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Lutz Weis

Publications and source records attributed to Lutz Weis.

At least 19 recordsLinked to original sources
arXiv2026

Necessary conditions for deterministic and stochastic maximal regularity

We study the role of Banach space geometry in deterministic and stochastic maximal regularity. We first construct an example showing that the UMD assumption in the characterisation of maximal $L^p$-regularity in terms of $R$-sectoriality cannot be omitted. Combining this construction with an equivalence between stochastic maximal regularity and deterministic maximal regularity on the $2$-concavification of the underlying space, we obtain an operator on a UMD Banach function space of type $2$ that has a bounded $H^\infty$-calculus of angle zero, but fails stochastic maximal $L^p$-regularity (SMR$_p$) for every $p\in[2,\infty)$. Motivated by this example, we study the Banach space geometry hypothesis underlying SMR$_p$ more closely. This is an $R$-boundedness condition $(S_p)$ for stochastic convolution operators. For UMD spaces $X$ of type $2$, we show that this condition is not only sufficient, but also necessary for two canonical test operators: a diagonal multiplier on a Rademacher space and, for $q>2$, the Laplacian on $L^q(\mathbb R^d;X)$. Finally, we prove that its interval-kernel and exponential-kernel formulations are equivalent and that, at the endpoint $p=2$, condition $(S_2)$ holds if and only if $X$ is isomorphic to a Hilbert space.

math.FA↗
arXiv2023

Euclidean structures and operator theory in Banach spaces

We present a general method to extend results on Hilbert space operators to the Banach space setting by representing certain sets of Banach space operators $Γ$ on a Hilbert space. Our assumption on $Γ$ is expressed in terms of $α$-boundedness for a Euclidean structure $α$ on the underlying Banach space $X$. This notion is originally motivated by $\mathcal{R}$- or $γ$-boundedness of sets of operators, but, for example, any operator ideal from the Euclidean space $\ell^2_n$ to $X$ defines such a structure. Therefore, our method is quite flexible. Conversely we show that $Γ$ has to be $α$-bounded for some Euclidean structure $α$ to be representable on a Hilbert space. By choosing the Euclidean structure $α$ accordingly, we get a unified and more general approach to classical factorization and extension theorems. Furthermore we use these Euclidean structures to build vector-valued function spaces and define an interpolation method based on these spaces, which has formulations modelled after both the real and the complex interpolation method. Using our representation theorem we prove a transference principle for sectorial operators on a Banach space, enabling us to extend Hilbert space results for sectorial operators to the Banach space setting. We define generalizations of the classical square function estimates in $L^p$-spaces and establish, via the $H^\infty$-calculus, a version of Littlewood-Paley theory and associated spaces of fractional smoothness for a rather large class of sectorial operators. Our results for sectorial operators lead to some sophisticated counterexamples.

math.FA↗
arXiv2018

Spectral multiplier theorems and averaged R-boundedness

Let $A$ be a $0$-sectorial operator with a bounded $H^\infty(Σ\_σ)$-calculus for some $σ\in (0,π),$ e.g. a Laplace type operator on $L^p(Ω),\: 1 < p < \infty,$ where $Ω$ is a manifold or a graph. We show that $A$ has a H{ö}rmander functional calculus if and only if certain operator families derived from the resolvent $(λ- A)^{-1},$ the semigroup $e^{-zA},$ the wave operators $e^{itA}$ or the imaginary powers $A^{it}$ of $A$ are $R$-bounded in an $L^2$-averaged sense. If $X$ is an $L^p(Ω)$ space with $1 \leq p < \infty,$ $R$-boundedness reduces to well-known estimates of square sums.

math.FA↗
arXiv2018

Spectral multiplier theorems via $H^\infty$ calculus and $R$-bounds

We prove spectral multiplier theorems for Hörmander classes $\mathcal{H}^α\_p$ for 0-sectorial operators A on Banach spaces assuming a bounded $H^\infty(Σ\_σ)$ calculus for some $σ\in (0,π)$ and norm and certain R-bounds on one of the following families of operators: the semigroup $e^{--zA}$ on $\mathbb{C}\_+$, the wave operators $e^{isA}$ for $s \in \mathbb{R}$, the resolvent $(λ-- A)^{-1}$ on $\mathbb{C} \backslash \mathbb{R}$, the imaginary powers $A^{it}$ for $t \in \mathbb{R}$ or the Bochner-Riesz means $(1-A/u)^α\_+$ for $u > 0.$ In contrast to the existing literature we neither assume that A operates on an Lp scale nor that A is self-adjoint on a Hilbert space. Furthermore, we replace (generalized) Gaussian or Poisson bounds and maximal estimates by the weaker notion of R-bounds, which allow for a unified approach to spectral multiplier theorems in a more general setting. In this setting our results are close to being optimal. Moreover, we can give a characterization of the (R-bounded) $\mathcal{H}^α\_1$ calculus in terms of R-boundedness of Bochner-Riesz means.

math.FA↗
arXiv2018

Martingale solutions for the stochastic nonlinear Schrödinger equation in the energy space

We consider a stochastic nonlinear Schrödinger equation with multiplicative noise in an abstract framework that covers subcritical focusing and defocusing stochastic NLS in $H^1$ on compact manifolds and bounded domains. We construct a martingale solution using a modified Faedo-Galerkin-method based on the Littlewood-Paley-decomposition. For 2d manifolds with bounded geometry, we use Strichartz estimates to show pathwise uniqueness.

math.PR↗
arXiv2018

Uniqueness of martingale solutions for the stochastic nonlinear Schrödinger equation on 3d compact manifolds

We prove pathwise uniqueness for solutions of the nonlinear Schrödinger equation with conservative multiplicative noise on compact 3D manifolds. In particular, we generalize the result by Burq, Gérard and Tzvetkov (N. Burq, P. Gérard, and N. Tzvetkov. Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds. American Journal of Mathematics, 126 (3):569--605, 2004) to the stochastic setting. The proof is based on deterministic and stochastic Strichartz estimates and the Littlewood-Paley decomposition.

math.AP↗
arXiv2016

Paley-Littlewood decomposition for sectorial operators and interpolation spaces

We prove Paley-Littlewood decompositions for the scales of fractional powers of $0$-sectorial operators $A$ on a Banach space which correspond to Triebel-Lizorkin spaces and the scale of Besov spaces if $A$ is the classical Laplace operator on $L^p(\mathbb{R}^n).$We use the $H^\infty$-calculus, spectral multiplier theorems and generalized square functions on Banach spaces and apply our results to Laplace-type operators on manifolds and graphs, Schrödinger operators and Hermite expansion.We also give variants of these results for bisectorial operators and for generators of groups with a bounded $H^\infty$-calculus on strips.

math.FA↗
arXiv2015

Estimates for vector-valued holomorphic functions and Littlewood-Paley-Stein theory

In this paper we consider generalized square function norms of holomorphic functions with values in a Banach space. One of the main results is a characterization of embeddings of the form \[L^p(X)\subseteq γ(X) \subseteq L^q(X),\] in terms of the type $p$ and cotype $q$ for the Banach space $X$. As an application we prove $L^p$-estimates for vector-valued Littlewood-Paley-Stein $g$-functions and derive an embedding result for real and complex interpolation spaces under type and cotype conditions.

math.FA↗
arXiv2015

The $H^{\infty}$-Functional Calculus and Square Function Estimates

Using notions from the geometry of Banach spaces we introduce square functions $γ(Ω,X)$ for functions with values in an arbitrary Banach space $X$. We show that they have very convenient function space properties comparable to the Bochner norm of $L_2(Ω,H)$ for a Hilbert space $H$. In particular all bounded operators $T$ on $H$ can be extended to $γ(Ω,X)$ for all Banach spaces $X$. Our main applications are characterizations of the $H^{\infty}$--calculus that extend known results for $L_p$--spaces from \cite{CowlingDoustMcIntoshYagi}. With these square function estimates we show, e. g., that a $c_0$--group of operators $T_s$ on a Banach space with finite cotype has an $H^{\infty}$--calculus on a strip if and only if $e^{-a|s|}T_s$ is $R$--bounded for some $a > 0$. Similarly, a sectorial operator $A$ has an $H^{\infty}$--calculus on a sector if and only if $A$ has $R$--bounded imaginary powers. We also consider vector valued Paley--Littlewood $g$--functions on $UMD$--spaces.

math.FA↗
arXiv2015

$R$-boundedness versus $γ$-boundedness

It is well-known that in Banach spaces with finite cotype, the $R$-bounded and $γ$-bounded families of operators coincide. If in addition $X$ is a Banach lattice, then these notions can be expressed as square function estimates. It is also clear that $R$-boundedness implies $γ$-boundedness. In this note we show that all other possible inclusions fail. Furthermore, we will prove that $R$-boundedness is stable under taking adjoints if and only if the underlying space is $K$-convex.

math.FA↗
arXiv2014

Maximal gamma-regularity

In this paper we prove maximal regularity estimates in "square function spaces" which are commonly used in harmonic analysis, spectral theory, and stochastic analysis. In particular, they lead to a new class of maximal regularity results for both deterministic and stochastic equations in $L^p$-spaces with $1<p<\infty$. For stochastic equations, the case $1<p<2$ was not covered in the literature so far. Moreover, the "square function spaces" allow initial values with the same roughness as in the $L^2$-setting.

math.FA↗
arXiv2014

On the R-boundedness of stochastic convolution operators

The $R$-boundedness of certain families of vector-valued stochastic convolution operators with scalar-valued square integrable kernels is the key ingredient in the recent proof of stochastic maximal $L^p$-regularity, $2<p<\infty$, for certain classes of sectorial operators acting on spaces $X=L^q(μ)$, $2\le q<\infty$. This paper presents a systematic study of $R$-boundedness of such families. Our main result generalises the afore-mentioned $R$-boundedness result to a larger class of Banach lattices $X$ and relates it to the $\ell^{1}$-boundedness of an associated class of deterministic convolution operators. We also establish an intimate relationship between the $\ell^{1}$-boundedness of these operators and the boundedness of the $X$-valued maximal function. This analysis leads, quite surprisingly, to an example showing that $R$-boundedness of stochastic convolution operators fails in certain UMD Banach lattices with type $2$.

math.FA↗
arXiv2014

Stochastic integration in Banach spaces - a survey

This paper presents a brief survey of the theory of stochastic integration in Banach spaces. Expositions of the stochastic integrals in martingale type 2 spaces and UMD spaces are presented, as well as some applications of the latter to vector-valued Malliavin calculus and the stochastic maximal regularity problem. A new proof of the stochastic maximal regularity theorem is included.

math.PR↗
arXiv2012

Stochastic maximal $L^p$-regularity

In this article we prove a maximal $L^p$-regularity result for stochastic convolutions, which extends Krylov's basic mixed $L^p(L^q)$-inequality for the Laplace operator on ${\mathbb{R}}^d$ to large classes of elliptic operators, both on ${\mathbb{R}}^d$ and on bounded domains in ${\mathbb{R}}^d$ with various boundary conditions. Our method of proof is based on McIntosh's $H^{\infty}$-functional calculus, $R$-boundedness techniques and sharp $L^p(L^q)$-square function estimates for stochastic integrals in $L^q$-spaces. Under an additional invertibility assumption on $A$, a maximal space--time $L^p$-regularity result is obtained as well.

math.PR↗
arXiv2012

Maximal $L^p$-regularity for stochastic evolution equations

We prove maximal $L^p$-regularity for the stochastic evolution equation \[\{{aligned} dU(t) + A U(t)\, dt& = F(t,U(t))\,dt + B(t,U(t))\,dW_H(t), \qquad t\in [0,T], U(0) & = u_0, {aligned}.\] under the assumption that $A$ is a sectorial operator with a bounded $H^\infty$-calculus of angle less than $\frac12π$ on a space $L^q(\mathcal{O},μ)$. The driving process $W_H$ is a cylindrical Brownian motion in an abstract Hilbert space $H$. For $p\in (2,\infty)$ and $q\in [2,\infty)$ and initial conditions $u_0$ in the real interpolation space $\XAp $ we prove existence of unique strong solution with trajectories in \[L^p(0,T;\Dom(A))\cap C([0,T];\XAp),\] provided the non-linearities $F:[0,T]\times \Dom(A)\to L^q(\mathcal{O},μ)$ and $B:[0,T]\times \Dom(A) \to \g(H,\Dom(A^{\frac12}))$ are of linear growth and Lipschitz continuous in their second variables with small enough Lipschitz constants. Extensions to the case where $A$ is an adapted operator-valued process are considered as well. Various applications to stochastic partial differential equations are worked out in detail. These include higher-order and time-dependent parabolic equations and the Navier-Stokes equation on a smooth bounded domain $\OO\subseteq \R^d$ with $d\ge 2$. For the latter, the existence of a unique strong local solution with values in $(H^{1,q}(\OO))^d$ is shown.

math.PR↗
arXiv2010

A note on maximal estimates for stochastic convolutions

In stochastic partial differential equations it is important to have pathwise regularity properties of stochastic convolutions. In this note we present a new sufficient condition for the pathwise continuity of stochastic convolutions in Banach spaces.

math.PR↗
arXiv2008

The Banach space -valued BMO, Carleson's condition, and paraproducts

We define a scale of L^q Carleson norms, all of which characterize the membership of a function in BMO. The phenomenon is analogous to the John-Nirenberg inequality, but on the level of Carleson measures. The classical Carleson condition corresponds to the L^2 case in our theory. The result is applied to give a new proof for the L^p-boundedness of paraproducts with a BMO symbol. A novel feature of the argument is that all p are covered at once in a completely interpolation-free manner. This is achieved by using the L^1 Carleson norm, and indicates the usefulness of this notion. Our approach is chosen so that all these results extend in a natural way to the case of X-valued functions, where X is a Banach space with the UMD property.

math.FA↗