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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^{itL} satisfies a typical 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.
Original language  English 

Pages (fromto)  13931426 
Number of pages  34 
Journal  Mathematische Annalen 
Volume  375 
Issue number  34 
Early online date  15 Jun 2019 
DOIs  
Publication status  Published  Dec 2019 
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Projects
 1 Finished

Multiparameter Harmonic Analysis: Weighted Estimates for Singular Integrals
Duong, X., Ward, L., Li, J., Lacey, M., Pipher, J. & MQRES, M.
16/02/16 → 30/06/20
Project: Research