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

Kriegler, Christoph; Weis, Lutz

doi:10.1007/s00209-017-1957-1

Cited by 17 publications

(63 citation statements)

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“…. Note that for any θ ∈ (0, π), the space [47]. Since W α p → H α p , by the above density, we get a bounded mapping W α p → B(X) extending the H ∞ calculus.…”

Section: Definition 29mentioning

confidence: 95%

“…Note that there is a partial converse of Corollary (R d , h 2 κ ; Y ))), according to [47]. Another application of Theorem 3.13 is the following spectral decomposition of Paley-Littlewood type.…”

mentioning

confidence: 95%

“…Then f β R belongs to H α q (with uniform norm bound for R > 0) if and only if β > α − 1 q , for q ∈ [1, ∞) and α > 1 q , see [20, p. 11] and [47]. Thus the boundedness of (1.2) yields boundedness of the Bochner-Riesz means for f β R (A) if β > α − 1 q .…”

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confidence: 99%

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Dunkl spectral multipliers with values in UMD lattices

Deleaval

Kriegler

2017

Journal of Functional Analysis

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We show a Hörmander spectral multiplier theorem for A = A0 ⊗ IdY acting on the Bochner space L p (R d , h 2 κ ; Y), where A0 is the Dunkl Laplacian, h 2 κ a weight function invariant under the action of a reflection group and Y is a UMD Banach lattice. We follow hereby a transference method developed by Bonami-Clerc and Dai-Xu, passing through a Marcinkiewicz multiplier theorem on the sphere. We hereby generalize works for A0 = −∆ acting on L p (R d , dx) by Girardi-Weis, Hytönen and others before. We apply our main result to maximal regularity for Cauchy problems involving A.

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“…. Note that for any θ ∈ (0, π), the space [47]. Since W α p → H α p , by the above density, we get a bounded mapping W α p → B(X) extending the H ∞ calculus.…”

Section: Definition 29mentioning

confidence: 95%

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confidence: 95%

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Dunkl spectral multipliers with values in UMD lattices

Deleaval

Kriegler

2017

Journal of Functional Analysis

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“…This statement makes sense in an arbitrary Banach space X and a set τ ⊂ B(X) is called R-bounded if (1.2) holds for all S i ∈ τ and x i ∈ X. Using R-boundedness in place of kernel estimates and the holomorphic H ∞ (Σ σ ) calculus instead of the spectral theorem for selfadjoint operators, one can develop a theory of spectral multiplier theorems for 0-sectorial operators on Banach spaces (see [21,22,23,24,25]). Again, R-bounds for one of the operator families listed above are sufficient to secure H α 2 (R + ) spectral theorems for such operators A.…”

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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“…One possible way was outlined by Kriegler and Weis (see [32], [35], and [36]). They considered 0-sectorial and 0-strip type operators A on a general Banach space X.…”

Section: Chapter 1 Introductionmentioning

confidence: 99%

The Holomorphic Hörmander Functional Calculus

Pannasch¹

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TThe topic of this thesis is functional calculus in connection with abstract multiplier theorems. In 1960, Hörmander showed how the uniform boundedness of certain integral means of a function m in L ∞ (R^d) and its weak derivatives imply that m yields a bounded Lp -Fourier multiplier. Nowadays, this is known as the Hörmander multiplier theorem, sometimes Hörmander--Mikhlin multiplier theorem. A noteworthy detail is that a radial function m(|x|) satisfies Hörmander's condition if and only if m (|x|²) does. Hence, Hörmander's theorem is also a result on the functional calculus of the negative Laplacian -Δ. Hörmander's result has inspired a lot of research, and authors have also proven similar results for other operators such as certain Schrödinger operators, Sublaplacians on Lie groups, and later certain differential operators on spaces of homogeneous type. For us, the work of Kriegler and Weis is of particular interest. Starting with the PhD thesis of Kriegler in 2009, they showed how abstract multiplier theorems can be proven in a more general context. Namely, considering a certain class of 0-sectorial and 0-strip type operators on a general Banach space, one can construct an abstract Hörmander functional calculus based on the classical holomorphic calculus. Then, by using probalistic techniques from Banach space geometry involving so-called R-boundedness one can derive multiplier results in this generalized setting. In 2001, García-Cuerva, Mauceri, Meda, Sjögren, and Torrea proved an abstract multiplier theorem for generators of symmetric contraction semigroups, where a bounded Hörmander calculus is inferred from growth conditions on the imaginary powers of the generator. As the considered operators need not be 0-sectorial, this result is not covered by the methods of Kriegler and Weis. However, the result is based on Meda's earlier work, where he derived a bounded Hörmander if the given imaginary powers only grow polynomially fast. In this case, the operator is 0-sectorial, and Kriegler and Weis were able to recover the result while improving the order of the calculus. In this thesis, we introduce a generalized class of Hörmander functions defined on strips and sectors. Based on this and the classical holomorphic calculus, we construct a holomorphic Hörmander calculus for a class of operators which may also have strip type or angle of sectoriality greater than zero. The main result is a generalization of the multiplier theorem of García-Cuerva et al. to Banach spaces of finite cotype and Banach spaces with Pisier's property (α), where we retain and even improve the order given by Kriegler and Weis for the 0-sectorial case.

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

Cited by 17 publications

References 52 publications

Dunkl spectral multipliers with values in UMD lattices

Dunkl spectral multipliers with values in UMD lattices

Spectral multiplier theorems and averaged R-boundedness

The Holomorphic Hörmander Functional Calculus

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