Searcharxiv⌕ Search

arXiv subjects

Christoph Kriegler

Publications and source records attributed to Christoph Kriegler.

At least 19 recordsLinked to original sources
arXiv2026

Functional calculus and semilinear evolution equations for the Taibleson operator on non-Archimedean local fields

For any non-Archimedean local field $\mathbb{K}$ and any integer $n \geq 1$, we show that the Taibleson operator admits a bounded $\mathrm{H}^\infty(Σ_θ)$ functional calculus on the Bochner space $\mathrm{L}^p(\mathbb{K}^n,Y)$ for any $\mathrm{UMD}$ Banach function space $Y$ and any angle $θ> 0$, where $Σ_θ=\{ z \in \mathbb{C}^*: |\arg z| < θ\}$ and $1 < p < \infty$. Moreover, we prove that it even admits a bounded Hörmander functional calculus of order $\frac{3}{2}$. In our study, we explore harmonic analysis on locally compact Spector-Vilenkin groups and establish the $R$-boundedness of a family of convolution operators. Our results contribute to the theory of functional calculi for operators acting on vector-valued $\mathrm{L}^p$-spaces over totally disconnected spaces. As an application, we obtain maximal regularity results and well-posedness for a class of evolution equations driven by the Taibleson operator.

math.CA↗
arXiv2026

Positive isometric Fourier multipliers on non-commutative $L^p$-spaces

For a locally compact group \(G\), let \(\mathcal{L}G\) denote its left group von Neumann algebra and let \(L^p(\mathcal{L}G)\), \(1 \le p \le \infty\), be the corresponding non-commutative \(L^p\)-space. Given \(ϕ\in L^\infty(G)\), we study the Fourier multiplier \(M_{ϕ,p}\) acting on \(L^p(\mathcal{L}G)\). We prove that for any \(p \neq 2\), the operator \(M_{ϕ,p}\) is a positive surjective isometry if and only if \(ϕ\) coincides locally almost everywhere with a continuous character of \(G\). This characterization extends results obtained recently (jointly with C.~Arhancet) in the unimodular setting.

math.OA↗
arXiv2025

The harmonic oscillator on the Moyal-Groenewold plane: an approach via Lie groups and twisted Weyl tuples

This paper investigates the functional calculus of the harmonic oscillator on each Moyal-Groenewold plane, the noncommutative phase space which is a fundamental object in quantum mechanics. Specifically, we show that the harmonic oscillator admits a bounded $\mathrm{H}^\infty(Σ_ω)$ functional calculus for any angle $0 < ω< \fracπ{2}$ and even a bounded Hörmander functional calculus on the associated noncommutative $\mathrm{L}^p$-spaces, where $Σ_ω=\{ z \in \mathbb{C}^*: |\arg z| <ω\}$. To achieve these results, we develop a connection with the theory of 2-step nilpotent Lie groups by introducing a notion of twisted Weyl tuple and connecting it to some semigroups of operators previously investigated by Robinson via group representations. Along the way, we demonstrate that $\mathrm{L}^p$-square-max decompositions lead to new insights between noncommutative ergodic theory and $R$-boundedness, and we prove a twisted transference principle, which is of independent interest. Our approach accommodates the presence of a constant magnetic field and they are indeed new even in the framework of magnetic Weyl calculus on classical $\mathrm{L}^p$-spaces. Our results contribute to the understanding of functional calculi on noncommutative spaces and have implications for the maximal regularity of the most basic evolution equations associated to the harmonic oscillator.

math.FA↗
arXiv2025

Decomposable Fourier Multipliers and an Operator-Algebraic Characterization of Amenability

We study the algebra $\mathfrak{M}^{\infty,\mathrm{dec}}(G)$ of decomposable Fourier multipliers on the group von Neumann algebra $\mathrm{VN}(G)$ of a locally compact group $G$, and its relation to the Fourier-Stieltjes algebra $\mathrm{B}(G)$. For discrete groups, we prove that these two algebras coincide isometrically. In contrast, we show that the identity $\mathfrak{M}^{\infty,\mathrm{dec}}(G) = \mathrm{B}(G)$ fails for various classes of non-discrete groups, and that, among second-countable unimodular groups, inner amenability ensures the equality. Our approach relies on the existence of contractive projections preserving complete positivity from the space of completely bounded weak* continuous operators on $\mathrm{VN}(G)$ onto the subspace of completely bounded Fourier multipliers. We show that such projections exist in the inner amenable case. As an application, we obtain a new operator-algebraic characterization of amenability. We also investigate the analogous problem for the space of completely bounded Fourier multipliers on the noncommutative $\mathrm{L}^p$-spaces $\mathrm{L}^p(\mathrm{VN}(G))$, for $1 \leq p \leq \infty$. Using Lie group theory and results stemming from the solution to Hilbert's fifth problem, we prove that second-countable unimodular finite-dimensional amenable locally compact groups admit compatible projections at $p = 1$ and $p = \infty$. These results reveal new structural links between harmonic analysis, operator algebras, and the geometry of locally compact groups.

math.FA↗
arXiv2024

$q$-variational H{ö}rmander functional calculus and Schr{ö}dinger and wave maximal estimates

This article is the continuation of the work [DK] where we had proved maximal estimates $$\left\|\sup_{t > 0} |m(tA)f| \right\|_{L^p(Ω,Y)} \leq C \|f\|_{L^p(Ω,Y)}$$ for sectorial operators $A$ acting on $L^p(Ω,Y)$ ($Y$ being a UMD lattice) and admitting a Hörmander functional calculus(a strengthening of the holomorphic $H^\infty$ calculus to symbols $m$ differentiable on $(0,\infty)$ in a quantified manner), and $m : (0, \infty) \to \mathbb{C}$ being a Hörmander class symbol with certain decay at $\infty$.In the present article, we show that under the same conditions as above, the scalar function $t \mapsto m(tA)f(x,ω)$ is of finite $q$-variation with $q > 2$, a.e. $(x,ω)$.This extends recent works by [BMSW,HHL,HoMa1,HoMa,JSW,LMX] who have considered among others $m(tA) = e^{-tA}$ the semigroup generated by $-A$.As a consequence, we extend estimates for spherical means in euclidean space from [JSW] to the case of UMD lattice-valued spaces.A second main result yields a maximal estimate $$\left\|\sup_{t > 0} |m(tA) f_t| \right\|_{L^p(Ω,Y)} \leq C \|f_t\|_{L^p(Ω,Y(Λ^β))}$$ for the same $A$ and similar conditions on $m$ as above but with $f_t$ depending itself on $t$ such that $t \mapsto f_t(x,ω)$ belongs to a Sobolev space $Λ^β$ over $(\mathbb{R}_+, \frac{dt}{t})$.We apply this to show a maximal estimate of the Schrödinger (case $A = -Δ$) or wave (case $A = \sqrt{-Δ}$) solution propagator $t \mapsto \exp(itA)f$.Then we deduce from it variants of Carleson's problem of pointwise convergence [Car]\[ \exp(itA)f(x,ω) \to f(x,ω) \text{ a. e. }(x,ω) \quad (t \to 0+)\]for $A$ a Fourier multiplier operator or a differential operator on an open domain $Ω\subseteq \mathbb{R}^d$ with boundary conditions.

math.CA↗
arXiv2023

Separating Fourier and Schur multipliers

Let $G$ be a locally compact unimodular group, let $1\leq p<\infty$,let $ϕ\in L^\infty(G)$ and assume that the Fourier multiplier $M_ϕ$associated with $ϕ$ is bounded on the noncommutative $L^p$-space $L^p(VN(G))$.Then $M_ϕ\colon L^p(VN(G))\to L^p(VN(G))$ is separating (that is,$\{a^*b=ab^*=0\}\Rightarrow\{M_ϕ(a)^* M_ϕ(b)=M_ϕ(a)M_ϕ(b)^*=0\}$for any $a,b\in L^p(VN(G))$) if and only if thereexists $c\in\mathbb C$ and a continuouscharacter $ψ\colon G\to\mathbb C$ such that $ϕ=cψ$ locally almost everywhere. This provides a characterization of isometricFourier multipliers on $L^p(VN(G))$, when $p\not=2$. Next, let $Ω$ be a $σ$-finite measure space, let $ϕ\in L^\infty(Ω^2)$and assume that the Schur multiplier associated with $ϕ$ is bounded on the Schatten space $S^p(L^2(Ω))$. We prove that this multiplier is separating if and only if there exist a constant $c\in\mathbb C$ and two unitaries $α,β\in L^\infty(Ω)$ such that $ϕ(s,t) =c\, α(s)β(t)$ a.e. on $Ω^2.$ This provides acharacterization of isometric Schur multiplierson $S^p(L^2(Ω))$, when $p\not=2$.

math.CA↗
arXiv2022

Projections, multipliers and decomposable maps on noncommutative $\mathrm{L}^p$-spaces

We introduce a noncommutative analogue of the absolute value of a regular operator acting on a noncommutative $\mathrm{L}^p$-space. We equally prove that two classical operator norms, the regular norm and the decomposable norm are identical. We also describe precisely the regular norm of several classes of regular multipliers. This includes Schur multipliers and Fourier multipliers on some unimodular locally compact groups which can be approximated by discrete groups in various senses. A main ingredient is to show the existence of a bounded projection from the space of completely bounded $\mathrm{L}^p$ operators onto the subspace of Schur or Fourier multipliers, preserving complete positivity. On the other hand, we show the existence of bounded Fourier multipliers which cannot be approximated by regular operators, on large classes of locally compact groups, including all infinite abelian locally compact groups. We finish by introducing a general procedure for proving positive results on selfadjoint contractively decomposable Fourier multipliers, beyond the amenable case.

math.OA↗
arXiv2022

Maximal Hörmander Functional Calculus on Lp Spaces and UMD Lattices

Let $A$ be a generator of an analytic semigroup having a H{ö}rmander functional calculus on $X = L^p(Ω,Y)$, where $Y$ is a UMD lattice. Using methods from Banach space geometry in connection with functional calculus, we show that for H{ö}rmander spectral multipliers decaying sufficiently fast at $\infty$, there holds a maximal estimate $\| \sup_{t \geq 0} |m(tA)f|\, \|_{L^p(Ω,Y)} \lesssim \|f\|_{L^p(Ω,Y)}$. We also show square function estimates $\left\| \left( \sum_k \sup _{t \geq 0} |m_k(tA)f_k|^2 \right)^{\frac12} \right\|_{L^p(Ω,Y)} \lesssim \left\| \left( \sum _k |f_k|^2 \right)^{\frac12} \right\|_{L^p(Ω,Y)}$ for suitable families of spectral multipliers $m_k$, which are even new for the euclidean Laplacian on scalar valued $L^p(\mathbb{R}^d)$. As corollaries, we obtain maximal estimates for wave propagators and Bochner--Riesz means. Finally, we illustrate the results by giving several examples of operators $A$ that admit a H{ö}rmander functional calculus on some $L^p(Ω,Y)$ and discuss examples of lattices $Y$ and non-self-adjoint operators $A$ fitting our context.

math.CA↗
arXiv2020

Riesz transforms, Hodge-Dirac operators and functional calculus for multipliers I

In this work, we solve the problem explicitly stated at the end of a paper of Junge, Mei and Parcet [JEMS2018, Problem C.5] for a large class of groups including all amenable groups and free groups. More precisely, we prove that the Hodge-Dirac operator of the canonical "hidden" noncommutative geometry associated with a Markov semigroup $(T_t)_{t \geq 0}$ of Fourier multipliers is bisectorial and admits a bounded $\mathrm{H}^\infty$ functional calculus on a bisector which implies a positive answer to the quoted problem. Our result can be seen as a strengthening of the dimension free estimates of Riesz transforms of the above authors and also allows us to provide Hodge decompositions. A part of our proof relies on a new transference argument between multipliers which is of independent interest. Our results are even new for the Poisson semigroup on $\mathbb{T}^n$. We also provide a similar result for Markov semigroups of Schur multipliers and dimension free estimates for noncommutative Riesz transforms associated with these semigroups. Along the way, we also obtain new Khintchine type equivalences for $q$-Gaussians in $\mathrm{L}^p$-spaces associated to crossed products. Our results allow us to introduce new spectral triples (i.e. noncommutative manifolds) and new quantum (locally) compact metric spaces, in connection with the carré du champ, which summarize the underlying geometry of our setting. Finally, our examples lead us to introduce a Banach space variant of the notion of spectral triple suitable for our context.

math.OA↗
arXiv2019

Hörmander functional calculus on UMD lattice valued $L^p$ spaces under generalised Gaussian estimates

We consider self-adjoint semigroups $T_t = \exp(-tA)$ acting on $L^2(Ω)$ and satisfying (generalised) Gaussian estimates, where $Ω$ is a metric measure space of homogeneous type of dimension $d$. The aim of the article is to show that $A \otimes \mathrm{Id}_Y$ admits a Hörmander type $\mathcal{H}^β_2$ functional calculus on $L^p(Ω;Y)$ where $Y$ is a UMD lattice, thus extending the well-known Hörmander calculus of $A$ on $L^p(Ω)$. We show that if $T_t$ is lattice positive (or merely admits an $H^\infty$ calculus on $L^p(Ω;Y)$) then this is indeed the case. Here the derivation exponent has to satisfy $β> α\cdot d + \frac12$, where $α\in (0,1)$ depends on $p$, and on convexity and concavity exponents of $Y$. A part of the proof is the new result that the Hardy-Littlewood maximal operator is bounded on $L^p(Ω;Y)$. Moreover, our spectral multipliers satisfy square function estimates in $L^p(Ω;Y)$. In a variant, we show that if $e^{itA}$ satisfies a dispersive $L^1(Ω) \to L^\infty(Ω)$ estimate, then $β> \frac{d+1}{2}$ above is admissible independent of convexity and concavity of $Y$. Finally, we illustrate these results in a variety of examples.

math.FA↗
arXiv2019

$H^\infty$ calculus for submarkovian semigroups on weighted $L^2$ spaces

Let $(T_t)_{t \geq 0}$ be a markovian (resp. submarkovian) semigroup on some $σ$-finite measure space $(Ω,μ)$. We prove that its negative generator $A$ has a bounded $H^\infty(Σ_θ)$ calculus on the weighted space $L^2(Ω,wdμ)$ as long as the weight $w : Ω\to (0,\infty)$ has finite characteristic defined by $Q^A_2(w) = \sup_{t > 0} \left\| T_t(w) T_t \left(w^{-1} \right) \right\|_{L^\infty(Ω)}$ (resp. by a variant for submarkovian semigroups). Some additional technical conditions on the semigroup have to be imposed and their validity in examples is discussed. Any angle $θ> \fracπ{2}$ is admissible in the above $H^\infty$ calculus, and for some semigroups also certain $θ= θ_w < \fracπ{2}$ depending on the size of $Q^A_2(w)$. The norm of the $H^\infty(Σ_θ)$ calculus is linear in the $Q^A_2$ characteristic for $θ> \fracπ{2}$. We also discuss negative results on angles $θ< \fracπ{2}$. Namely we show that there is a markovian semigroup on a probability space and a $Q^A_2$ weight $w$ without Hörmander functional calculus on $L^2(Ω,w dμ)$.

math.CA↗
arXiv2018

H{ö}rmander Functional Calculus for Poisson Estimates

The aim of the article is to show a H{ö}rmander spectral multiplier theorem for an operator $A$ whose kernel of the semigroup $\exp(-zA)$ satisfies certain Poisson estimates for complex times $z.$ Here $\exp(-zA)$ acts on $L^p(Ω),\,1 < p < \infty,$ where $Ω$ is a space of homogeneous type with the additional condition that the measure of annuli is controlled. In most of the known H{ö}rmander type theorems in the literature, Gaussian bounds and self-adjointness for the semigroup are needed, whereas here the new feature is that the assumptions are the to some extend weaker Poisson bounds, and $\HI$ calculus in place of self-adjointness. The order of derivation in our H{ö}rmander multiplier result is typically $\frac{d}{2},$ $d$ being the dimension of the space $Ω.$ Moreover the functional calculus resulting from our H{ö}rmander theorem is shown to be $R$-bounded. Finally, the result is applied to some examples.

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↗
arXiv2017

Dimension free bounds for the vector-valued Hardy-Littlewood maximal operator

In this article, Fefferman-Stein inequalities in $L^p(\mathbb R^d;\ell^q)$ withbounds independent of the dimension $d$ are proved, for all $1 \textless{} p, q \textless{} + \infty.$This result generalizes in a vector-valued setting the famous one by Steinfor the standard Hardy-Littlewood maximal operator. We then extendour result by replacing $\ell^q$ with an arbitrary UMD Banach lattice. Finally,we prove similar dimensionless inequalities in the setting of the Grushinoperators.

math.FA↗
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↗
arXiv2012

Spectral multipliers for wave operators

A classical theorem of Mihlin yields Lp estimates for spectral multipliers Lp(R^d) -> Lp(R^d); g -> F^{-1}[f(| |^2) Fg] in terms of L^\infty bounds of the multiplier function f and its weighted derivatives up to an order > d/2. This theorem, which is a functional calculus for the standard Laplace operator, has generalisations in several contexts such as elliptic operators on domains and manifolds, Schrödinger operators and sublaplacians on Lie groups. However, for the wave equation functions f (s) = (1 + s)^{-α} e^{its} a better estimate is available, in the standard case (works of Miyachi and Peral) and on Heisenberg Lie groups (Müller and Stein). By a transference method for polynomially bounded regularized groups, we obtain a new class of spectral multipliers for operators that have these better wave spectral multipliers and that admit a spectral decomposition of Paley-Littlewood type.

math.FA↗