Boundedness of Differential Transforms for Heat Semigroups Generated by Schrödinger Operators (original) (raw)

The ρ-variation of the heat semigroup in the Hermitian setting: behaviour inL∞

Proceedings of the Edinburgh Mathematical Society, 2011

Let$\mathcal{V}_\rho(\re^{-tH})$, ρ > 2, be the ρ-variation of the heat semigroup associated to the harmonic oscillatorH= ½(−Δ + |x|2). We show that iff∈L∞(ℝ), the$\mathcal{V}_\rho(\re^{-tH})$(f)(x) < ∞, a.e.x∈ ℝ. However, we find a functionG∈L∞(ℝ), such that$\mathcal{V}_\rho(\re^{-tH})$(G)(x) ∉L∞(ℝ). We also analyse the local behaviour inL∞of the operator$\mathcal{V}_\rho(\re^{-tH})$. We find that its growth is smaller than that of a standard singular integral operator. As a by-product of our work we obtain anL∞(ℝ) functionF, such that the square functiona.e.x∈ ℝ, whereis the classical Poisson kernal in ℝ.

On Dunkl Schr\"odinger semigroups with Green bounded potentials

2022

On R N equipped with a normalized root system R, a multiplicity function k(α) > 0, and the associated measure dw(x) = α∈R | x, α | k(α) dx, we consider a Dunkl Schrödinger operator L = −∆ k + V , where ∆ k is the Dunkl Laplace operator and V ∈ L 1 loc (dw) is a non-negative potential. Let h t (x, y) and k {V } t (x, y) denote the Dunkl heat kernel and the integral kernel of the semigroup generated by −L respectively. We prove that k {V } t (x, y) satisfies the following heat kernel lower bounds: there are constants C, c > 0 such that h ct (x, y) ≤ Ck {V } t (x, y) if and only if sup x∈R N ∞ 0 R N V (y)w(B(x, √ t)) −1 e − x−y 2 /t dw(y) dt < ∞, where B(x, √ t) stands for the Euclidean ball centered at x ∈ R N and radius √ t.

General Kernel estimates of Schr\"odinger type operators with unbounded diffusion terms

Cornell University - arXiv, 2022

We prove first that the realization A min of A := div(Q∇) − V in L 2 (R d) with unbounded coefficients generates a symmetric sub-Markovian and ultracontractive semigroup on L 2 (R d) which coincides on L 2 (R d) ∩ C b (R d) with the minimal semigroup generated by a realization of A on C b (R d). Moreover, using time dependent Lyapunov functions, we prove pointwise upper bounds for the heat kernel of A and deduce some spectral properties of A min in the case of polynomially and exponentially diffusion and potential coefficients.

Functional calculus of operators with heat kernel bounds on non-doubling manifolds with ends

Indiana University Mathematics Journal, 2020

Let ∆ be the Laplace-Beltrami operator acting on a non-doubling manifold with two ends Ê m ♯R n with m > n ≥ 3. Let t (x, y) be the kernels of the semigroup e −t∆ generated by ∆. We say that a non-negative self-adjoint operator L on L 2 (Ê m ♯R n) has a heat kernel with upper bound of Gaussian type if the kernel h t (x, y) of the semigroup e −tL satisfies h t (x, y) ≤ C αt (x, y) for some constants C and α. This class of operators includes the Schrödinger operator L = ∆ + V where V is an arbitrary non-negative potential. We then obtain upper bounds of the Poisson semigroup kernel of L together with its time derivatives and use them to show the weak type (1, 1) estimate for the holomorphic functional calculus Å(√ L) where Å(z) is a function of Laplace transform type. Our result covers the purely imaginary powers L is , s ∈ Ê, as a special case and serves as a model case for weak type (1, 1) estimates of singular integrals with non-smooth kernels on non-doubling spaces.

The Schrödinger equation in LpL^{p}Lp spaces for operators with heat kernel satisfying Poisson type bounds

Journal of the Mathematical Society of Japan, 2021

Let L be a non-negative self-adjoint operator acting on L 2 (X) where X is a space of homogeneous type with a dimension n. In this paper, we study sharp endpoint L p-Sobolev estimates for the solution of the initial value problem for the Schrödinger equation i∂ t u + Lu = 0 and show that for all f ∈ L p (X), 1 < p < ∞, e itL (I + L) −σn f p ≤ C(1 + |t|) σn f p , t ∈ R, σ ≥ 1 2 − 1 p , where the semigroup e −tL generated by L satisfies a Poisson type upper bound.

Heat-diffusion maximal operators for Laguerre semigroups with negative parameters

Journal of Functional Analysis, 2005

We prove L p boundedness for the maximal operator of the heat semigroup associated to the Laguerre functions, {L α k } k , when the parameter α is greater than −1. Namely, the maximal operator is of strong type (p, p) if p > 1 and 2 2+α < p < 2 −α , when −1 < α < 0. If α ≥ 0 there is strong type for 1 < p ≤ ∞. The behavior at the end points is studied in detail.