Rs-bounded H∞-calculus for sectorial operators via generalized Gaussian estimates

Kunstmann, Peer Christian; Ullmann, Alexander

doi:10.1002/mana.201300132

Cited by 3 publications

(5 citation statements)

References 37 publications

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“…and the dual estimate due to self-adjointness of the semigroup, the generalised Gaussian estimates (2.5) imply the hypotheses of [KuUl,Theorem 2.3]. Then [KuUl,Theorem 2.3]…”

Section: Corollary 43 1 Let the Assumptions Of Theorem 42 Be Satisfie...mentioning

confidence: 82%

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Hörmander functional calculus on UMD lattice valued $L^p$ spaces under generalised Gaussian estimates

Deleaval,

Kemppainen,

Kriegler

2018

Preprint

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We consider self-adjoint semigroups Tt = exp(−tA) acting on L 2 (Ω) and satisfying (generalised) Gaussian estimates, where Ω is a metric measure space of homogeneous type of dimension d. The aim of the article is to show that A ⊗ IdY admits a Hörmander type H β 2 functional calculus on L p (Ω; Y ) where Y is a UMD lattice, thus extending the wellknown Hörmander calculus of A on L p (Ω). We show that if Tt is lattice positive (or merely admits an H ∞ calculus on L p (Ω; Y )) then this is indeed the case. Here the derivation exponent has to satisfy β > α • d + 1 2 , where α ∈ (0, 1) depends on p, and on convexity and concavity exponents of Y . A part of the proof is the new result that the Hardy-Littlewood maximal operator is bounded on L p (Ω; Y ). Moreover, our spectral multipliers satisfy square function estimates in L p (Ω; Y ). In a variant, we show that if e itA satisfies a dispersive L 1 (Ω) → L ∞ (Ω) estimate, then β > d+1 2 above is admissible independent of convexity and concavity of Y . Finally, we illustrate these results in a variety of examples.

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“…and the dual estimate due to self-adjointness of the semigroup, the generalised Gaussian estimates (2.5) imply the hypotheses of [KuUl,Theorem 2.3]. Then [KuUl,Theorem 2.3]…”

Section: Corollary 43 1 Let the Assumptions Of Theorem 42 Be Satisfie...mentioning

confidence: 82%

“…Consequently, A has a H β 2 calculus on L p (R d ; L s (Ω ′ )) for any p 0 < p, s < p ′ 0 and β as in (5.2). We refer to [Bl,Section 2], [KuUl,Section 3] and the references therein for detailed explanations of the two preceding paragraphs and more examples.…”

Section: Examplesmentioning

confidence: 99%

Hörmander functional calculus on UMD lattice valued $L^p$ spaces under generalised Gaussian estimates

Deleaval,

Kemppainen,

Kriegler

2018

Preprint

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show abstract

“…and the dual estimate due to self-adjointness of the semigroup, the generalized Gaussian estimates (2.5) imply the hypotheses of [KuUl,Theorem 2.3]. Then [KuUl,Theorem 2.3] implies that A has an R s -bounded H ∞ ( ω ) calculus on L p ( ) for s, p ∈ (p 0 , p ′ 0 ) and ω > 0, R s boundedness being defined in that article. By [KuUl,Theorem 2.1], A has then a bounded H ∞ ( ω ) calculus on L p ( ;…”

Section: Now Decompose H Into the Sum Of Two Functionsmentioning

confidence: 86%

“…Then [KuUl,Theorem 2.3] implies that A has an R s -bounded H ∞ ( ω ) calculus on L p ( ) for s, p ∈ (p 0 , p ′ 0 ) and ω > 0, R s boundedness being defined in that article. By [KuUl,Theorem 2.1], A has then a bounded H ∞ ( ω ) calculus on L p ( ;…”

Section: Now Decompose H Into the Sum Of Two Functionsmentioning

confidence: 99%

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Hörmander functional calculus on UMD lattice valued Lp spaces under generalized Gaussian estimates

Deleaval

Kemppainen²,

Kriegler

2021

JAMA

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We consider self-adjoint semigroups T t = exp(−tA) acting on L 2 ( ) and satisfying (generalized) Gaussian estimates, where is a metric measure space of homogeneous type of dimension d. The aim of the article is to show that A ⊗ Id Y admits a Hörmander type H β 2 functional calculus on L p ( ; Y) where Y is a UMD lattice, thus extending the well-known Hörmander calculus of A on L p ( ). We show that if T t is lattice positive (or merely admits an H ∞ calculus on L p ( ; Y)) then this is indeed the case. Here the derivation exponent has to satisfy β > α•d+ 1 2 , where α ∈ (0, 1) depends on p, and on convexity and concavity exponents of Y. A part of the proof is the new result that the Hardy-Littlewood maximal operator is bounded on L p ( ; Y). Moreover, our spectral multipliers satisfy square function estimates in L p ( ; Y). In a variant, we show that if e itA satisfies a dispersive L 1 ( ) → L ∞ ( ) estimate, then β > d+1 2 above is admissible independent of convexity and concavity of Y. Finally, we illustrate these results in a variety of examples.

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Maximal Hörmander Functional Calculus onLpSpaces and UMD Lattices

Deleaval

Kriegler

2022

International Mathematics Research Notices

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Let $A$ be a generator of an analytic semigroup having a Hörmander functional calculus on $X = L^p(\Omega ,Y)$, where $Y$ is a UMD lattice. Using methods from Banach space geometry in connection with functional calculus, we show that for Hörmander spectral multipliers decaying sufficiently fast at $\infty $, there holds a maximal estimate $\| \sup _{t> 0} |m(tA)f|\, \|_{L^p(\Omega ,Y)} \lesssim \|f\|_{L^p(\Omega ,Y)}$. We also show square function estimates $\left \| \left ( \sum _k \sup _{t> 0} |m_k(tA)f_k|^2 \right )^{\frac 12} \right \|_{L^p(\Omega ,Y)} \lesssim \left \| \left ( \sum _k |f_k|^2 \right )^{\frac 12} \right \|_{L^p(\Omega ,Y)}$ for suitable families of spectral multipliers $m_k$, which are even new for the euclidean Laplacian on scalar valued $L^p(\ensuremath {{\mathbb {R}}}^d)$. As corollaries, we obtain maximal estimates for wave propagators and Bochner–Riesz means. Finally, we illustrate the results by giving several examples of operators $A$ that admit a Hörmander functional calculus on some $L^p(\Omega ,Y)$ and discuss examples of lattices $Y$ and non-self-adjoint operators $A$ fitting our context.

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Rs-bounded H∞-calculus for sectorial operators via generalized Gaussian estimates

Cited by 3 publications

References 37 publications

Hörmander functional calculus on UMD lattice valued $L^p$ spaces under generalised Gaussian estimates

Hörmander functional calculus on UMD lattice valued $L^p$ spaces under generalised Gaussian estimates

Hörmander functional calculus on UMD lattice valued Lp spaces under generalized Gaussian estimates

Maximal Hörmander Functional Calculus onLpSpaces and UMD Lattices

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