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

Kunstmann, Peer Christian; Weis, Lutz

doi:10.1007/978-3-540-44653-8_2

Cited by 436 publications

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“…(The symbol Σ ϕ denotes the double sector of angle ϕ, see below.) For proofs see also [32,Theorem 10.7] and Remark 3.8 below.…”

Section: Introductionmentioning

confidence: 99%

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A transference principle for general groups and functional calculus on UMD spaces

Haase

2009

Math. Ann.

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Let −i A be the generator of a C 0 -group (U (s)) s∈R on a Banach space X and ω > θ(U ), the group type of U . We prove a transference principle that allows to estimateIf X is a Hilbert space this yields new proofs of important results of McIntosh and Boyadzhiev-de Laubenfels. If X is a UMD space, one obtains a bounded H ∞ 1 -calculus of A on horizontal strips. Related results for sectorial and parabola-type operators follow. Finally it is proved that each generator of a cosine function on a UMD space has bounded H ∞ -calculus on sectors.

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“…(The symbol Σ ϕ denotes the double sector of angle ϕ, see below.) For proofs see also [32,Theorem 10.7] and Remark 3.8 below.…”

Section: Introductionmentioning

confidence: 99%

“…In [19], using the notion of R-boundedness, Clément and Prüss extended the transference principle and Theorem 1.2 to the operator-valued setting and gave applications to the maximal regularity problem. See [19,22,32,40] for more information.…”

Section: Introductionmentioning

confidence: 99%

A transference principle for general groups and functional calculus on UMD spaces

Haase

2009

Math. Ann.

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“…The next proposition collects some preliminary results which are needed in the sequel (see also [5,7,12,19]). Proposition 2.1.…”

Section: Preparationsmentioning

confidence: 99%

“…During the past few years a theory of L p -multipliers for operator valued functions has been developed by means of the notion of R-boundedness of sets of operators, see for example [1,2,3,4,5,6,7,8,10,12,13,17,18,19]. This theory has been applied to study maximal regularity of certain abstract evolution equations.…”

Section: Introductionmentioning

confidence: 99%

On R-boundedness of Unions of Sets of Operators

Gaans

2006

Partial Differential Equations and Functional Analysis

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It is shown that the union of a sequence T 1 , T 2 , . . . of R-bounded sets of operators from X into Y with R-bounds τ 1 , τ 2 , . . ., respectively, is Rbounded if X is a Banach space of cotype q, Y a Banach space of type p, and ∞ k=1 τ r k < ∞, where r = pq/(q−p) if q < ∞ and r = p if q = ∞. Here 1 ≤ p ≤ 2 ≤ q ≤ ∞ and p = q. The power r is sharp. Each Banach space that contains an isomorphic copy of c 0 admits operators T 1 , T 2 , . . . such that T k = 1/k, k ∈ N, and {T 1 , T 2 , . . .} is not R-bounded. Further it is shown that the set of positive linear contractions in an infinite dimensional L p is R-bounded only if p = 2.

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“…The H ∞ -calculus was introduced by McIntosh in [18]. We refer the reader to Denk, Hieber, Prüss [5] or Kunstmann, Weis [17] for recent and extensive surveys. converges absolutely and defines an element in L(Y ).…”

Section: Introductionmentioning

confidence: 99%

BoundedH_∞-Calculus for Differential Operators on Conic Manifolds with Boundary

Coriasco

Schrohe

Seiler

2007

Communications in Partial Differential Equations

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Abstract. We derive conditions that ensure the existence of a bounded H∞-calculus in weighted Lp-Sobolev spaces for closed extensions A T of a differential operator A on a conic manifold with boundary, subject to differential boundary conditions T . In general, these conditions ask for a particular pseudodifferential structure of the resolvent (λ − A T ) −1 in a sector Λ ⊂ C. In case of the minimal extension they reduce to parameter-ellipticity of the boundary value problem A T . Examples concern the Dirichlet and Neumann Laplacians.

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

Cited by 436 publications

References 0 publications

A transference principle for general groups and functional calculus on UMD spaces

A transference principle for general groups and functional calculus on UMD spaces

On R-boundedness of Unions of Sets of Operators

BoundedH_∞-Calculus for Differential Operators on Conic Manifolds with Boundary

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

Cited by 436 publications

References 0 publications

A transference principle for general groups and functional calculus on UMD spaces

A transference principle for general groups and functional calculus on UMD spaces

On R-boundedness of Unions of Sets of Operators

BoundedH∞-Calculus for Differential Operators on Conic Manifolds with Boundary

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Product

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BoundedH_∞-Calculus for Differential Operators on Conic Manifolds with Boundary