Multipliers for eigenfunction expansions of some Schrödinger operators

Stempak, Krzysztof

doi:10.1090/s0002-9939-1985-0774006-9

Cited by 5 publications

(5 citation statements)

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“…By Note that the auxiliary calculus in the preceding Theorem was only assumed to be able to define the Bochner-Riesz means as closed densely-defined operators, it can be omitted if A is self-adjoint on L 2 (U) and X = L p (U). To our knowledge, R-bounded Bochner-Riesz means have been considered for the first time in [7] and [57]. In [57, Theorem A], see also [7,Théorème (7.2)], a functional calculus similar to our H β 1 calculus is established, with the somewhat stronger norm,…”

Section: Assumption 82mentioning

confidence: 99%

Spectral multiplier theorems via $$H^\infty $$ H ∞ calculus and R-bounds

Kriegler

Weis

2017

Math. Z.

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We prove spectral multiplier theorems for Hörmander classes H α p for 0-sectorial operators A on Banach spaces assuming a bounded H ∞ (Σ σ ) calculus for some σ ∈ (0, π) and norm and certain R-bounds on one of the following families of operators: the semigroup e −zA on C + , the wave operators e isA for s ∈ R, the resolvent (λ − A) −1 on C\R, the imaginary powers A it for t ∈ R or the Bochner-Riesz means (1 − A/u) α + for u > 0. In contrast to the existing literature we neither assume that A operates on an L p scale nor that A is self-adjoint on a Hilbert space. Furthermore, we replace (generalized) Gaussian or Poisson bounds and maximal estimates by the weaker notion of R-bounds, which allow for a unified approach to spectral multiplier theorems in a more general setting. In this setting our results are close to being optimal. Moreover, we can give a characterization of the (R-bounded) H α 1 calculus in terms of R-boundedness of Bochner-Riesz means.

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Section: Assumption 82mentioning

confidence: 99%

Spectral multiplier theorems via $$H^\infty $$ H ∞ calculus and R-bounds

Kriegler

Weis

2017

Math. Z.

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“…L p where x i ∈ L p (Ω) and the S i are members of one of the families listed above (see e.g. [3,34] for an early appearance of this square sum estimate in the context of spectral multiplier theorems). If (r n ) is a sequence of Rademacher functions on [0, 1] one can reformulate (1.1) equivalently as (1.2)…”

Section: Introductionmentioning

confidence: 99%

Spectral multiplier theorems and averaged R-boundedness

Kriegler

Weis

2017

Semigroup Forum

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Let A be a 0-sectorial operator with a bounded H ∞ (Σ σ )-calculus for some σ ∈ (0, π), e.g. a Laplace type operator on L p (Ω), 1 < p < ∞, where Ω is a manifold or a graph. We show that A has a H α 2 (R + ) Hörmander functional calculus if and only if certain operator families derived from the resolvent (λ−A) −1 , the semigroup e −zA , the wave operators e itA or the imaginary powerswe denote the R-bound of this set. In a Hilbert space X, R[L 2 (J)]-boundedness reduces to the simple estimate J | N(t)x, y | 2 dt 1 2 ≤ C x y for all x, y ∈ H.Assume now that A is a 0-sectorial operator with an H ∞ (Σ σ ) calculus for some σ ∈ (0, π) on a Banach space isomorphic to a subspace of an L p (Ω) space with 1 ≤ p < ∞ (or more generally, let X have Pisier's property (α)). Then our main results, Theorems 6.1 and 6.4 show (among other statements), that the following conditions on the operator function above are essentially equivalent:1. A has an R-bounded H α 2 spectral calculus, i.e.

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“…The Properties of these multipliers in L p -spaces have been investigated in the references S. Bagchi, S. Thangavelu [1], J. Epperson [15], K. Stempak and J.L. Torrea [26,27,28], S. Thangavelu [29,30] and references therein. Hermite expansions for distributions can be found in B. Simon [25].…”

Section: Introductionmentioning

confidence: 99%