Intrinsic Ultracontractivity for a class of Schroedinger Semigroups in $\mathrm{L}^{2}\left( \mathbb{R}^{n} \right)$ using Log-Sobolev-inequalities and duality arguments
University of Rostock（罗斯托克大学）；Cadi Ayyad University（卡迪阿亚德国立大学）
We present a class of potentials $q \colon \mathbb{R}^{n} \to (0,\infty)$ that implies the weighted Schrödinger semigroup $φ^{-1}\mathrm{e}^{-tH}φ$ to map a weighted Lebesgue function space $\mathrm{L}_μ^{1}(\mathbb{R}^{n})$ into a weighted Lebesgue function space $\mathrm{L}_μ^{2}(\mathbb{R}^{n})$ continously at every time $t>0$ by Logarithmic Sobolev inequalities for $H=-Δ+ q(x)$ with it's strictly positive ground state $φ\colon \mathbb{R}^{n} \to (0,\infty)$. We use the self-adjointness of $\mathrm{e}^{-tH}$ in $\mathrm{L}^{2}(\mathbb{R}^{n})$ to infer an intrinsic ultracontractivity, i.e.
$\forall t>0 \ \exists C_{t} > 0 \ : \ \left| \mathrm{e}^{-tH} u (x) \right| \ \leq \ C_{t} φ(x) \| u \|_{2}$
for every $u \in \mathrm{L}^{2}\left( \mathbb{R}^{n} \right)$ almost everywhere in $\mathbb{R}^{n}$.
We introduce heat semigroup-based Besov classes in the general framework of Dirichlet spaces. General properties of those classes are studied and quantitative regularization estimates for the heat semigroup in this scale of spaces are obtained. As a highlight of the paper, we obtain a far reaching $L^p$-analogue, $p \ge 1$, of the Sobolev inequality that was proved for $p=2$ by N. Varopoulos under the assumption of ultracontractivity for the heat semigroup. The case $p=1$ is of special interest since it yields isoperimetric type inequalities.
We prove mixed norm space-time estimates for solutions of the Schroedinger equation, with initial data in $L^p$ Sobolev or Besov spaces, and clarify the relation with adjoint restriction.
Function spaces associated with Schroedinger operators: the Poeschl-Teller potential
我们解决了与Schroedinger算子H相关的函数空间理论。讨论的重点是量子物理中的Poeschl-Teller势。利用双正交并矢系统，我们引入了与H相关联的Besov空间和Triebel-Lizorkin空间（包括索伯列夫空间）.然后，我们使用插值方法，以确定这些空间与经典的p，q> 1的一定范围内。一个物理含义是，相应的波函数$ψ(t,x)=e^{-i t H }f(x)$承认适当的时间衰减在贝索夫空间尺度。
For $α>1$ we consider the initial value problem for the dispersive equation $i\partial_t u +(-Δ)^{α/2} u= 0$. We prove an endpoint $L^p$ inequality for the maximal function $\sup_{t\in[0,1]}|u(\cdot,t)|$ with initial values in $L^p$-Sobolev spaces, for $p\in(2+4/(d+1),\infty)$. This strengthens the fixed time estimates due to Fefferman and Stein, and Miyachi. As an essential tool we establish sharp $L^p$ space-time estimates (local in time) for the same range of $p$.
We define Hölder classes $Λ_α$ associated with a Markovian semigroup and prove that, when the semigroup satisfies the $Γ^2 \geq 0$ condition, the Riesz transforms are bounded between the Hölder classes. As a consequence, this bound holds in manifolds with nonnegative Ricci curvature. We also show, without the need for extra assumptions on the semigroup, that a version of the Morrey inequalities is equivalent to the ultracontractivity property. This result extends the semigroup approach to the Sobolev inequalities laid by Varopoulos. After that, we study certain families of operators between th…
We prove the existence of quasi-periodic solutions for Schroedinger equations with a multiplicative potential on T^d, d \geq 1, merely differentiable nonlinearities, and tangential frequencies constrained along a pre-assigned direction. The solutions have only Sobolev regularity both in time and space. If the nonlinearity and the potential are infinitely many times differentiable so are the solutions. The proofs are based on an improved Nash-Moser iterative scheme, which assumes the weakest tame estimates for the inverse linearized operators ("Green functions") along scales of Sobolev spaces. …
In the Euclidean space of any dimension d, we consider the heat semigroup generated by the magnetic Schroedinger operator from which an inverse-square potential is subtracted in order to make the operator critical in the magnetic-free case. Assuming that the magnetic field is compactly supported, we show that the polynomial large-time behaviour of the heat semigroup is determined by the eigenvalue problem for a magnetic Schroedinger operator on the (d-1)-dimensional sphere whose vector potential reflects the behaviour of the magnetic field at the space infinity. From the spectral problem on th…
We establish boundedness estimates for solutions of generalized porous medium equations of the form $$ \partial_t u+(-\mathfrak{L})[u^m]=0\quad\quad\text{in $\mathbb{R}^N\times(0,T)$}, $$ where $m\geq1$ and $-\mathfrak{L}$ is a linear, symmetric, and nonnegative operator. The wide class of operators we consider includes, but is not limited to, Lévy operators. Our quantitative estimates take the form of precise $L^1$--$L^\infty$-smoothing effects and absolute bounds, and their proofs are based on the interplay between a dual formulation of the problem and estimates on the Green function of $-\m…
In this paper we prove that the heat kernel $k$ associated to the operator $A:= (1+|x|^α)Δ+b|x|^{α-1}\frac{x}{|x|}\cdot\nabla -|x|^β$ satisfies $$ k(t,x,y) \leq c_1e^{λ_0 t+ c_2t^{-γ}}\left(\frac{1+|y|^α}{1+|x|^α}\right)^{\frac{b}{2α}} \frac{(|x||y|)^{-\frac{N-1}{2}-\frac{1}{4}(β-α)}}{1+|y|^α} e^{-\frac{\sqrt{2}}{β-α+2}\left(|x|^{\frac{β-α+2}{2}}+ |y|^{\frac{β-α+2}{2}}\right)} $$ for $t>0,\,|x|,\,|y|\ge 1$, where $b\in\mathbb{R}$, $c_1,\,c_2$ are positive constants, $λ_0$ is the largest eigenvalue of the operator $A$, and $γ=\frac{β-α+2}{β+α-2}$, in the case where $N>2,\,α>2$ and $β>α-2$. The …
Nowadays we have many methods allowing to exploit the regularising properties of the linear part of a nonlinear dispersive equation (such as the KdV equation, the nonlinear wave or the nonlinear Schroedinger equations) in order to prove well-posedness in low regularity Sobolev spaces. By well-posedness in low regularity Sobolev spaces we mean that less regularity than the one imposed by the energy methods is required (the energy methods do not exploit the dispersive properties of the linear part of the equation). In many cases these methods to prove well-posedness in low regularity Sobolev spa…
