinger Operators with Morrey Traces

Let L be a Schrodinger operator of the form L = -△ ＋ V acting on L2（Rn） where the nonnegative potential V belongs to the reverse Holder class Bq for some q _〉 n. In this article we will show that a function f∈ L2,λ（Rn）, 0 〈λ 〈 n, is the trace of the solution of Lu = -utt ＋ Lu = O, u（x, 0） = f（x）, where u satisfies a Carleson type condition sup t-λB xB,rB∫τB 0∫B（xB,τB）t{ u（x,t）}2dxdt≤C〈∞.Its proof heavily relies on investigate the intrinsic relationship between the classical Morrey spaces and the new Campanato spaces .L2,λL（Rn） associated to the operator L, i.e. Conversely, this Carleson type condition characterizes all the L-harmonic functions whose traces belong to the space L2,λ（Rn） for all 0 〈λ〈 n.

Sharp lower bound of spectral gap for Schrodinger operator and related results

We give an easy proof of Andrews and Clutterbuck＇s main results [J. Amer. Math. Soc., 2011, 24（3）： 899-916], which gives both a sharp lower bound for the spectral gap of a Schr5dinger operator and a sharp modulus of concavity for the logarithm of the corresponding first eigenfunction. We arrive directly at the same estimates by the ＇double coordinate＇ approach and asymptotic behavior of parabolic flows. Although using the techniques appeared in the above paper, we partly simplify the method and argument. This maybe help to provide an easy way for estimating spectral gap. Besides, we also get a new lower bound of spectral gap for a class of SchSdinger operator.

Maximal function characterizations of Musielak-Orlicz-Hardy spaces associated with magnetic Schriidinger operators

Let φ be a growth function, and let A ：= -（V- ia）. （V- ia）＋ V be a magnetic SchrSdinger operator on L2（Rn）, n≥ 2, where a ：= （a1, a2... an） ∈ r L1 loc（Rn） We establish the equivalent characteriza- L2 1oc（Rn, Rn） and 0 ≤ V ∈Lloc（Rn） tions of the Musielak-Orlicz-Hardy space HA,^（IRn）, defined by the Lusin area function associated with {e-t2A}t〉0, in terms of the Lusin area function associated with {e-t√A}t〉0, the radial maximal functions and the non- tangential maximal functions associated with {e-t2A}t〉o and {e-t√A}t〉0, respectively. The boundedness of the Riesz transforms LkA-U1/2, k ∈ {1, 2... n}, from HA,φ（Rn） to Lφ（Rn） is also presented, where Lk is the closure of δ/δxk iak in L2（Rn）. These results are new even when φ（x,t） ：= w（x）tp for all x ∈Rn and t∈ （0, ＋∞） with p ∈ （0, 1] and ω∈ A∞（Rn） （the class of Muckenhoupt weights on Rn）.

Decay estimates of discretized Green's functions for Schrdinger type operators

For a sparse non-singular matrix A, generally A- 1 is a dense matrix. However, for a class of matrices, A-1 can be a matrix with off-diagonal decay properties, i.e., ｜Aij^-1｜ decays fast to 0 with respect to the increase of a properly defined distance between i and j. Here we consider the off-diagonal decay properties of discretized Green＇s functions for SchrSdinger type operators. We provide decay estimates for discretized Green＇s functions obtained from the finite difference discretization, and from a variant of the pseudo-spectral discretization. The asymptotic decay rate in our estimate is independent of the domain size and of the discretization parameter. We verify the decay estimate with numerical results for one-dimensional Schr6dinger type operators.

Hlder Continuity for a Class of Strongly Degenerate Schrdinger Operator Formed by Vector Fields

In this paper we obtain the H61der continuity property of the solutions for a class of degenerate Schr6dinger equation generated by the vector fields：∑i,j=1^m Xj^＊（aij（x）Xiu）＋bXu=uu=0,where X = {X1,.-. ,Xm} is a family of C^∞ vector fields satisfying the H6rmander condition, and the lower order terms belong to an appropriate Morrey type space.