Holomorphic Semigroup Research Articles

$L^p$-bounds for Stein's square functions associated to operators and applications to spectral multipliers

Let $(X, d, \mu)$ be a metric measure space endowed with a metric $d$ and a nonnegative Borel doubling measure $\mu$. Let $L$ be a non-negative self-adjoint operator of order $m$ on $X$. Assume that $L$ generates a holomorphic semigroup $e^{-tL}$ whose kernels $p_t(x,y)$ satisfy Gaussian upper bounds but without any assumptions on the regularity of space variables $x$ and $y$. Also assume that $L$ satisfies a Plancherel type estimate. Under these conditions, we show the $L^p$ bounds for Stein's square functions arising from Bochner-Riesz means associated to the operator $L$. We then use the $L^p$ estimates on Stein's square functions to obtain a Hormander-type criterion for spectral multipliers of $L$. These results are applicable for large classes of operators including sub-Laplacians acting on Lie groups of polynomial growth and Schrodinger operators with rough potentials.

Nonlinear perturbations of a class of holomorphic semigroups of growth order [formula omitted] by comparison theorems for Volterra equations

This paper studies nonlinear perturbation of a holomorphic semigroup of growth order α in a Banach space X. The generator A of a holomorphic semigroup is assumed to be almost sectorial and it is also assumed that the part A0 in the closure X0 of the domain D(A) is sectorial. A nonlinear perturbing operator B is assumed to be locally continuous from a subset of a real interpolation space between X0 and D(A0) into X. Existence and uniqueness of mild solutions to the associated Cauchy problems are proved by using comparison theorems for integral equations of Volterra type. The result obtained is applied to show the global well-posedness of drift–diffusion systems with no-flux boundary conditions in the two dimensional case.

Regularization for ill-posed parabolic evolution problems

Abstract. We prove regularization for a certain ill-posed parabolic evolution problem in a Banach space X by obtaining Hölder-continuous dependence of its solution on modeling. In particular, we consider the generally ill-posed problem , , with initial data , in X where is the infinitesimal generator of a bounded holomorphic semigroup of angle on X, and for each with for . Assuming there exists a solution of the problem adhering to certain stabilizing conditions, we approximate by the solution of an approximate well-posed problem. We then use this estimate to prove the existence of a family of regularizing operators for the given ill-posed problem. The theory has applications to the backwards heat equation and other ill-posed partial differential equations in , , with time-dependent coefficients.

Littlewood–Paley and Spectral Multipliers on Weighted L p Spaces

Let L be a non-negative self-adjoint operator acting on L 2(X), where X is a space of homogeneous type. Assume that L generates a holomorphic semigroup e −tL whose kernel p t (x,y) has a Gaussian upper bound but there is no assumption on the regularity in variables x and y. In this article we study weighted L p -norm inequalities for spectral multipliers of L. We show that a weighted Hormander-type spectral multiplier theorem follows from weighted L p -norm inequalities for the Lusin and Littlewood–Paley functions, Gaussian heat kernel bounds, and appropriate L 2 estimates of the kernels of the spectral multipliers.

Laguerre semigroup and Dunkl operators

AbstractWe construct a two-parameter family of actionsωk,aof the Lie algebra 𝔰𝔩(2,ℝ) by differential–difference operators on ℝN∖{0}. Herekis a multiplicity function for the Dunkl operators, anda>0 arises from the interpolation of the two 𝔰𝔩(2,ℝ) actions on the Weil representation ofMp(N,ℝ) and the minimal unitary representation of O(N+1,2). We prove that this actionωk,alifts to a unitary representation of the universal covering ofSL(2,ℝ) , and can even be extended to a holomorphic semigroup Ωk,a. In thek≡0 case, our semigroup generalizes the Hermite semigroup studied by R. Howe (a=2) and the Laguerre semigroup studied by the second author with G. Mano (a=1) . One boundary value of our semigroup Ωk,aprovides us with (k,a) -generalized Fourier transforms ℱk,a, which include the Dunkl transform 𝒟k(a=2) and a new unitary operator ℋk (a=1) , namely a Dunkl–Hankel transform. We establish the inversion formula, a generalization of the Plancherel theorem, the Hecke identity, the Bochner identity, and a Heisenberg uncertainty relation for ℱk,a. We also find kernel functions for Ωk,aand ℱk,afora=1,2 in terms of Bessel functions and the Dunkl intertwining operator.

Weighted norm inequalities, Gaussian bounds and sharp spectral multipliers

Let L be a non-negative self-adjoint operator acting on L 2 ( X ) where X is a space of homogeneous type. Assume that L generates a holomorphic semigroup e − t L whose kernels p t ( x , y ) have Gaussian upper bounds but there is no assumption on the regularity in variables x and y. In this article, we study weighted L p -norm inequalities for spectral multipliers of L. We show that sharp weighted Hörmander-type spectral multiplier theorems follow from Gaussian heat kernel bounds and appropriate L 2 estimates of the kernels of the spectral multipliers. These results are applicable to spectral multipliers for large classes of operators including Laplace operators acting on Lie groups of polynomial growth or irregular non-doubling domains of Euclidean spaces, elliptic operators on compact manifolds and Schrödinger operators with non-negative potentials.

Stein spaces characterized by their endomorphisms

Finite-dimensional Stein spaces admitting a proper holomorphic embedding of the complex line are characterized, among all complex spaces, by their holomorphic endomorphism semigroup in the sense that any semigroup isomorphism induces either a biholomorphic or an antibiholomorphic map between them.

Generalized Fourier transforms [formula omitted

We construct a two-parameter family of actions ω k , a of the Lie algebra sl ( 2 , R ) by differential-difference operators on R N . Here, k is a multiplicity-function for the Dunkl operators, and a > 0 arises from the interpolation of the Weil representation and the minimal unitary representation of the conformal group. The action ω k , a lifts to a unitary representation of the universal covering of SL ( 2 , R ) , and can even be extended to a holomorphic semigroup Ω k , a . Our semigroup generalizes the Hermite semigroup studied by R. Howe ( k ≡ 0 , a = 2 ) and the Laguerre semigroup by T. Kobayashi and G. Mano ( k ≡ 0 , a = 1 ). The boundary value of our semigroup Ω k , a provides us with ( k , a ) - generalized Fourier transforms F k , a , which includes the Dunkl transform D k ( a = 2 ) and a new unitary operator H k ( a = 1 ) as a Dunkl-type generalization of the classical Hankel transform. To cite this article: S. Ben Saïd et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).

Perturbation theorems for holomorphic semigroups

The concept of the gap function is used to give new perturbation results for generators of holomorphic semigroups. In particular, we show that if A is the generator of a holomorphic semigroup on a Banach space and \({M_{A}:=\limsup_{|\lambda| \rightarrow \infty, \lambda \in \mathbb {C}_+}\|\lambda R(\lambda , A) \|}\) , then every closed linear operator C such that \({(\omega,\infty)\subset\rho(C)}\) for some \({\omega\in \mathbb {R}}\) and $$\limsup_{\lambda \rightarrow \infty}\|\lambda R(\lambda ,A)- \lambda R(\lambda ,C)\|< \frac{1}{2+\sqrt{3}}\left( 1+ M_{A}^2 \right)^{-\frac{1}{2}}$$ generates a holomorphic semigroup, too. Moreover, we obtain an analogue of this result for differences of semigroups. If T is a holomorphic semigroup and \({k_T:=(\limsup_{t\rightarrow 0^+}\|(T(t)+I)^{-1}\|)^{-1}}\) , then every C0-semigroup S with $$\limsup\limits_{t\rightarrow 0^+}\|T(t)-S(t)\|< k_T$$ is holomorphic. We also give certain estimates for the constants MA and kT appearing in the above conditions.

Robust multivariable PI-controllers for linear systems in Banach state spaces

In this paper we give a generalization of the PI-controller design theory proposed by Kobayashi [T.A. Kobayashi, Digital PI-controller for distributed parameter systems, SIAM J. Control Optim. 26 (1988) 1399–1414] and later by Xu and Jerbi [C.Z. Xu, H. Jerbi, A robust PI-controller for infinite-dimensional systems, Internat. J. Control 61 (1995) 33–45] to stabilize and regulate a large class of infinite-dimensional linear systems in Banach space. We prove that even if the dynamic operator A of the open loop system does not generate a holomorphic semigroup, the invertibility of C A −1 B (where C is an unbounded observation operator and B is a bounded control operator for the system) is still a necessary and sufficient condition for the existence of a robust PI-controller.

Criteria for analyticity of subordinate semigroups

Let ψ be a Bernstein function. A. Carasso and T. Kato obtained necessary and sufficient conditions for ψ to have the property that ψ(A) generates a quasibounded holomorphic semigroup for every generator A of a bounded C0-semigroup in a Banach space, in terms of some convolution semigroup of measures associated with ψ. We give an alternative to Carasso-Kato’s criterion, and derive several sufficient conditions for ψ to have the above-mentioned property.

GENERATORS OF REGULAR SEMIGROUPS

AbstractA regular semigroup (cf. [4, p. 38]) is a C0-semigroup T(⋅) that has an extension as a holomorphic semigroup W(⋅) in the right halfplane $\Bbb C^+$, such that ||W(⋅)|| is bounded in the ‘unit rectangle’ Q:=(0, 1]× [−1, 1]. The important basic facts about a regular semigroup T(⋅) are: (i) it possesses a boundary groupU(⋅), defined as the limit lims → 0+W(s+i⋅) in the strong operator topology; (ii) U(⋅) is a C0-group, whose generator is iA, where A denotes the generator of T(⋅); and (iii) W(s+it)=T(s)U(t) for all s+it ∈$\Bbb C^+$ (cf. Theorems 17.9.1 and 17.9.2 in [3]). The following converse theorem is proved here. Let A be the generator of a C0-semigroup T(⋅). If iA generates a C0-group, U(⋅), then T(⋅) is a regular semigroup, and its holomorphic extension is given by (iii). This result is related to (but not included in) known results of Engel (cf. Theorem II.4.6 in [2]), Liu [7] and the author [6] for holomorphic extensions into arbitrary sectors, of C0-semigroups that are bounded in every proper subsector. The method of proof is also different from the method used in these references. Criteria for generators of regular semigroups follow as easy corollaries.

Dissipative operators in the Krein space. Invariant subspaces and properties of restrictions

We prove that a dissipative operator in the Krein space has a maximal nonnegative invariant subspace provided that the operator admits matrix representation with respect to the canonical decomposition of the space and the upper right operator in this representation is compact relative to the lower right operator. Under the additional assumption that the upper and lower left operators are bounded (the so-called Langer condition), this result was proved (in increasing order of generality) by Pontryagin, Krein, Langer, and Azizov. We relax the Langer condition essentially and prove under the new assumptions that a maximal dissipative operator in the Krein space has a maximal nonnegative invariant subspace such that the spectrum of its restriction to this subspace lies in the left half-plane. Sufficient conditions are found for this restriction to be the generator of a holomorphic semigroup or a C 0-semigroup.

The Stokes Operator with Robin Boundary Conditions in Solenoidal Subspaces of L 1(ℝ n +) and L ∞(ℝ n +)

We prove that the Stokes operator with Robin boundary conditions is the generator of a bounded holomorphic semigroup on , which is even strongly continuous on the space BUCσ(ℝ n +). Contrary to that result it is also proved that there exists no Stokes semigroup on , except if we assume the special case of Neumann boundary conditions. Nevertheless, we also obtain gradient estimates for the solution of the Stokes equations in for all types of Robin boundary conditions.

MAPPING PROPERTIES OF HEAT KERNELS, MAXIMAL REGULARITY, AND SEMI-LINEAR PARABOLIC EQUATIONS ON NONCOMPACT MANIFOLDS

Let [Formula: see text] be a second order, uniformly elliptic, positive semi-definite differential operator on a complete Riemannian manifold of bounded geometry M, acting between sections of a vector bundle with bounded geometry E over M. We assume that the coefficients of L are uniformly bounded. Using finite speed of propagation for L, we investigate properties of operators of the form [Formula: see text]. In particular, we establish results on the distribution kernels and mapping properties of e-tLand (μ + L)s. We show that L generates a holomorphic semigroup that has the usual mapping properties between the Ws,p-Sobolev spaces on M and E. We also prove that L satisfies maximal Lp–Lq-regularity for 1 &lt; p, q &lt; ∞. We apply these results to study parabolic systems of semi-linear equations of the form ∂tu + Lu = F(t, x, u, ∇ u).

Continuous dependence results for inhomogeneous ill-posed problems in Banach space

We apply semigroup theory and other operator-theoretic methods to prove Hölder-continuous dependence on modeling for the inhomogeneous ill-posed Cauchy problem in Banach space. The inhomogeneous ill-posed Cauchy problem is given by d u d t = A u ( t ) + h ( t ) , u ( 0 ) = χ , 0 ⩽ t < T ; where − A is the infinitesimal generator of a holomorphic semigroup on a Banach space X, χ ∈ X , and h : [ 0 , T ) → X . For a suitable function f, the approximate problem is given by d v d t = f ( A ) v ( t ) + h ( t ) , v ( 0 ) = χ . Under certain stabilizing conditions, we prove that ‖ u ( t ) − v ( t ) ‖ ⩽ C ˜ β 1 − ω ( t ) M ω ( t ) for a related norm, where C ˜ and M are computable constants independent of β, 0 < β < 1 , and ω ( t ) is a harmonic function. These results extend earlier work of Ames and Hughes on the homogeneous ill-posed problem.

Discrete Simulation of a Class of Distributed Systems Using Functional Analytic Methods

This paper investigates the discrete simulation of the solution of initial-boundary-value problems that typically arise in technical areas. Since many of them lead to unbounded and non-self-adjoint differential operators, we have to use a rather general theory as a mathematical basis. For the class of sectorial operators with a compact resolvent operator, the solution of initial-boundary-value problem can be represented by means of a certain holomorphic semigroup. It is shown that the solution can be expanded with respect to the canonical system of the considered operator. Such an expansion corresponds to a multi-dimensional functional transformation in the frequency domain. This fact leads to simple structures for the realization of the resulting system. Computationally efficient numerical algorithms can be derived by proper methods well-known from the theory of digital signal processing.

Bounded Solutions of Parabolic Equations in Continuous Function Spaces

This paper is concerned with the existence of bounded mild solutions to equations of the form u′(t) = Au(t) + f(t), where A generates a holomorphic semigroup that is not necessarily strongly continuous, and f is a bounded function. This problem arises when one considers a parabolic equation in spaces of continuous functions. The obtained results, that are stated in terms of spectral properties of the spectrum of A and the uniform spectrum of f, extend previous ones.

Maximal operators for the holomorphic Laguerre semigroup

For each $p$ in $[1,\infty)$ let $\mathbf{E}_p$ denote the closure of the region of holomorphy of the Laguerre semigroup $\{M^{\alpha}_t:t&gt;0\}$ on $L^p$ with respect to the Laguerre measure $\mu_{\alpha}$. We prove weak type and strong type estimates for the maximal operator $f\mapsto \sup\{|M^{\alpha}_z f|:z\in \mathbf{E}_p\}$. In particular, we give a new proof for the weak type $1$ estimate for the maximal operator associated to $M^{\alpha}_t$. Our starting point is the well-known relationship between the Laguerre semigroup of half-integer parameter and the Ornstein-Uhlenbeck semigroup.

A maximal theorem for holomorphic semigroups

Let X be a closed linear subspace of the Lebesgue space L^p(Omega ; mu); let -A be an invertible linear operator that is the generator of abounded holomorphic semigroup T_t on X. The for each 0<a<1 the maximal operator sup |T_tf(x)| belongs to L^p for each f in the domain of A^a. If moreover iA generates a bounded C_0 group and A has spectrum contained in the positive real semi axis, then A has a bounded H infinity functional calculus.