arXiv subjects

Christoph Schwerdt

Publications and source records attributed to Christoph Schwerdt.

5 recordsLinked to original sources

arXiv2026

Ultracontractivity of heat semigroups with non-local Robin boundary conditions via two-sided bounds for a positive eigenfunction

We study heat semigroups on bounded Lipschitz domains $Ω\subset \mathbb{R}^{d}$ with dimension $d>2$ under non-local Robin boundary conditions. The boundary operator $B \in \mathcal{L}( \mathrm{L}^{2}(\partialΩ))$ is allowed to destroy the positivity of the semigroup. We assume that there exists a positive operator $C \in \mathcal{L}(\mathrm{L}^{2}(\partialΩ))$ such that $$ |Bu| \leq C|u| \quad\text{for every }u\in \mathrm{L}^{2}(\partialΩ), \qquad C{\bf 1}\in \mathrm{L}^{\infty}(\partialΩ). $$ Under this assumption, we prove that the semigroup generated by the corresponding uniformly elliptic operator is ultracontractive. More precisely, its norm from $\mathrm{L}^{2}(Ω)$ to $\mathrm{L}^{\infty}(Ω)$ has short-time order $t^{-d/4}$. The main step is the construction of a positive eigenfunction $ϕ$ of a dominating elliptic operator such that $$ 0 \ < \ δ\ \leq \ ϕ\ \leq \ M $$ almost everywhere in $Ω$. The upper bound is obtained by power truncations and an iteration of Sobolev exponents. The lower bound follows by comparison with the Neumann semigroup. The two-sided estimate gives an $\mathrm{L}^{\infty}$-bound for the comparison semigroup. Nash's inequality and duality then yield ultracontractivity.

math.AP↗

arXiv2026

Ultracontractivity of Heat semigroups in $\mathrm{L}^{2}\left( Ω\right)$ with non-local Robin boundary conditions using Nash's inequality

We study heat equations $\frac{\partial u}{\partial t} - \operatorname{div} \left( A \nabla u \right) = 0$ on bounded Lipschitz domains $Ω$ in $\mathbb{R}^{d}$ for $d \in \mathbb{N}$, where $-\operatorname{div} \left( A \nabla \cdot \right)$ is a second-order uniformly elliptic operator with generalised Robin boundary conditions. These boundary conditions are formally given by $ν\cdot A \nabla u + Bu = 0$ where $ν$ is the outer unit normal on $\partialΩ$ and $B \in \mathcal{L} \left( \mathrm{L}^{2}\left( \partial Ω\right) \right)$ is a general operator which is allowed to destroy the positivity preserving property of the solution semigroup. Ultracontractivity of the solution semigroup is shown by using Nash's inequality on the Sobolev space $H^{1}( Ω)$.

math.AP↗

arXiv2026

Newman's Tauberian theorem, the Riemann-Lebesgue Lemma, and abstract analytic number theory

We give a generalized and effective version of Bekehermes' improvement of Newman's Tauberian theorem. To do so we prove an effective version of the Riemann-Lebesgue Lemma for functions of bounded $p$-variation. We apply our Tauberian theorem to abstract analytic semigroups and prove a version of the prime number theorem as well as an estimate for Mertens' function with explicit error term.

math.CV↗

arXiv2026

Intrinsic Ultracontractivity for a class of Schroedinger Semigroups in $L^{2}(\mathbb{R}^{n})$ by Logarithmic Sobolev inequalities

In the first part of this article we present a growth condition on the potential $q$ in the Schrödinger operator $H=-Δ+ q(x)$ in $\mathrm{L}^{2}\left( \mathbb{R}^{n} \right)$ that implies Rosen inequalities for the ground state $φ$ of $H$, i.e. $\forall \varepsilon > 0 \exists γ(\varepsilon) > 0 \ : \ - \ln\left( φ(x) \right) \leq \varepsilon q(x) + γ(\varepsilon)$. While these inequalities are not particularly interesting in themselves, they offer Logarithmic Sobolev inequalities which are absolutely essential to prove an intrinsic ultracontractivity of the associated Schrödinger semigroup $\mathrm{e}^{-tH}$, i.e. $\forall t>0 \exists C_{t} > 0 \ : \ \left| \mathrm{e}^{-tH} u (x) \right| \ \leq \ C_{t} φ(x) \| u \|_{2}$ holds for every $u \in \mathrm{L}^{2}\left( \mathbb{R}^{n} \right)$ almost everywhere in $\mathbb{R}^{n}$ which we prove in the second part of this article. For proving Rosen inequalities we focus on solving a radial Schrödinger inequality and use Agmon's version of the comparison principle and Young's inequality for increasing functions. We follow the classic method proving intrinsic ultracontractivity of $\mathrm{e}^{-tH}$ by using weighted Sobolev function spaces, weighted Schrödinger semigroups and Logarithmic Sobolev inequalities.

math.AP↗

arXiv2026

Intrinsic Ultracontractivity for a class of Schroedinger Semigroups in $\mathrm{L}^{2}\left( \mathbb{R}^{n} \right)$ using Log-Sobolev-inequalities and duality arguments

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 $\mathbb{R}^{n}$.

math.AP↗