### A note on Schrödinger operators with polynomial potentials

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For a Schrödinger operator A = -Δ + V, where V is a nonnegative polynomial, we define a Hardy ${H}_{A}^{1}$ space associated with A. An atomic characterization of ${H}_{A}^{1}$ is shown.

Let A = -Δ + V be a Schrödinger operator on ${\mathbb{R}}^{d}$, d ≥ 3, where V is a nonnegative potential satisfying the reverse Hölder inequality with an exponent q > d/2. We say that f is an element of ${H}_{A}^{p}$ if the maximal function $su{p}_{t>0}\left|{T}_{t}f\left(x\right)\right|$ belongs to ${L}^{p}\left({\mathbb{R}}^{d}\right)$, where ${{T}_{t}}_{t>0}$ is the semigroup generated by -A. It is proved that for d/(d+1) < p ≤ 1 the space ${H}_{A}^{p}$ admits a special atomic decomposition.

Let ${{K}_{t}}_{t>0}$ be the semigroup of linear operators generated by a Schrödinger operator -L = Δ - V with V ≥ 0. We say that f belongs to $H{\xb9}_{L}$ if $\left|\right|su{p}_{t>0}|{K}_{t}{f\left(x\right)\left|\right||}_{L\xb9\left(dx\right)}<\infty $. We state conditions on V and ${K}_{t}$ which allow us to give an atomic characterization of the space $H{\xb9}_{L}$.

Let {T} be the semigroup of linear operators generated by a Schrödinger operator -A = Δ - V, where V is a nonnegative potential that belongs to a certain reverse Hölder class. We define a Hardy space H by means of a maximal function associated with the semigroup {T}. Atomic and Riesz transforms characterizations of H are shown.

The aim of this paper is to demonstrate how a fairly simple nilpotent Lie algebra can be used as a tool to study differential operators on ${\mathbb{R}}^{n}$ with polynomial coefficients, especially when the property studied depends only on the degree of the polynomials involved and/or the number of variables.

Let L be a positive Rockland operator of homogeneous degree d on a graded homogeneous group G and let ${p}_{t}$ be the convolution kernels of the semigroup generated by L. We prove that if τ(x) is a Riemannian distance of x from the unit element, then there are constants c>0 and C such that $|{p}_{1}\left(x\right)|\le Cexp(-c\tau {\left(x\right)}^{d/(d-1)})$. Moreover, if G is not stratified, more precise estimates of ${p}_{1}$ at infinity are given.

Let 𝓓 be a symmetric Siegel domain of tube type and S be a solvable Lie group acting simply transitively on 𝓓. Assume that L is a real S-invariant second order operator that satisfies Hörmander's condition and annihilates holomorphic functions. Let H be the Laplace-Beltrami operator for the product of upper half planes imbedded in 𝓓. We prove that if F is an L-Poisson integral of a BMO function and HF = 0 then F is pluriharmonic. Some other related results are also considered.

Let G be a compactly generated, locally compact group with polynomial growth and let ω be a weight on G. We look for general conditions on the weight which allow us to develop a functional calculus on a total part of L(G,ω). This functional calculus is then used to study harmonic analysis properties of L(G,ω), such as the Wiener property and Domar's theorem.

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