### Abstract

Let *X* be a metric space with a doubling measure satisfying μ(B)≳r_{B}^{n} for any ball *B* with any radius r_{B}> 0. Let *L* be a non negative selfadjoint operator on *L*^{2}(X). We assume that e^{-}* ^{t}^{L}* satisfies a Gaussian upper bound and that the flow e

*satisfies a typical*

^{itL}*L*

^{1}-

*L*

^{∞}dispersive estimate of the form

‖e^{itL}‖_{L1→L}^{∞}≲|t|^{-n/2}.

Then we prove a similar *L*^{1}- *L*^{∞} dispersive estimate for a general class of flows e^{i}^{t}^{ϕ}^{(}^{L}^{)}, with *φ*(r) of power type near 0 and near ∞. In the case of fractional powers φ(*L*) = *L*^{ν}, ν∈ (0 , 1) , we deduce dispersive estimates for e^{itL}^{ν} with data in Sobolev, Besov or Hardy spaces H_{L}^{p} with p∈ (0 , 1] , associated to the operator *L*.

Language | English |
---|---|

Pages | 1393-1426 |

Number of pages | 34 |

Journal | Mathematische Annalen |

Volume | 375 |

Issue number | 3-4 |

Early online date | 15 Jun 2019 |

DOIs | |

Publication status | Published - Dec 2019 |

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### Cite this

*Mathematische Annalen*,

*375*(3-4), 1393-1426. https://doi.org/10.1007/s00208-019-01857-w

}

*Mathematische Annalen*, vol. 375, no. 3-4, pp. 1393-1426. https://doi.org/10.1007/s00208-019-01857-w

**On the flows associated to selfadjoint operators on metric measure spaces.** / Bui, The Anh; D'Ancona, Piero; Duong, Xuan Thinh; Müller, Detlef.

Research output: Contribution to journal › Article › Research › peer-review

TY - JOUR

T1 - On the flows associated to selfadjoint operators on metric measure spaces

AU - Bui, The Anh

AU - D'Ancona, Piero

AU - Duong, Xuan Thinh

AU - Müller, Detlef

PY - 2019/12

Y1 - 2019/12

N2 - Let X be a metric space with a doubling measure satisfying μ(B)≳rBn for any ball B with any radius rB> 0. Let L be a non negative selfadjoint operator on L2(X). We assume that e-tL satisfies a Gaussian upper bound and that the flow eitL satisfies a typical L1- L∞ dispersive estimate of the form ‖eitL‖L1→L∞≲|t|-n/2.Then we prove a similar L1- L∞ dispersive estimate for a general class of flows eitϕ(L), with φ(r) of power type near 0 and near ∞. In the case of fractional powers φ(L) = Lν, ν∈ (0 , 1) , we deduce dispersive estimates for eitLν with data in Sobolev, Besov or Hardy spaces HLp with p∈ (0 , 1] , associated to the operator L.

AB - Let X be a metric space with a doubling measure satisfying μ(B)≳rBn for any ball B with any radius rB> 0. Let L be a non negative selfadjoint operator on L2(X). We assume that e-tL satisfies a Gaussian upper bound and that the flow eitL satisfies a typical L1- L∞ dispersive estimate of the form ‖eitL‖L1→L∞≲|t|-n/2.Then we prove a similar L1- L∞ dispersive estimate for a general class of flows eitϕ(L), with φ(r) of power type near 0 and near ∞. In the case of fractional powers φ(L) = Lν, ν∈ (0 , 1) , we deduce dispersive estimates for eitLν with data in Sobolev, Besov or Hardy spaces HLp with p∈ (0 , 1] , associated to the operator L.

UR - http://www.scopus.com/inward/record.url?scp=85067783279&partnerID=8YFLogxK

UR - http://purl.org/au-research/grants/arc/DP160100153

U2 - 10.1007/s00208-019-01857-w

DO - 10.1007/s00208-019-01857-w

M3 - Article

VL - 375

SP - 1393

EP - 1426

JO - Mathematische Annalen

T2 - Mathematische Annalen

JF - Mathematische Annalen

SN - 0025-5831

IS - 3-4

ER -