Peer Christian Kunstmann

Maximal Lp-regularity for Parabolic Equations, Fourier Multiplier Theorems and \(H^\infty\)-functional Calculus

In these lecture notes we report on recent breakthroughs in the functional analytic approach to maximal regularity for parabolic evolution equations, which set off a wave of activity in the last years and allowed to establish maximal L p -regularity for large classes of classical partial differential operators and systems.

Calderón-Zygmund theory for non-integral operators and the

We modify Hormanders well-known weak type (1,1) condition for integral operators (in a weakened version due to Duong and McIn- tosh) and present a weak type (p,p) condition for arbitrary operators. Given an operatorA on L2 with a bounded H ∞ calculus, we show as an application the Lr-boundedness of the H ∞ calculus for all r ∈ (p,q), provided the semigroup (e −tA ) satisfies suitable weighted Lp → Lq-norm estimates with 2 ∈ (p,q ). This generalizes results due to Duong, McIntosh and Robinson for the special case (p,q )=( 1, ∞) 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 im- provement is still applicable, e.g. if A is a Schrodinger operator with a singular potential, an elliptic higher order operator with bounded measurable coefficients or an elliptic second order operator with sin- gular lower order terms.

H^{\infty}

Preface.- Giuseppe Da Prato: An Introduction to Markov Semigroups.- Peer C. Kunstmann and Lutz Weis: Maximal

functional calculus

L_p -regularity for Parabolic Equations, Fourier Multiplier Theorems and

Functional analytic methods for evolution equations

H^\infty

Distribution semigroups and abstract Cauchy problems

-functional Calculus.- Irena Lasiecka: Optimal Control Problems and Riccati Equations for Systems with Unbounded Controls and Partially Analytic Generators-Applications to Boundary and Point Control Problems.- Alessandra Lunardi: An Introduction to Parabolic Moving Boundary Problems.- Roland Schnaubelt: Asymptotic Behaviour of Parabolic Nonautonomous Evolution Equations.

Weighted Admissibility and Wellposedness of Linear Systems in Banach Spaces

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 δ′ ⊗ I − δ ⊗ A has a fundamental solution in D′(L(E, D)) where D denotes the domain of A supplied with the graph norm and I denotes the inclusion D → E; (c) A generates a local integrated semigroup. We also show that every generator of a distribution semigroup generates a regularized semigroup. Introduction Distribution semigroups in Banach spaces and their generators were introduced by J. L. Lions in [16] as a generalization of C0-semigroups and their generators. Like the generators of C0-semigroups the generators of the distribution semigroups in [16] are densely defined. It was then shown in [16] that the densely defined closed linear operators A that generate a distribution semigroup are exactly those for which the convolution operator δ′ ⊗ I − δ ⊗ A where I denotes the inclusion D(A) → E has a fundamental solution. Treating Cauchy problems as convolution equations has turned out to be very useful (see [6], [12], [13] and [14]). But in doing so the assumption that D(A) is dense in E is unnecessary in most of the results. On the other hand in the last years there has been an interest in dealing with operators that are not densely defined, starting perhaps with [7]. Reasons for this interest may be that it might not be so easy to decide whether a given operator is densely defined and the difficulties that arise when passing from E to E∗ or l∞(E). So the question how the theory of distribution semigroups introduced in [16] can be extended to cover non-densely defined generators in such a way that the above characterization via fundamental solutions holds seems natural. It turns out that this indeed is possible in a way that even simplifies the original definitions given in [16]. Moreover, the local strongly continuous representations (see section 4) of the distribution semigroups thus obtained are exactly the local integrated semigroups (for the distribution semigroups in the sense of [16] a relation to local integrated semigroups has been obtained in [23], Theorem 5.6; see also [15], Theorem 7.6 or [2], Theorem 7.2). So the theory presented here Received by the editors October 17, 1995 and, in revised form, February 6, 1997. 1991 Mathematics Subject Classification. Primary 47D03, 34G10, 47A10, 46F10. c ©1999 American Mathematical Society

Heat Kernel Estimates and LP Spectral Independence of Elliptic Operators

We study linear control systems in infinite-dimensional Banach spaces governed by analytic semigroups. For

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

p\in[1,\infty]

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

and