Peer Christian Kunstmann

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Maximal L p -regularity for Parabolic Equations, Fourier Multiplier Theorems and $H^\infty$ -functional Calculus

2004-01-01

Peer Christian Kunstmann , Lutz Weis

Calderón-Zygmund theory for non-integral operators and the $H^{\infty}$ functional calculus

2003-12-31

Sönke Blunck , Peer Christian Kunstmann

We modify Hörmander's well-known weak type (1,1) condition for integral operators (in a weakened version due to Duong and McIntosh) and present a weak type (p,p) condition for arbitrary operators. … We modify Hörmander's well-known weak type (1,1) condition for integral operators (in a weakened version due to Duong and McIntosh) and present a weak type (p,p) condition for arbitrary operators. Given an operator A on L_2 with a bounded H^\infty calculus, we show as an application the L_r -boundedness of the H^\infty calculus for all r\in(p,q) , provided the semigroup (e^{-tA}) satisfies suitable weighted L_p\to L_q -norm estimates with 2\in(p,q) . This generalizes results due to Duong, McIntosh and Robinson for the special case (p,q)=(1,\infty) where these weighted norm estimates are equivalent to Poisson-type heat kernel bounds for the semigroup (e^{-tA}) . Their results fail to apply in many situations where our improvement is still applicable, e.g. if A is a Schrödinger operator with a singular potential, an elliptic higher order operator with bounded measurable coefficients or an elliptic second order operator with singular lower order terms.

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Functional Analytic Methods for Evolution Equations

2004-01-01

Giuseppe Da Prato , Peer Christian Kunstmann , Lutz Weis +3 more

Erratum to: Perturbation and interpolation theorems for the H ∞-calculus with applications to differential operators

2013-08-16

Peer Christian Kunstmann , Lutz Weis

In our article [2], some of the general boundedness results in Sects.7 and 8 for the H ∞ -functional calculus may not be correct as stated since when applying [4, … In our article [2], some of the general boundedness results in Sects.7 and 8 for the H ∞ -functional calculus may not be correct as stated since when applying [4, 1.2.4] our proofs use implicitly certain inclusions of interpolation spaces, which were not stated as assumptions and which do not hold in all generality.We would like to point out here that, with these assumptions added we obtain correct results.As a consequence, in an application to the Stokes operator in Sect.9 we have to strengthen the regularity assumption on the underlying domain to ensure that our additional assumption is satisfied. Abstract resultsTheorem 7.9 in [2] should read: Theorem 1.1 (cf.[2, Thm.7.9]) Let Y be a complemented subspace of a B-convex Banach space X .Let A have an H ∞ -calculus on X and let B be almost R-sectorial on Y .If P( Ẋβ j ,A ) = Ẏβ j ,B and Ẏβ j ,B → Ẋβ j ,A for two different β 1 , β 2 = 0 with |β j | ≤ m then B has an H ∞ -calculus on Y .

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Weak type (p,p) estimates for Riesz transforms

2004-04-07

Sönke Blunck , Peer Christian Kunstmann

Perturbation and Interpolation Theorems for the H ∞-Calculus with Applications to Differential Operators

2006-08-08

N. J. Kalton , Peer Christian Kunstmann , Lutz Weis

Weighted norm estimates and maximal regularity

2002-01-01

Sönke Blunck , Peer Christian Kunstmann

We give a sufficient condition for maximal regularity of the evolution equation $u'(t) - Au(t) = f(t) ,\ t\ge0 ,\ u(0)=0,$ in $L_p$-spaces. Our condition is a weighted norm estimate … We give a sufficient condition for maximal regularity of the evolution equation $u'(t) - Au(t) = f(t) ,\ t\ge0 ,\ u(0)=0,$ in $L_p$-spaces. Our condition is a weighted norm estimate for the semigroup $(e^{tA})$ and it is strictly weaker than the assumption that the $e^{tA}$ are integral operators whose kernels satisfy Gaussian estimates. As an application we present new results for the maximal regularity of Schr\"odinger operators with singular potentials, elliptic higher order operators with bounded measurable coefficients, and elliptic second order operators with singular lower order terms. Moreover, we prove a similar result for maximal regularity of the discrete time evolution equation $ u_{n+1} - Tu_n = f_n ,$ $n\in\mathbb N_0 ,$ $u_0=0 $.

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Distribution semigroups and abstract Cauchy problems

1999-01-01

Peer Christian Kunstmann

We present a new definition of distribution semigroups, covering in particular non-densely defined generators. We show that for a closed operator $A$ in a Banach space $E$ the following assertions … We present a new definition of distribution semigroups, covering in particular non-densely defined generators. We show that for a closed operator $A$ in a Banach space $E$ the following assertions are equivalent: (a) $A$ generates a distribution semigroup; (b) the convolution operator $\delta '\otimes I-\delta \otimes A$ has a fundamental solution in ${\mathcal D}'(L(E,D))$ where $D$ denotes the domain of $A$ supplied with the graph norm and $I$ denotes the inclusion $D\to E$; (c) $A$ generates a local integrated semigroup. We also show that every generator of a distribution semigroup generates a regularized semigroup.

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Spectral multiplier theorems of Hormander type on Hardy and Lebesgue spaces

2015-02-01

Peer Christian Kunstmann , Matthias Uhl

Let X be a space of homogeneous type and let L be an injective, non-negative, selfadjoint operator on L 2 (X) such that the semigroup generated by -L fulfills Davies-Gaffney … Let X be a space of homogeneous type and let L be an injective, non-negative, selfadjoint operator on L 2 (X) such that the semigroup generated by -L fulfills Davies-Gaffney estimates of arbitrary order.We prove that the operator F (L), initially defined on H 1 L (X) ∩ L 2 (X), acts as a bounded linear operator on the Hardy space H 1 L (X) associated with L whenever F is a bounded, sufficiently smooth function.Based on this result, together with interpolation, we establish Hörmander type spectral multiplier theorems on Lebesgue spaces for non-negative, self-adjoint operators satisfying generalized Gaussian estimates in which the required differentiability order is relaxed compared to all known spectral multiplier results.

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Perturbation theorems for maximal $L_p$-regularity

2001-01-01

Peer Christian Kunstmann , Lutz Weis

In this paper we prove perturbation theorems for R-sectorial operators. Via the characterization of maximal Lp-regularity in terms of R-boundedness due to the second author we obtain perturbation theorems for … In this paper we prove perturbation theorems for R-sectorial operators. Via the characterization of maximal Lp-regularity in terms of R-boundedness due to the second author we obtain perturbation theorems for maximal Lp-regularity in UMD -spaces. We prove that R-sectoriality of A is preserved by A-small perturbations and by perturbations that are bounded in a fractional scale and small in a certain sense. Here, our method seems to give new results even for sectorial operators. We apply our results to uniformly elliptic systems with bounded uniformly continuous coefficients, to Schr6dinger operators with bad potentials, to the perturbation of boundary conditions, and to pseudo-differential operators with non-smooth symbols.

Admissibility of Unbounded Operators and Wellposedness of Linear Systems in Banach Spaces

2005-12-20

Bernhard H. Haak , Peer Christian Kunstmann

Heat Kernel Estimates and L P Spectral Independence of Elliptic Operators

1999-05-01

Peer Christian Kunstmann

Runge–Kutta Time Discretization of Nonlinear Parabolic Equations Studied via Discrete Maximal Parabolic Regularity

2017-08-10

Peer Christian Kunstmann , Buyang Li , Christian Lubich

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Weighted Admissibility and Wellposedness of Linear Systems in Banach Spaces

2007-01-01

Bernhard H. Haak , Peer Christian Kunstmann

We study linear control systems in infinite‐dimensional Banach spaces governed by analytic semigroups. For $p\in[1,\infty]$ and $\alpha\in\mathbb{R}$ we introduce the notion of $L^p$‐admissibility of type $\alpha$ for unbounded observation and … We study linear control systems in infinite‐dimensional Banach spaces governed by analytic semigroups. For $p\in[1,\infty]$ and $\alpha\in\mathbb{R}$ we introduce the notion of $L^p$‐admissibility of type $\alpha$ for unbounded observation and control operators. Generalizing earlier work by Le Merdy [J. London Math. Soc. (2), 67 (2003), pp. 715–738] and Haak and Le Merdy [Houston J. Math., 31 (2005), pp. 1153–1167], we give conditions under which $L^p$‐admissibility of type $\alpha$ is characterized by boundedness conditions which are similar to those in the well‐known Weiss conjecture. We also study $L^p$‐wellposedness of type $\alpha$ for the full system. Here we use recent ideas due to Pruss and Simonett [Arch. Math. (Basel), 82 (2004), pp. 415–431]. 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 in [C. I. Byrnes et al., J. Dynam. Control Systems, 8 (2002), pp. 341–370] to non‐Hilbertian settings and to $p\neq2$.

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New criteria for the $${H^\infty}$$ H ∞ -calculus and the Stokes operator on bounded Lipschitz domains

2016-09-22

Peer Christian Kunstmann , Lutz Weis

Perturbation, interpolation, and maximal regularity

2006-01-01

Bernhard H. Haak , Markus Haase , Peer Christian Kunstmann

We prove perturbation theorems for sectoriality and $R$--sectoriality in Banach spaces, which yield results on perturbation of generators of analytic semigroups and on perturbation of maximal $L^p$--regularity. For a given … We prove perturbation theorems for sectoriality and $R$--sectoriality in Banach spaces, which yield results on perturbation of generators of analytic semigroups and on perturbation of maximal $L^p$--regularity. For a given sectorial or $R$--sectorial operator $A$ in a Banach space $X$ we give conditions on intermediate spaces $Z$ and $W$ such that, for an operator $S: Z\to W$ of small norm, the perturbed operator $A+S$ is again sectorial or $R$--sectorial, respectively. These conditions are obtained by factorising the perturbation as $S= -BC$, where $B$ acts on an auxiliary Banach space $Y$ and $C$ maps into $Y$. Our results extend previous work on perturbations in the scale of fractional domain spaces associated with $A$ and allow for a greater flexibility in choosing intermediate spaces for the action of perturbation operators. At the end we illustrate our results with several examples, in particular with an application to a ``rough'' boundary-value problem.

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On maximal regularity of type<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msup><mml:mi>L</mml:mi><mml:mi>p</mml:mi></mml:msup><mml:mtext>–</mml:mtext><mml:msup><mml:mi>L</mml:mi><mml:mi>q</mml:mi></mml:msup></mml:math>under minimal assumptions for elliptic non-divergence operators

2008-10-17

Peer Christian Kunstmann

Rs-Sectorial Operators and Generalized Triebel-Lizorkin Spaces

2014-01-01

Peer Christian Kunstmann , Alexander Ullmann

We introduce a notion of generalized Triebel-Lizorkin spaces associated with sectorial operators in Banach function spaces. Our approach is based on holo- morphic functional calculus techniques. Using the concept of … We introduce a notion of generalized Triebel-Lizorkin spaces associated with sectorial operators in Banach function spaces. Our approach is based on holo- morphic functional calculus techniques. Using the concept of Rs-sectorial operators, which in turn is based on the notion of Rs-bounded sets of operators introduced by Lutz Weis, we obtain a neat theory including equivalence of various norms and a pre- cise description of real and complex interpolation spaces. Another main result of this article is that an Rs-sectorial operator always has a bounded H ∞ -functional calculus in its associated generalized Triebel-Lizorkin spaces.

Spectral multiplier theorems of H\"ormander type on Hardy and Lebesgue spaces

2012-09-03

Peer Christian Kunstmann , Matthias Uhl

Let $X$ be a space of homogeneous type and let $L$ be an injective, non-negative, self-adjoint operator on $L^2(X)$ such that the semigroup generated by $-L$ fulfills Davies-Gaffney estimates of … Let $X$ be a space of homogeneous type and let $L$ be an injective, non-negative, self-adjoint operator on $L^2(X)$ such that the semigroup generated by $-L$ fulfills Davies-Gaffney estimates of arbitrary order. We prove that the operator $F(L)$, initially defined on $H^1_L(X)\cap L^2(X)$, acts as a bounded linear operator on the Hardy space $H^1_L(X)$ associated with $L$ whenever $F$ is a bounded, sufficiently smooth function. Based on this result, together with interpolation, we establish Hormander type spectral multiplier theorems on Lebesgue spaces for non-negative, self-adjoint operators satisfying generalized Gaussian estimates in which the required differentiability order is relaxed compared to all known spectral multiplier results.

On the existence of global solutions of the one-dimensional cubic NLS for initial data in the modulation space<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="double-struck">R</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>

2017-07-12

Leonid Chaichenets , Dirk Hundertmark , Peer Christian Kunstmann +1 more

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$L_p$ -spectral properties of the Neumann Laplacian on horns, comets and stars

2002-02-01

Peer Christian Kunstmann

Maximal L p Regularity for Second Order Elliptic Operators with Uniformly Continuous Coefficients on Domains

2003-01-01

Peer Christian Kunstmann

Maximal regularity of type L p is an important tool when dealing with quasi-linear equations of parabolic type (see, e.g., [1], [3]). If the closed linear operator A is the … Maximal regularity of type L p is an important tool when dealing with quasi-linear equations of parabolic type (see, e.g., [1], [3]). If the closed linear operator A is the generator of a bounded analytic C o-semigroup (T t) in a Banach space X and p ∈ (1, ∞) the A is sais to have maximal L p -regularity (which we denote by A ∈ MR p (X)) if for any f ∈ L p ((0,∞), X) the solution u=T * f of the equation u′ = Au + f, u(0) = (0) satisfies u′ ∈ L p ((0,∞), X) and Au ∈ L p ((0,∞), X).By the closed graph theorem this is equivalent to the existence of a C > 0 such that 1 $$ \left\| {u\prime } \right\|_{L_p \left( {\left( {{\text{0,}}\infty } \right),X} \right)} + \left\| {Au} \right\|_{L_p \left( {\left( {{\text{0,}}\infty } \right),X} \right)} \leqslant C\left\| f \right\|_{L_p \left( {\left( {{\text{0,}}\infty } \right),X} \right)} . $$

On Kato’s Method for Navier–Stokes Equations

2008-07-05

Bernhard H. Haak , Peer Christian Kunstmann

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$\mathcal {R}_{s}$ -Sectorial Operators and Generalized Triebel-Lizorkin Spaces

2013-12-09

Peer Christian Kunstmann , Alexander Ullmann

H ∞-calculus for the Stokes operator on unbounded domains

2008-07-26

Peer Christian Kunstmann

Stability of dispersion managed solitons for vanishing average dispersion

2015-02-21

Dirk Hundertmark , Peer Christian Kunstmann , Roland Schnaubelt

Lp-Spectral Multipliers for some Elliptic Systems

2014-10-10

Peer Christian Kunstmann , Matthias Uhl

Abstract We show results on L p -spectral multipliers for Maxwell operators with bounded measurable coefficients. We also present similar results for the Stokes operator with Hodge boundary conditions and … Abstract We show results on L p -spectral multipliers for Maxwell operators with bounded measurable coefficients. We also present similar results for the Stokes operator with Hodge boundary conditions and the Lamé system. Here, we rely on resolvent estimates recently established by Mitrea and Monniaux.

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Uniformly elliptic operators with maximal Lp-spectrum in planar domains

2001-05-01

Peer Christian Kunstmann

Navier-stokes equations on unbounded domains with rough initial data

2010-06-01

Peer Christian Kunstmann

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Universally divergent Fourier series via Landau's extremal functions

2015-05-16

Gerd Herzog , Peer Christian Kunstmann

We prove the existence of functions $f\in A(\mathbb D)$, the Fourier series of which being universally divergent on countable subsets of $\mathbb T = \partial \mathbb D$. The proof is … We prove the existence of functions $f\in A(\mathbb D)$, the Fourier series of which being universally divergent on countable subsets of $\mathbb T = \partial \mathbb D$. The proof is based on a uniform estimate of the Taylor polynomials of Landau's extremal functions on $\mathbb T\setminus\{1\}$.

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Cwikel’s bound reloaded

2022-09-05

Dirk Hundertmark , Peer Christian Kunstmann , Tobias Ried +1 more

Abstract There are several proofs by now for the famous Cwikel–Lieb–Rozenblum (CLR) bound, which is a semiclassical bound on the number of bound states for a Schrödinger operator, proven in … Abstract There are several proofs by now for the famous Cwikel–Lieb–Rozenblum (CLR) bound, which is a semiclassical bound on the number of bound states for a Schrödinger operator, proven in the 1970s. Of the rather distinct proofs by Cwikel, Lieb, and Rozenblum, the one by Lieb gives the best constant, the one by Rozenblum does not seem to yield any reasonable estimate for the constants, and Cwikel’s proof is said to give a constant which is at least about 2 orders of magnitude off the truth. This situation did not change much during the last 40+ years. It turns out that this common belief, i.e, Cwikel’s approach yields bad constants, is not set in stone: We give a substantial refinement of Cwikel’s original approach which highlights a natural but overlooked connection of the CLR bound with bounds for maximal Fourier multipliers from harmonic analysis. Moreover, it gives an astonishingly good bound for the constant in the CLR inequality. Our proof is also quite flexible and leads to rather precise bounds for a large class of Schrödinger-type operators with generalized kinetic energies.

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Weak Neumann implies $H^\infty$ for Stokes

2015-01-01

Matthias Geißert , Peer Christian Kunstmann

Let $\Omega\subset {\mathbb R}^n$ be a domain with uniform $C^3$ boundary and assume that the Helmholtz decomposition exists in ${\mathbb L}^q(\Omega):=L^q(\Omega)^n$ for some $q\in(1,\infty)$. We show that a suitable translate … Let $\Omega\subset {\mathbb R}^n$ be a domain with uniform $C^3$ boundary and assume that the Helmholtz decomposition exists in ${\mathbb L}^q(\Omega):=L^q(\Omega)^n$ for some $q\in(1,\infty)$. We show that a suitable translate of the Stokes operator admits a bounded ${\cal H}^\infty$-calculus in ${\mathbb L}_\sigma^p(\Omega)$ for $p\in(\min\{q,q'\},\max\{q,q'\})$. For the proof we use a recent maximal regularity result for the Stokes operator on such domains ([GHHS12]) and an abstract result for the ${\cal H}^\infty$-calculus in complemented subspaces ([KKW06], [KW13]).

Riesz transform on locally symmetric spaces and Riemannian manifolds with a spectral gap

2009-09-11

Lizhen Ji , Peer Christian Kunstmann , Andreas Weber⋆

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Classes of distribution semigroups

2008-01-01

Peer Christian Kunstmann , Modrag Mijatović , Stevan Pilipović

We introduce various classes of distribution semigroups distinguished by their behavior at the origin. We relate them to quasi-distribution semigroups and integrated semigroups. A class of such semigroups, called strong … We introduce various classes of distribution semigroups distinguished by their behavior at the origin. We relate them to quasi-distribution semigroups and integrated semigroups. A class of such semigroups, called strong distribution semigroups, is charact

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Microlocal analysis of imaging operators for effective common offset seismic reconstruction

2018-08-22

Christine Grathwohl , Peer Christian Kunstmann , Eric Todd Quinto +1 more

The elliptic Radon transform (eRT) integrates functions over ellipses in 2D and ellipsoids of revolution in 3D. It thus serves as a model for linearized seismic imaging under the common … The elliptic Radon transform (eRT) integrates functions over ellipses in 2D and ellipsoids of revolution in 3D. It thus serves as a model for linearized seismic imaging under the common offset scanning geometry where sources and receivers are offset by a constant vector. As an inversion formula of eRT is unknown we propose certain imaging operators (generalized backprojection operators) which allow to reconstruct some singularities of the searched-for reflectivity function from seismic measurements. We calculate and analyze the principal symbols of these imaging operators as pseudo-differential operators to understand how they map, emphasize or de-emphasize singularities. We use this information to develop local reconstruction operators that reconstruct relatively independently of depth and offset. Numerical examples illustrate the theoretical findings.

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$L^p$-spectral multipliers for some elliptic systems

2012-09-04

Peer Christian Kunstmann , Matthias Uhl

We show results on $L^p$-spectral multipliers for Maxwell operators with bounded measurable coefficients. We also present similar results for the Stokes operator with Hodge boundary conditions and the Lam\'e system. … We show results on $L^p$-spectral multipliers for Maxwell operators with bounded measurable coefficients. We also present similar results for the Stokes operator with Hodge boundary conditions and the Lam\'e system. Here we rely on resolvent estimates established recently by M. Mitrea and S. Monniaux.

Cwikel's bound reloaded

2018-01-01

Dirk Hundertmark , Peer Christian Kunstmann , Tobias Ried +1 more

There are a couple of proofs by now for the famous Cwikel--Lieb--Rozenblum (CLR) bound, which is a semiclassical bound on the number of bound states for a Schrödinger operator, proven … There are a couple of proofs by now for the famous Cwikel--Lieb--Rozenblum (CLR) bound, which is a semiclassical bound on the number of bound states for a Schrödinger operator, proven in the 1970s. Of the rather distinct proofs by Cwikel, Lieb, and Rozenblum, the one by Lieb gives the best constant, the one by Rozenblum does not seem to yield any reasonable estimate for the constants, and Cwikel's proof is said to give a constant which is at least about 2 orders of magnitude off the truth. This situation did not change much during the last 40+ years. It turns out that this common belief, i.e, Cwikel's approach yields bad constants, is not set in stone: We give a drastic simplification of Cwikel's original approach which leads to an astonishingly good bound for the constant in the CLR inequality. Our proof is also quite flexible and leads to rather precise bounds for a large class of Schrödinger-type operators with generalized kinetic energies. Moreover, it highlights a natural but overlooked connection of the CLR bound with bounds for maximal Fourier multipliers from harmonic analysis.

$H^\infty $-calculus for submarkovian generators

2003-02-05

Peer Christian Kunstmann , Željko Štrkalj

Let $-A$ be the generator of a symmetric submarkovian semigroup in $L_2(\Omega )$. In this note we show that on $L_p(\Omega ), 1<p<\infty ,$ the operator $A$ admits a bounded … Let $-A$ be the generator of a symmetric submarkovian semigroup in $L_2(\Omega )$. In this note we show that on $L_p(\Omega ), 1<p<\infty ,$ the operator $A$ admits a bounded $H^\infty$ functional calculus on the sector $\Sigma (\phi )=\{z\in \mathbb {C}\setminus \{0\}:|\mbox {arg} z|<\phi \}$ for each $\phi >\psi _p^*$ with \[ \psi _p^*=\frac {\pi }{2}|\frac {1}{p}-\frac {1}{2}| +(1-|\frac {1}{p}-\frac {1}{2}|)\arcsin (\frac {|p-2|}{2p-|p-2|}). \] This improves a result due to M. Cowling. We apply our result to obtain maximal regularity for parabolic equations and evolutionary integral equations.

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A $$T(1)$$ -Theorem for non-integral operators

2013-01-31

Dorothee Frey , Peer Christian Kunstmann

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Rs-bounded H∞-calculus for sectorial operators via generalized Gaussian estimates

2015-03-16

Peer Christian Kunstmann , Alexander Ullmann

We show that, for negative generators of analytic semigroups, a bounded -calculus self-improves to an -bounded -calculus in an appropriate scale of -spaces if the semigroup satisfies suitable generalized Gaussian … We show that, for negative generators of analytic semigroups, a bounded -calculus self-improves to an -bounded -calculus in an appropriate scale of -spaces if the semigroup satisfies suitable generalized Gaussian estimates. As application of our result we obtain that large classes of differential operators have an -bounded -calculus.

Weighted norm estimates and Lp-spectral independence of linear operators

2007-01-01

Peer Christian Kunstmann , Hendrik Vogt

We investigate the $L_p$-spectrum of linear operators defined consistently on $L_p({\mit\Omega} )$ for $p_0\le p\le p_1$, where $({\mit\Omega} ,\mu )$ is an arbitrary $\sigma $-finite measure space and $1\le p_0< … We investigate the $L_p$-spectrum of linear operators defined consistently on $L_p({\mit\Omega} )$ for $p_0\le p\le p_1$, where $({\mit\Omega} ,\mu )$ is an arbitrary $\sigma $-finite measure space and $1\le p_0< p_1\le \infty $. We prove $p$-independence

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Nonlinear Schrödinger equation, differentiation by parts and modulation spaces

2019-04-02

Leonid Chaichenets , Dirk Hundertmark , Peer Christian Kunstmann +1 more

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Knocking Out Teeth in One-Dimensional Periodic Nonlinear Schrödinger Equation

2019-01-01

Leonid Chaichenets , Dirk Hundertmark , Peer Christian Kunstmann +1 more

We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic nonlinear Schrödinger equation in one dimension with initial data $u_{0}$ in $H^{s_{1}}(\mathbb … We show the existence of weak solutions in the extended sense of the Cauchy problem for the cubic nonlinear Schrödinger equation in one dimension with initial data $u_{0}$ in $H^{s_{1}}(\mathbb R)+H^{s_{2}}(\mathbb T), 0\leq s_{1}\leq s_{2}.$ In addition, we show that if $u_{0}\in H^{s}(\mathbb R)+H^{\frac12+ε}(\mathbb T)$ where $ε>0$ and $\frac16\leq s\leq\frac12$ the solution is unique in $H^{s}(\mathbb R)+H^{\frac12+ε}(\mathbb T).$ Our main tool is a normal form type reduction via the use of the differentiation by parts technique.

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A Fixed Point Theorem for Decreasing Functions

2013-04-01

Gerd Herzog , Peer Christian Kunstmann

Abstract Let E be a Banach spaces ordered by a cone K. We prove a fixed point theorem for Lipschitz continuous monotone decreasing functions f: K → K, which proves … Abstract Let E be a Banach spaces ordered by a cone K. We prove a fixed point theorem for Lipschitz continuous monotone decreasing functions f: K → K, which proves the existence of a unique fixed point in cases where the Lipschitz constant of f is bigger than 1. This fixed point theorem can be applied to Hammerstein integral equations in a quite natural way. Keywords: Fixed pointsHammerstein equationsMonotone decreasing functionsOrdered Banach spacesAMS Subject Classification: 46B4047H0447H10

Modulation type spaces for generators of polynomially bounded groups and Schrödinger equations

2019-03-14

Peer Christian Kunstmann

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A new interpolation approach to spaces of Triebel–Lizorkin type

2015-01-01

Peer Christian Kunstmann

We introduce in this paper new interpolation methods for closed subspaces of Banach function spaces. For $q\in[1,\infty]$, the $l^{q}$-interpolation method allows to interpolate linear operators that have bounded $l^{q}$-valued extensions. … We introduce in this paper new interpolation methods for closed subspaces of Banach function spaces. For $q\in[1,\infty]$, the $l^{q}$-interpolation method allows to interpolate linear operators that have bounded $l^{q}$-valued extensions. For $q=2$ and if the Banach function spaces are $r$-concave for some $r<\infty$, the method coincides with the Rademacher interpolation method that has been used to characterize boundedness of the $H^{\infty}$-functional calculus. As a special case, we obtain Triebel–Lizorkin spaces $F^{2\theta}_{p,q}(\mathbb{R}^{d})$ by $l^{q}$-interpolation between $L^{p}(\mathbb{R}^{d})$ and $W^{2}_{p}(\mathbb{R}^{d})$ where $p\in(1,\infty)$. A similar result holds for the recently introduced generalized Triebel–Lizorkin spaces associated with $R_{q}$-sectorial operators in Banach function spaces. So, roughly speaking, for the scale of Triebel–Lizorkin spaces our method thus plays the role the real interpolation method plays in the theory of Besov spaces.

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$${\varvec{L}}^{\varvec{q}}$$ L q -Helmholtz Decomposition on Periodic Domains and Applications to Navier–Stokes Equations

2017-12-18

Jens Babutzka , Peer Christian Kunstmann

On Elliptic Non-divergence Operators with Measurable Coefficients

2006-02-08

Peer Christian Kunstmann

We study properties of the coefficient matrices of non-divergence operators on ℝn aiming at sectoriality and R-sectoriality of these operators. In particular, we present results on approximation, scaling, and the … We study properties of the coefficient matrices of non-divergence operators on ℝn aiming at sectoriality and R-sectoriality of these operators. In particular, we present results on approximation, scaling, and the behavior in the L p -scale.

Post-Widder Inversion for Laplace Transforms of Hyperfunctions

2007-01-01

Peer Christian Kunstmann

We prove a Post-Widder inversion formula for the Laplace transform of hyperfunctions with compact support in [0,∞). We observe that any hyperfunction with support in [0,∞) has Laplace transforms which … We prove a Post-Widder inversion formula for the Laplace transform of hyperfunctions with compact support in [0,∞). We observe that any hyperfunction with support in [0,∞) has Laplace transforms which are analytic on the right half-plane ℂ+, and we extend the Post-Widder inversion formula to suitably bounded representatives of arbitrary hyperfunctions with support in [0,∞).

Nonuniqueness and wellposedness of abstract Cauchy problems in a Fréchet space

2001-02-01

Peer Christian Kunstmann

Suppose that A is a closed linear operator in a Fréchet space X . We show that there always is a maximal subspace Z containing all x ∈ X for … Suppose that A is a closed linear operator in a Fréchet space X . We show that there always is a maximal subspace Z containing all x ∈ X for which the abstract Cauchy problem has a mild solution, which is a Fréchet space for a stronger topology. The space Z is isomorphic to a quotient of a Fréchet space F , and the part A z of A in Z is similar to the quotient of a closed linear operator B on F for which the abstract Cauchy problem is well-posed. If mild solutions of the Cauchy problem for A in X are unique it is not necessary to pass to a quotient, and we reobtain a result due to R. deLaubenfels. Moreover, we obtain a continuous selection operator for mild solutions of the inhomogeneous equation.

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Inequalities characterizing differential operators

2025-04-16

Gerd Herzog , Peer Christian Kunstmann

A Perron–Frobenius type result in Banach algebras via asymptotic closeness to a cone

2024-06-12

Gerd Herzog , Peer Christian Kunstmann

Abstract For an element a of a Banach algebra (scaled to spectral radius 1) we prove that the spectral radius is contained in the spectrum, if the sequence of powers … Abstract For an element a of a Banach algebra (scaled to spectral radius 1) we prove that the spectral radius is contained in the spectrum, if the sequence of powers $$(a^k)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>a</mml:mi> <mml:mi>k</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> is asymptotically not too far from a normal cone.

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Minimal periods for semilinear parabolic equations

2024-04-12

Gerd Herzog , Peer Christian Kunstmann

Abstract We show that, if $$-A$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>A</mml:mi> </mml:mrow> </mml:math> generates a bounded holomorphic semigroup in a Banach space X , $$\alpha \in [0,1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> … Abstract We show that, if $$-A$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>A</mml:mi> </mml:mrow> </mml:math> generates a bounded holomorphic semigroup in a Banach space X , $$\alpha \in [0,1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , and $$f:D(A)\rightarrow X$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>:</mml:mo> <mml:mi>D</mml:mi> <mml:mo>(</mml:mo> <mml:mi>A</mml:mi> <mml:mo>)</mml:mo> <mml:mo>→</mml:mo> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> satisfies $$\Vert f(x)-f(y)\Vert \le L\Vert A^\alpha (x-y)\Vert $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow> <mml:mo>‖</mml:mo> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>-</mml:mo> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>‖</mml:mo> <mml:mo>≤</mml:mo> <mml:mi>L</mml:mi> <mml:mo>‖</mml:mo> </mml:mrow> <mml:msup> <mml:mi>A</mml:mi> <mml:mi>α</mml:mi> </mml:msup> <mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>-</mml:mo> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>‖</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , then a non-constant T -periodic solution of the equation $${\dot{u}}+Au=f(u)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mover> <mml:mi>u</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> <mml:mo>+</mml:mo> <mml:mi>A</mml:mi> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> satisfies $$LT^{1-\alpha }\ge K_\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>L</mml:mi> <mml:msup> <mml:mi>T</mml:mi> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mi>α</mml:mi> </mml:mrow> </mml:msup> <mml:mo>≥</mml:mo> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>α</mml:mi> </mml:msub> </mml:mrow> </mml:math> where $$K_\alpha >0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>α</mml:mi> </mml:msub> <mml:mo>></mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> is a constant depending on $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> and the semigroup. This extends results by Robinson and Vidal-Lopez, which have been shown for self-adjoint operators $$A\ge 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> in a Hilbert space. For the latter case, we obtain - with a conceptually new proof - the optimal constant $$K_\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>K</mml:mi> <mml:mi>α</mml:mi> </mml:msub> </mml:math> , which only depends on $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> , and we also include the case $$\alpha =1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . In Hilbert spaces H and for $$\alpha =0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , we present a similar result with optimal constant where Au in the equation is replaced by a possibly unbounded gradient term $$\nabla _H{\mathscr {E}}(u)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>∇</mml:mi> <mml:mi>H</mml:mi> </mml:msub> <mml:mi>E</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . This is inspired by applications with bounded gradient terms in a paper by Mawhin and Walter.

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Eventually positive elements in ordered Banach algebras

2024-03-19

Gerd Herzog , Peer Christian Kunstmann

In ordered Banach algebras, we introduce eventually and asymptotically positive elements. We give conditions for the following spectral properties: the spectral radius belongs to the spectrum (Perron--Frobenius property); the spectral … In ordered Banach algebras, we introduce eventually and asymptotically positive elements. We give conditions for the following spectral properties: the spectral radius belongs to the spectrum (Perron--Frobenius property); the spectral radius is the only element in the peripheral spectrum; there are positive (approximate) eigenvectors for the spectral radius. Recently such types of results have been shown for operators on Banach lattices. Our results can be viewed as a complement, since our structural assumptions on the ordered Banach algebra are much weaker.

Analyticity in space-time of solutions to the Navier-Stokes equations via parameter trick based on maximal regularity

2023-03-07

Hideo Kozono , Peer Christian Kunstmann , Senjo Shimizu

A weak differential inequality

2023-01-01

Gerd Herzog , Peer Christian Kunstmann

Cwikel’s bound reloaded

2022-09-05

Dirk Hundertmark , Peer Christian Kunstmann , Tobias Ried +1 more

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Global wellposedness of NLS in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.svg"><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="double-struck">R</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo linebreak="goodbreak" linebreakstyle="after">+</mml:mo><mml:msup><mml:mrow><mml:mi>H</mml:mi></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msup><mml:mo stretchy…

2022-05-25

Friedrich Klaus , Peer Christian Kunstmann

Global wellposedness of NLS in $H^1(\mathbb{R})+H^s(\mathbb{T})$

2021-09-23

Friedrich Klaus , Peer Christian Kunstmann

We show global wellposedness for the defocusing cubic nonlinear Schrodinger equation (NLS) in $H^1(\mathbb{R}) + H^{3/2+}(\mathbb{T})$, and for the defocusing NLS with polynomial nonlinearities in $H^1(\mathbb{R}) + H^{5/2+}(\mathbb{T})$. This complements … We show global wellposedness for the defocusing cubic nonlinear Schrodinger equation (NLS) in $H^1(\mathbb{R}) + H^{3/2+}(\mathbb{T})$, and for the defocusing NLS with polynomial nonlinearities in $H^1(\mathbb{R}) + H^{5/2+}(\mathbb{T})$. This complements local results for the cubic NLS [6] and global results for the quadratic NLS [8] in this hybrid setting.

Counting bound states with maximal Fourier multipliers

2021-05-18

Dirk Hundertmark , Peer Christian Kunstmann , Tobias Ried +1 more

We report on a version of Cwikel's proof of the famous Cwikel–Lieb–Rozenblum (CLR) inequality which highlights the connection of the CLR inequality to maximal Fourier multipliers. This new approach enables … We report on a version of Cwikel's proof of the famous Cwikel–Lieb–Rozenblum (CLR) inequality which highlights the connection of the CLR inequality to maximal Fourier multipliers. This new approach enables us to get a constant at least ten times better than Cwikels in all dimensions. In dimensions $d\geq 5$ our results are better than all previously known ones.

Unconditional uniqueness of higher order nonlinear Schrödinger equations

2021-04-20

Friedrich Klaus , Peer Christian Kunstmann , Nikolaos Pattakos

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On the global well-posedness of the quadratic NLS on $$H^1({{\mathbb {T}}}) + L^2({{\mathbb {R}}})$$

2021-01-30

Leonid Chaichenets , Dirk Hundertmark , Peer Christian Kunstmann +1 more

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Quadratic inequalities for functionals in l^{∞}

2021-01-01

Gerd Herzog , Peer Christian Kunstmann

For a class of operators T on l ∞ and T -invariant functionals ϕ we prove inequalities between ϕ(x), ϕ(x 2 ) and the upper density of the setsApplications are … For a class of operators T on l ∞ and T -invariant functionals ϕ we prove inequalities between ϕ(x), ϕ(x 2 ) and the upper density of the setsApplications are given to Banach limits and integrals.

Global wellposedness of NLS in $H^1(\mathbb{R}) + H^s(\mathbb{T})$

2021-01-01

Friedrich Klaus , Peer Christian Kunstmann

We show global wellposedness for the defocusing cubic nonlinear Schrödinger equation (NLS) in $H^1(\mathbb{R}) + H^{3/2+}(\mathbb{T})$, and for the defocusing NLS with polynomial nonlinearities in $H^1(\mathbb{R}) + H^{5/2+}(\mathbb{T})$. This complements … We show global wellposedness for the defocusing cubic nonlinear Schrödinger equation (NLS) in $H^1(\mathbb{R}) + H^{3/2+}(\mathbb{T})$, and for the defocusing NLS with polynomial nonlinearities in $H^1(\mathbb{R}) + H^{5/2+}(\mathbb{T})$. This complements local results for the cubic NLS and global results for the quadratic NLS in this hybrid setting.

On continuity properties of semigroups in real interpolation spaces

2020-12-22

Peer Christian Kunstmann

Abstract Starting from a bi-continuous semigroup in a Banach space X (which might actually be strongly continuous), we investigate continuity properties of the semigroup that is induced in real interpolation … Abstract Starting from a bi-continuous semigroup in a Banach space X (which might actually be strongly continuous), we investigate continuity properties of the semigroup that is induced in real interpolation spaces between X and the domain D ( A ) of the generator. Of particular interest is the case $$(X,D(A))_{\theta ,\infty }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:mi>X</mml:mi><mml:mo>,</mml:mo><mml:mi>D</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>A</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mi>θ</mml:mi><mml:mo>,</mml:mo><mml:mi>∞</mml:mi></mml:mrow></mml:msub></mml:math> . We obtain topologies with respect to which the induced semigroup is bi-continuous, among them topologies induced by a variety of norms. We illustrate our results with applications to a nonlinear Schrödinger equation and to the Navier–Stokes equations on $$\mathbb {R}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi>R</mml:mi></mml:mrow><mml:mi>d</mml:mi></mml:msup></mml:math> .

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Asymptotic Constants in Averaged Hölder Inequalities

2020-11-01

Gerd Herzog , Peer Christian Kunstmann

For $0 \lt q \le p$ we take $L^\beta$-averages over the Hölder inequality between the $l^q$-norm and the $l^p$-norm in $\mathbb{R}^n$. We obtain precise limits as $n\to\infty$ for the $l^p$-unit … For $0 \lt q \le p$ we take $L^\beta$-averages over the Hölder inequality between the $l^q$-norm and the $l^p$-norm in $\mathbb{R}^n$. We obtain precise limits as $n\to\infty$ for the $l^p$-unit ball and, in case $p\ge1$, also for the $l^p$-unit sphere, which coincide and are independent of $\beta>0$. These are consequences of more general results on the asymptotic behavior of corresponding integrals over balls and spheres of certain bounded measurable functions.

Correction to: Local Well-Posedness for the Nonlinear Schrödinger Equation in the Intersection of Modulation Spaces $$M_{p, q}^s({\mathbb {R}}^d) \cap M_{\infty , 1}({\mathbb {R}}^d)$$

2020-01-01

Leonid Chaichenets , Dirk Hundertmark , Peer Christian Kunstmann +1 more

Local Well-Posedness for the Nonlinear Schrödinger Equation in the Intersection of Modulation Spaces $$M_{p, q}^s({\mathbb {R}}^d) \cap M_{\infty , 1}({\mathbb {R}}^d)$$

2020-01-01

Leonid Chaichenets , Dirk Hundertmark , Peer Christian Kunstmann +1 more

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Korovkin’s Theorem for Functionals and Limits for Box Integrals

2019-05-10

Gerd Herzog , Peer Christian Kunstmann

We prove a version of Korovkin's theorem for functionals that is well suited to obtain limits for integrals of mean values and has applications to limits of certain box integrals. We prove a version of Korovkin's theorem for functionals that is well suited to obtain limits for integrals of mean values and has applications to limits of certain box integrals.

Laplace Transform Theory for Logarithmic Regions

2019-04-24

Peer Christian Kunstmann

We define new spaces of test functions and distributions admitting a Laplace transform in the classical sense, i.e. by evaluation at exponentials. We use these spaces to give a characterization … We define new spaces of test functions and distributions admitting a Laplace transform in the classical sense, i.e. by evaluation at exponentials. We use these spaces to give a characterization of the Laplace inverses of analytic functions which are polynomially bounded on logarithmic regions and have values in a Banach space. As an illustration we give a conceptually new proof of the characterization of tempered convolution operators that have a distributional fundamental solution. Connections to asymptotic Lapalce transforms and Laplace transforms of (Laplace) hyperfunctions are sketched. AMS Subject Classification: 44 A 10, 46 F 05, 44 A 35.

Nonlinear Schrödinger equation, differentiation by parts and modulation spaces

2019-04-02

Leonid Chaichenets , Dirk Hundertmark , Peer Christian Kunstmann +1 more

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Modulation type spaces for generators of polynomially bounded groups and Schrödinger equations

2019-03-14

Peer Christian Kunstmann

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Knocking Out Teeth in One-Dimensional Periodic Nonlinear Schrödinger Equation

2019-01-01

Leonid Chaichenets , Dirk Hundertmark , Peer Christian Kunstmann +1 more

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Microlocal analysis of imaging operators for effective common offset seismic reconstruction

2018-08-22

Christine Grathwohl , Peer Christian Kunstmann , Eric Todd Quinto +1 more

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Boundary value problems with solutions in convex sets

2018-08-06

Gerd Herzog , Peer Christian Kunstmann

By means of the continuation method for contractions we prove the existence of solutions of Dirichlet boundary value problems in convex sets. As an application we prove the existence of … By means of the continuation method for contractions we prove the existence of solutions of Dirichlet boundary value problems in convex sets. As an application we prove the existence of concave solutions of certain boundary value problems in ordered Banach spaces.

Cwikel's bound reloaded

2018-01-01

Dirk Hundertmark , Peer Christian Kunstmann , Tobias Ried +1 more

$${\varvec{L}}^{\varvec{q}}$$ L q -Helmholtz Decomposition on Periodic Domains and Applications to Navier–Stokes Equations

2017-12-18

Jens Babutzka , Peer Christian Kunstmann

Runge–Kutta Time Discretization of Nonlinear Parabolic Equations Studied via Discrete Maximal Parabolic Regularity

2017-08-10

Peer Christian Kunstmann , Buyang Li , Christian Lubich

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On the existence of global solutions of the one-dimensional cubic NLS for initial data in the modulation space<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi>M</mml:mi></mml:mrow><mml:mrow><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi></mml:mrow></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mi mathvariant="double-struck">R</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>

2017-07-12

Leonid Chaichenets , Dirk Hundertmark , Peer Christian Kunstmann +1 more

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Note on Vector Translation with Respect to Thompson’s Metric

2017-04-27

Gerd Herzog , Peer Christian Kunstmann

It is known that vector translations are contractive with respect to Thompson’s part metric. Here, we give a simple proof, based on a representation of Thompson’s metric through positive functionals. … It is known that vector translations are contractive with respect to Thompson’s part metric. Here, we give a simple proof, based on a representation of Thompson’s metric through positive functionals. Moreover, we use contractivity of translations to prove a fixed point result for mappings that are Lipschitz continuous with respect to Thompson’s metric with Lipschitz constant r>1. The case r = 1 for order preserving or order reversing mappings has been recently studied by Lawson and Lim. We apply our result to a nonlinear boundary value problem.

Local well-posedness for the nonlinear Schr\"odinger equation in modulation spaces $M_{p,q}^{s}(\mathbb{R}^d)$

2016-10-26

Leonid Chaichenets , Dirk Hundertmark , Peer Christian Kunstmann +1 more

New criteria for the $${H^\infty}$$ H ∞ -calculus and the Stokes operator on bounded Lipschitz domains

2016-09-22

Peer Christian Kunstmann , Lutz Weis

Two Applications of a Generalization of an Asymptotic Fixed Point Theorem

2016-08-01

Gerd Herzog , Peer Christian Kunstmann

On the numerical range of generators of symmetric $${L_{\infty}}$$ L ∞ -contractive semigroups

2016-07-23

Markus Haase , Peer Christian Kunstmann , Hendrik Vogt

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Runge-Kutta time discretization of nonlinear parabolic equations studied via discrete maximal parabolic regularity

2016-06-12

Peer Christian Kunstmann , Buyang Li , Christian Lubich

For a large class of fully nonlinear parabolic equations, which include gradient flows for energy functionals that depend on the solution gradient, the semidiscretization in time by implicit Runge-Kutta methods … For a large class of fully nonlinear parabolic equations, which include gradient flows for energy functionals that depend on the solution gradient, the semidiscretization in time by implicit Runge-Kutta methods such as the Radau IIA methods of arbitrary order is studied. Error bounds are obtained in the $W^{1,\infty}$ norm uniformly on bounded time intervals and, with an improved approximation order, in the parabolic energy norm. The proofs rely on discrete maximal parabolic regularity. This is used to obtain $W^{1,\infty}$ estimates, which are the key to the numerical analysis of these problems.

Runge-Kutta time discretization of nonlinear parabolic equations studied via discrete maximal parabolic regularity

2016-01-01

Peer Christian Kunstmann , Buyang Li , Christian Lubich

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Asymptotic estimates in Thompson’s metric for semilinear parabolic equations

2015-10-23

Gerd Herzog , Peer Christian Kunstmann

Universally divergent Fourier series via Landau's extremal functions

2015-05-16

Gerd Herzog , Peer Christian Kunstmann

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Rs-bounded H∞-calculus for sectorial operators via generalized Gaussian estimates

2015-03-16

Peer Christian Kunstmann , Alexander Ullmann

Stability of dispersion managed solitons for vanishing average dispersion

2015-02-21

Dirk Hundertmark , Peer Christian Kunstmann , Roland Schnaubelt

Spectral multiplier theorems of Hormander type on Hardy and Lebesgue spaces

2015-02-01

Peer Christian Kunstmann , Matthias Uhl

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Weak Neumann implies $H^\infty$ for Stokes

2015-01-01

Matthias Geißert , Peer Christian Kunstmann

A new interpolation approach to spaces of Triebel–Lizorkin type

2015-01-01

Peer Christian Kunstmann

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Lp-Spectral Multipliers for some Elliptic Systems

2014-10-10

Peer Christian Kunstmann , Matthias Uhl

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Kuratowski's Measure of Noncompactness with Respect to Thompson's Metric

2014-07-02

Gerd Herzog , Peer Christian Kunstmann

It is known that the interior of a normal cone K in a Banach space is a complete metric space with respect to Thompson's metric d . We prove that … It is known that the interior of a normal cone K in a Banach space is a complete metric space with respect to Thompson's metric d . We prove that Kuratowski's measure of noncompactness \tau in (K°; d) has the Mazur-Darbo property and that, as a consequence, an analog of Darbo-Sadovskii's fixed point theorem is valid in (K°; d) . We show that the properties of \tau partly differ to the classical case. Among others \tau is nicely compatible with the multiplication in ordered Banach algebras.

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Inequalities and Means from a Cyclic Differential Equation

2014-06-01

Gerd Herzog , Peer Christian Kunstmann

Abstract We prove that the solution of the cyclic initial value problem u’ k = 1/2 - u k /(u k+1 + u k+2 ) (k Z/nZ), u(0)= x is … Abstract We prove that the solution of the cyclic initial value problem u’ k = 1/2 - u k /(u k+1 + u k+2 ) (k Z/nZ), u(0)= x is convergent to an equilibrium μ (x) (1,…,1), and study the properties of the function x → μ(x) and its relation to Shapiro’s inequality.

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Rs-Sectorial Operators and Generalized Triebel-Lizorkin Spaces

2014-01-01

Peer Christian Kunstmann , Alexander Ullmann

Global Asymptotic Stability for Semilinear Equations via Thompson's Metric

2014-01-01

Gerd Herzog , Peer Christian Kunstmann

In ordered Banach spaces we prove the global asymptotic stability of the unique strictly positive equilibrium of the semilinear equation u ' = Au+f(u), if A is the generator of … In ordered Banach spaces we prove the global asymptotic stability of the unique strictly positive equilibrium of the semilinear equation u ' = Au+f(u), if A is the generator of a positive and exponentially stable C0-semigroup and f is a contraction with respect to Thompson's metric. The given estimates show that convergence holds with a uniform exponential rate.

A local Landau type inequality for semigroup orbits

2014-01-01

Gerd Herzog , Peer Christian Kunstmann

Given a strongly continuous semigroup $(S(t))_{t\ge0}$ on a Banach space $X$ with generator $A$ and an element $f\in D(A^2)$ satisfying $\|S(t)f\|\le e^{-\omega t}\|f\|$ and $\|S(t)A^2f\|$ $\le e^{-\omega t}\|A^2f\|$ for all … Given a strongly continuous semigroup $(S(t))_{t\ge0}$ on a Banach space $X$ with generator $A$ and an element $f\in D(A^2)$ satisfying $\|S(t)f\|\le e^{-\omega t}\|f\|$ and $\|S(t)A^2f\|$ $\le e^{-\omega t}\|A^2f\|$ for all $t\ge0$ and some $\omega>0$

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$\mathcal {R}_{s}$ -Sectorial Operators and Generalized Triebel-Lizorkin Spaces

2013-12-09

Peer Christian Kunstmann , Alexander Ullmann

Coauthor	Papers Together
Gerd Herzog	19
Dirk Hundertmark	11
Nikolaos Pattakos	8
Leonid Chaichenets	7
Lutz Weis	6
Bernhard H. Haak	5
Friedrich Klaus	4
Matthias Uhl	4
Buyang Li	3
Sönke Blunck	3
Matthias Uhl	3
Semjon Vugalter	3
Tobias Ried	3
Christian Lubich	3
Alexander Ullmann	3
Markus Haase	2
Hendrik Vogt	2
Roland Schnaubelt	2
Dorothee Frey	2
Alessandra Lunardi	1
Modrag Mijatović	1
Matthias Geißert	1
Giuseppe Da Prato	1
Christine Grathwohl	1
Hideo Kozono	1
Željko Štrkalj	1
Jens Babutzka	1
Andreas Weber⋆	1
Senjo Shimizu	1
Eric Todd Quinto	1
Alexander Ullmann	1
Lizhen Ji	1
N. J. Kalton	1
Irena Lasiecka	1
Andreas Rieder	1
Stevan Pilipović	1

Erratum to: Perturbation and interpolation theorems for the H ∞-calculus with applications to differential operators

2013-08-16

Peer Christian Kunstmann , Lutz Weis

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Weighted norm estimates and maximal regularity

2002-01-01

Sönke Blunck , Peer Christian Kunstmann

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Operator–valued Fourier multiplier theorems and maximal $L_p$-regularity

2001-04-01

Lutz Weis

Maximal L p -regularity for Parabolic Equations, Fourier Multiplier Theorems and $H^\infty$ -functional Calculus

2004-01-01

Peer Christian Kunstmann , Lutz Weis

Banach space operators with a boundedH∞ functional calculus

1996-02-01

Michael Cowling , Ian Doust , Alan Micintosh +1 more

Abstract In this paper, we give a general definition for f(T) when T is a linear operator acting in a Banach space, whose spectrum lies within some sector, and which … Abstract In this paper, we give a general definition for f(T) when T is a linear operator acting in a Banach space, whose spectrum lies within some sector, and which satisfies certain resolvent bounds, and when f is holomorphic on a larger sector. We also examine how certain properties of this functional calculus, such as the existence of a bounded H ∈ functional calculus, bounds on the imaginary powers, and square function estimates are related. In particular we show that, if T is acting in a reflexive L p space, then T has a bounded H ∈ functional calculus if and only if both T and its dual satisfy square function estimates. Examples are given to show that some of the theorems that hold for operators in a Hilbert space do not extend to the general Banach space setting.

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Generalized Gaussian estimates and the Legendre transform

2005-01-01

Sönke Blunck , Pc Kunstmann

Perturbation and Interpolation Theorems for the H ∞-Calculus with Applications to Differential Operators

2006-08-08

N. J. Kalton , Peer Christian Kunstmann , Lutz Weis

The Functional Calculus for Sectorial Operators

2006-01-01

Markus Haase

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Uniformly Elliptic Operators with Measurable Coefficients

1995-08-01

E. B. Davies

Calderón-Zygmund theory for non-integral operators and the $H^{\infty}$ functional calculus

2003-12-31

Sönke Blunck , Peer Christian Kunstmann

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On square functions associated to sectorial operators

2004-01-01

Christian Le Merdy

We give new results on square functionsp associated to a sectorial operator A on L p for 1 < p < ∞.Under the assumption that A is actually R-sectorial, we … We give new results on square functionsp associated to a sectorial operator A on L p for 1 < p < ∞.Under the assumption that A is actually R-sectorial, we prove equivalences of the formfor suitable functions F, G.We also show that A has a bounded H ∞ functional calculus with respect to .F .Then we apply our results to the study of conditions under which we have an estimate ( ∞ 0 |Ce -tA (x)| 2 dt) 1/2 q ≤ M x p , when -A generates a bounded semigroup e -tA on L p and C : D(A) → L q is a linear mapping. Résumé (Sur les fonctions carrées associées aux opérateurs sectoriels)Nous obtenons de nouveaux résultats sur les fonctions carréesp associées à un opérateur sectoriel A sur L p pour 1 < p < ∞.Quand A est en fait R-sectoriel, on montre des équivalences de la forme K -1 x G ≤ x F ≤ K x G pour des fonctions F, G appropriées.On démontre également que A possède un calcul fonctionnel H ∞ borné par rapport à .F .Puis nous appliquons nos résultats à l'étude de conditions impliquant une inégalité du type ( ∞ 0 |Ce -tA (x)| 2 dt) 1/2 q ≤ M x p, où -A engendre un semigroupe borné e -tA sur L p et C : D(A) → L q est une application linéaire.

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The $H^{\infty}-$ calculus and sums of closed operators

2001-10-01

N. J. Kalton , Lutz Weis

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Square root problem for divergence operators and related topics

2018-11-06

Pascal Auscher , Philippe Tchamitchian

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The Functional Calculus for Sectorial Operators

2006-01-01

Markus Haase

In Section 2.1 the basic theory of sectorial operators is developed, including examples and the concept of sectorial approximation. In Section 2.2 we introduce some notation for certain spaces of … In Section 2.1 the basic theory of sectorial operators is developed, including examples and the concept of sectorial approximation. In Section 2.2 we introduce some notation for certain spaces of holomorphic functions on sectors. A functional calculus for sectorial operators is constructed in Section 2.3 along the lines of the abstract framework of Chapter 1. Fundamental properties like the composition rule are proved. In Section 2.5 we give natural extensions of the functional calculus to larger function spaces in the case where the given operator is bounded and/or invertible. In this way a panorama of functional calculi is developed. In Section 2.6 some mixed topics are discussed, e.g., adjoints and restrictions of sectorial operators and some fundamental boundedness and some first approximation results. Section 2.7 contains a spectral mapping theorem.

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Imaginary powers of elliptic second order differential operators in $L\sp p$-spaces

1993-01-01

Jan Prüß , Hermann Sohr

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Linear and Quasilinear Parabolic Problems

1995-01-01

Herbert Amann

In the first volume we give a thorough discussion of linear parabolic evolution equations in general Banach spaces.This is the abstract basis for the nonlinear theory.The second volume is devoted … In the first volume we give a thorough discussion of linear parabolic evolution equations in general Banach spaces.This is the abstract basis for the nonlinear theory.The second volume is devoted to concrete realizations of linear parabolic evolution equations by general parabolic systems.There we discuss the various function spaces that are needed and useful, and the generation of analytic semigroups by general elliptic boundary value problems.The last volume contains the abstract nonlinear theory as well as various applications to concrete systems, illustrating the scope and the flexibility of the general results.Of course, each one of the three volumes contains much material of independent interest related to our main subject.In writing this book I had help from many friends, collegues, and students.It is a pleasure to thank all of them, named or unnamed.I am particularly indebted to P. Quittner and G. Simonett, who critically and very carefully read, not only the whole manuscript of this first volume but also many earlier versions that were produced over the years and will never be published, and pointed out numerous mistakes and improvements.Large parts of the first volume, and of earlier versions as well, were also read and commented on by D. Daners, J. Escher, and P

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The global Cauchy problem for the NLS and NLKG with small rough data

2006-10-11

Baoxiang Wang , Henryk Hudzik

Bounded $H_\infty$-calculus for elliptic operators

1994-01-01

Herbert Amann , Matthias Hieber , Gieri Simonett

It is shown, in particular, that L p-realizations of general elliptic systems on Rn or on compact manifolds without boundaries possess bounded imaginary powers, provided rather mild regularity conditions are … It is shown, in particular, that L p-realizations of general elliptic systems on Rn or on compact manifolds without boundaries possess bounded imaginary powers, provided rather mild regularity conditions are satisfied.In addition, there are given some new perturbation theorems for operators possessing a bounded H00-calculus. Introduction.It is the main purpose of this paper to prove -under mild regularity assumptions-that Lp-realizations of elliptic differential operators acting on vector valued functions over JRn or on sections of vector bundles over compact manifolds without boundaries possess bounded imaginary powers.In fact, we shall prove a more general result guaranteeing that, given any elliptic operator A with a sufficiently large zero order term such that the spectrum of its principal symbol is contained in a sector of the form 8& 0 := {z E C; I atgz!::::; eo} U {0} for some 0 e0 E [0, n), and given any bounded holomorphic function f: S& ---7 C for some e E (e0, n), we can define a bounded linear operator j(A) on Lp, and an estimate of the form llf(A)II.ccLp) ::::; c llflloo is valid.This means that elliptic operators possess a bounded R 00-calculus in the sense of Mcintosh [16].Choosing, in particular, f(z) :=zit fortE JR, it follows that A possesses bounded imaginary powers ( cf.Section 2 below for more precise statements).There are two main reasons for our interest in this problem.First, it is known (cf.[22], [24]) that the complex interpolation spaces [E, D(A)]& coincide with the domains of the fractional powers A & for 0 < e < 1, provided A is a densely defined linear operator on the Banach space E possessing bounded imaginary powers.Second, by a result of Dore and Venni [1 OJ, the fact that A possesses bounded imaginary powers is intimately connected with 'maximal regularity results' for abstract evolution equations of the form u +Au = f (t).Both these results are of great use in the

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Local well-posedness of nonlinear dispersive equations on modulation spaces

2009-04-03

Árpád Bényi , Kasso A. Okoudjou

By using tools of time-frequency analysis, we obtain some improved local well-posedness results for the NLS, NLW and NLKG equations with Cauchy data in modulation spaces $M{p, 1}_{0,s}$. By using tools of time-frequency analysis, we obtain some improved local well-posedness results for the NLS, NLW and NLKG equations with Cauchy data in modulation spaces $M{p, 1}_{0,s}$.

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NLS in the Modulation Space $$M_{2,q}({\mathbb {R}})$$ M 2 , q ( R )

2018-12-10

Nikolaos Pattakos

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Involutive structures on the tangent bundle of symmetric spaces

2001-02-01

N. J. Kalton , Lutz Weis

Semigroups of Linear Operators and Applications to Partial Differential Equations

1983-01-01

A. Pazy

Singular integral operators with non-smooth kernels on irregular domains

1999-08-31

Xuan Thinh Duong , Alan McIntosh

Let \chi be a space of homogeneous type. The aims of this paper are as follows: i) Assuming that T is a bounded linear operator on L_2(\chi) we give a … Let \chi be a space of homogeneous type. The aims of this paper are as follows: i) Assuming that T is a bounded linear operator on L_2(\chi) we give a sufficient condition on the kernel of T so that T is of weak type (1,1) , hence bounded on L_p(\chi) for 1 < p ≤ 2 ; our condition is weaker than the usual Hörmander integral condition. ii) Assuming that T is a bounded linear operator on L_2(\Omega) where \Omega is a measurable subset of \chi , we give a sufficient condition on the kernel of T so that T is of weak type (1,1) , hence bounded on L_p(\Omega) for 1 0} | T_\epsilon u(x) | , to be L_p bounded, 1 < p < \infty . Applications include weak (1,1) estimates of certain Riesz transforms and L_p boundedness of holomorphic functional calculi of linear elliptic operators on irregular domains.

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Stability results on interpolation scales of quasi-Banach spaces and applications

1998-01-01

N. J. Kalton , Marius Mitrea

We investigate the stability of Fredholm properties on interpolation scales of quasi-Banach spaces. This analysis is motivated by problems arising in PDE's and several applications are presented. We investigate the stability of Fredholm properties on interpolation scales of quasi-Banach spaces. This analysis is motivated by problems arising in PDE's and several applications are presented.

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Weak type (p,p) estimates for Riesz transforms

2004-04-07

Sönke Blunck , Peer Christian Kunstmann

On the 1D Cubic Nonlinear Schrödinger Equation in an Almost Critical Space

2016-03-08

Shaoming Guo

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$H_\infty$-calculus for elliptic operators with nonsmooth coefficients

1997-01-01

Xuan Thinh Duong , Gieri Simonett

We show that general systems of elliptic di↵erential operators have a bounded H 1 -functional calculus in L p spaces, provided the coe cients satisfy only minimal regularity assumptions. We show that general systems of elliptic di↵erential operators have a bounded H 1 -functional calculus in L p spaces, provided the coe cients satisfy only minimal regularity assumptions.

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Perturbation theorems for maximal $L_p$-regularity

2001-01-01

Peer Christian Kunstmann , Lutz Weis

Limits onLpRegularity of Self-Adjoint Elliptic Operators

1997-03-01

E. B. Davies

New thoughts on old results of R.T. Seeley

2004-03-09

Robert Denk , Giovanni Dore , Matthias Hieber +2 more

Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes

1971-01-01

Ronald R. Coifman , Guido Weiss

H∞functional calculus in real interpolation spaces, II

2001-01-01

Giovanni Dore

Let $A$ be a linear closed one-to-one operator in a complex Banach space $X$, having dense domain and dense range. If $A$ is of type $\omega $ (i.e.the spectrum of … Let $A$ be a linear closed one-to-one operator in a complex Banach space $X$, having dense domain and dense range. If $A$ is of type $\omega $ (i.e.the spectrum of $A$ is contained in a sector of angle $2\omega $, symmetric about the real positive axis, a

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On the L-Theory of C0-Semigroups Associated with Second-Order Elliptic Operators, I

2002-08-01

Zeev Sobol , Hendrik Vogt

ℛ-boundedness, Fourier multipliers and problems of elliptic and parabolic type

2003-01-01

Robert Denk , Matthias Hieber , Jan Prüß

On the L-Theory of C0-Semigroups Associated with Second-Order Elliptic Operators, II

2002-08-01

Vitali Liskevich , Zeev Sobol , Hendrik Vogt

On necessary and sufficient conditions for $L^p$-estimates of Riesz transforms associated to elliptic operators on $\RR^n$ and related estimates

2005-01-01

Pascal Auscher

This article focuses on $L^p$ estimates for objects associated to elliptic operators in divergence form: its semigroup, the gradient of the semigroup, functional calculus, square functions and Riesz transforms. We … This article focuses on $L^p$ estimates for objects associated to elliptic operators in divergence form: its semigroup, the gradient of the semigroup, functional calculus, square functions and Riesz transforms. We introduce four critical numbers associated to the semigroup and its gradient that completely rule the ranges of exponents for the $L^p$ estimates. It appears that the case $p<2$ already treated earlier is radically different from the case $p>2$ which is new. We thus recover in a unified and coherent way many $L^p$ estimates and give further applications. The key tools from harmonic analysis are two criteria for $L^p$ boundedness, one for $p<2$ and the other for $p>2$ but in ranges different from the usual intervals $(1,2)$ and $(2,\infty)$.

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Maximal L p Regularity for Second Order Elliptic Operators with Uniformly Continuous Coefficients on Domains

2003-01-01

Peer Christian Kunstmann

On the Helmholtz decomposition in general unbounded domains

2007-02-19

Reinhard Farwig , Hideo Kozono , Hermann Sohr

Analytic Semigroups and Optimal Regularity in Parabolic Problems

1995-01-01

Alessandra Lunardi

Some Maximal Inequalities

1971-01-01

Charles Fefferman , E. M. Stein

Bounded holomorphic functional calculus for non-divergence form differential operators

2002-01-01

Xuan Thinh Duong , Yan Li

Let $L$ be a second-order elliptic partial differential operator of non-divergence form acting on ${\bf R^n}$ with bounded coefficients. We show that for each $1 < p_0 <2, L$ has … Let $L$ be a second-order elliptic partial differential operator of non-divergence form acting on ${\bf R^n}$ with bounded coefficients. We show that for each $1 < p_0 <2, L$ has a bounded $H_{\infty}$-functional calculus on $L^p({\bf R^n})$ for $p_0 <p <\infty$ if the $BMO$ norm of the coefficients is sufficiently small.

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An Lq-approach to Stokes and Navier-Stokes equations in general domains

2005-01-01

Reinhard Farwig , Hideo Kozono , Hermann Sohr

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H ∞ -calculus for the Stokes operator on L q -spaces

2003-07-01

André Noll , Jürgen Saal

Spectral multipliers, R-bounded homomorphisms and analytic diffusion semigroups

2009-12-04

Christoph Kriegler

Ce travail traite du calcul fonctionnel des op\'{e}rateurs dont le spectre est contenu dans les nombres r\'{e}els positifs. On s'int\'{e}resse en particulier aux th\'{e}or\`{e}mes de multiplicateurs spectraux. On aborde le … Ce travail traite du calcul fonctionnel des op\'{e}rateurs dont le spectre est contenu dans les nombres r\'{e}els positifs. On s'int\'{e}resse en particulier aux th\'{e}or\`{e}mes de multiplicateurs spectraux. On aborde le calcul abstrait et optimal, c'est-\`{a}-dire les homomorphismes $u : C(K) \to B(X)$. Si $X$ est un espace de Hilbert, alors l'extension naturelle $\hat{u} : C(K;[u]') \to B(X)$ de $u$ sur l'ensemble des op\'{e}rateurs est \`{a} nouveau born\'{e}e. En utilisant la $R$-bornitude, un renforcement de la bornitude uniforme, on donne une extension de ce r\'{e}sultat \`{a} des espaces de Banach g\'{e}n\'{e}raux $X$ et on l'applique au calcul $H$ infini et aux bases inconditionnelles dans des espaces On d\'{e}veloppe des calculs associ\'{e}s \`{a} des op\'{e}rateurs sectoriels. Les exemples classiques en sont les th\'{e}or\`{e}mes spectraux de Mihlin et H\{o}rmander donnant des classes de fonctions lisses qui forment des multiplicateurs de Fourier sur $L^p$. Ces th\'{e}or\`{e}mes ont d\'{e}j\`{a} \'{e}t\'{e} \'{e}tendus \`{a} une large classe d'op\'{e}rateurs de type Laplacien. On les regroupe sous une forme unifi\'{e}e gr\^{a}ce \`{a} la th\'{e}orie des op\'{e}rateurs: on compare le calcul de Mihlin et de H\{o}rmander \`{a} la bornitude des familles classiques associ\'{e}es \`{a} un op\'{e}rateur sectoriel. Pour la famille des puissances imaginaires, on donne une caract\'{e}risation de leur croissance polynomiale en fonction d'un calcul fonctionnel qui raffine le calcul de Mihlin. On \'{e}tudie des semi-groupes de diffusion qui agissent sur une \'{e}chelle d'espaces de Banach. Il est connu que le semi-groupe a une extension analytique sur un secteur dans le plan complexe si cette \'{e}chelle consiste des espaces $L^p$. On donne une g\'{e}n\'{e}ralisation de ce r\'{e}sultat \`{a} des espaces $L^p$ non commutatifs en utilisant la th\'{e}orie des espaces d'op\'{e}rateurs.

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Semigroup Kernels, Poisson Bounds, and Holomorphic Functional Calculus

1996-11-01

Xuan Thinh Duong , Derek W. Robinson

Unimodular Fourier multipliers for modulation spaces

2007-03-08

Árpád Bényi , Karlheinz Gröchenig , Kasso A. Okoudjou +1 more

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THE WEISS CONJECTURE FOR BOUNDED ANALYTIC SEMIGROUPS

2003-05-20

Christian Le Merdy

New results concerning the so-called Weiss conjecture on admissible operators for bounded analytic semigroups are given. Let ( T t ) t ⩾ 0 be a bounded analytic semigroup with … New results concerning the so-called Weiss conjecture on admissible operators for bounded analytic semigroups are given. Let ( T t ) t ⩾ 0 be a bounded analytic semigroup with generator –A on some Banach space X. It is proved that if A1/2 is admissible for A, that is, if there is an estimate ∫ 0 ∞ ‖ A 1 / 2 e − t A x ‖ 2 d t ⩽ M 2 ‖ x ‖ 2 , then any continuous mapping C : D(A) → Y valued in a Banach space Y is admissible for A provided that there is an estimate ‖ ( − Re ( λ ) ) 1 / 2 C ( λ − A ) − 1 ‖ ⩽ K . for λ ∈, Re(λ)<0. This holds in particular if ( T t ) t ⩾ 0 is a contractive (analytic) semigroup on Hilbert space. In the converse direction, it is shown that this may happen for a bounded analytic semigroup on Hilbert space that is not similar to a contractive one. Applications in non-Hilbertian Banach spaces are also given.