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Bernhard H. Haak

Publications and source records attributed to Bernhard H. Haak.

9 recordsLinked to original sources
arXiv2026

A weak type $(p,a)$ criterion for operators, and applications

Let $(X, d, μ)$ be a space of homogeneous type and $Ω$ an open subset of $X$. Given a bounded operator $T: L^p(Ω) \to L^q(Ω)$ for some $1 \le p \le q < \infty$, we give a criterion for $T$ to be of weak type $(p_0, a)$ for $p_0$ and $a$ such that $\frac{1}{p_0} - \frac{1}{a} = \frac{1}{p}-\frac{1}{q}$. These results are illustrated by several applications including estimates of weak type $(p_0, a)$ for Riesz potentials $L^{-\fracα{2}}$ or for Riesz transform type operators $\nabla Δ^{-\fracα{2}}$ as well as $L^p-L^q$ boundedness of spectral multipliers $F(L)$ when the heat kernel of $L$ satisfies a Gaussian upper bound or an off-diagonal bound. We also prove boundedness of these operators from the Hardy space $H^1_L$ associated with $L$ into $L^a(X)$. By duality this gives boundedness from $L^{a'}(X)$ into $\text{BMO}_L$.

math.FA↗
arXiv2025

A cheap way to closed operator sums

Let $A$ and $B$ be sectorial operators in a Banach space $X$ of angles $ω_A$ and $ω_B$, respectively, where $ω_A+ω_B<π$. We present a simple and common approach to results on closedness of the operator sum $A+B$, based on Littlewood-Paley type norms and tools from several interpolation theories. This allows us to give short proofs for the well-known results due to Da~Prato-Grisvard and Kalton-Weis. We prove a new result in $\ell^q$-interpolation spaces and illustrate it with a maximal regularity result for abstract parabolic equations. Our approach also yields a new proof for the Dore-Venni result.

math.FA↗
arXiv2024

Vector-Valued Holomorphic Functions and Abstract Fubini-Type Theorems

Let $f = f(z,t)$ be a function holomorphic in $z \in O \subseteq {\mathbb C}^d$ for fixed $t\in Ω$ and measurable in $t$ for fixed $z$ and such that$z \mapsto f(z,\cdot)$ is bounded with values in$E := L_{p}(Ω)$, $1\le p \le \infty$. It is proved (among other things) that \[ \langle t\mapsto φ( f(\cdot,t) ) , μ\rangle= φ(z \mapsto \langle f(z, \cdot) , μ\rangle )\] whenever $μ\in E'$ and $φ$ is a linear functional on $H^\infty(O)$ that is sequentially continuous with respect to bounded pointwise convergence in $H^\infty(O)$.

math.FA↗
arXiv2017

Mathematical Analysis of the Motion of a Rigid Body in a Compressible Navier-Stokes-Fourier Fluid

We study an initial and boundary value problem modelling the motion of a rigid body in a heat conducting gas. The solid is supposed to be a perfect thermal insulator. The gas is described by the compressible Navier-Stokes-Fourier equations, whereas the motion of the solid is governed by Newton's laws. The main results assert the existence of strong solutions, in an L p-L q setting, both locally in time and globally in time for small data. The proof is essentially using the maximal regularity property of associated linear systems. This property is checked by proving the R-sectoriality of the corresponding operators, which in turn is obtained by a perturbation method.

math.AP↗
arXiv2011

Uniformly gamma-radonifying families of operators and and the stochastic Weiss conjecture

We introduce the notion of uniform gamma-radonification of a family of operators, which unifies the notions of R-boundedness of a family of operators and gamma-radonification of an individual operator. We study the the properties of uniformly gamma-radonifying families of operators in detail and apply our results to the stochastic abstract Cauchy problem $dU(t) = AU(t) dt + B dW(t); U(0) = 0$ Here, $A$ is the generator of a strongly continuous semigroup of operators on a Banach space $E$, $B$ is a bounded linear operator from a separable Hilbert space $H$ into $E$, and $W$ is an $H$-cylindrical Brownian motion. When $A$ and $B$ are simultaneously diagonalisable, we prove that an invariant measure exists if and only if the family $ \{\sqrtλ R(λ, A)B : λ> 0\} $ is uniformly gamma-radonifying. This result can be viewed as a partial solution of a stochastic version of the Weiss conjecture in linear systems theory.

math.FA↗
arXiv2009

Admissibility and Controllability of diagonal Volterra equations with scalar inputs

This article studies Volterra evolution equations from the point of view of control theory, in the case that the generator of the underlying semigroup has a Riesz basis of eigenvectors. Conditions for admissibility of the system's control operator are given in terms of the Carleson embedding properties of certain discrete measures. Moreover, exact and null controllability are expressed in terms of a new interpolation question for analytic functions, providing a generalization of results known to hold for the standard Cauchy problem. The results are illustrated by examples involving heat conduction with memory.

math.OC↗
arXiv2008

On Kato's method for Navier--Stokes Equations

We investigate Kato's method for parabolic equations with a quadratic non-linearity in an abstract form. We extract several properties known from linear systems theory which turn out to be the essential ingredients for the method. We give necessary and sufficient conditions for these conditions and provide new and more general proofs, based on real interpolation. In application to the Navier-Stokes equations, our approach unifies several results known in the literature, partly with different proofs. Moreover, we establish new existence and uniqueness results for rough initial data on arbitrary domains in ${\mathbb R}^3$ and irregular domains in ${\mathbb R}^n$.

math.AP↗
arXiv2006

Weighted Admissibility and Wellposedness of linear systems in Banach spaces

We study linear control systems in infinite--dimensional Banach spaces governed by analytic semigroups. For $p\in[1,\infty]$ and $α\in\RR$ we introduce the notion of $L^p$--admissibility of type $α$ for unbounded observation and control operators. Generalising earlier work by Le Merdy and the first named author and Le Merdy we give conditions under which $L^p$--admissibility of type $α$ is characterised by boundedness conditions which are similar to those in the well--known Weiss conjecture. We also study $L^p$--wellposedness of type $α$ for the full system. Here we use recent ideas due to Pruess and Simonett. Our results are illustrated by a controlled heat equation with boundary control and boundary observation where we take Lebesgue and Besov spaces as state space. This extends the considerations from Byrnes, Gilliam, Shubov and Weiss to non--Hilbertian settings and to $p\neq 2$.

math.OC↗