Author:
Dickinson Detta,Gramchev Todor,Yoshino Masafumi
Abstract
AbstractThis paper concerns perturbations of smooth vector fields on $\mathbb{T}^n$ (constant if $n\geq3$) with zeroth-order $C^\infty$ and Gevrey $G^\sigma$, $\sigma\geq1$, pseudodifferential operators. Simultaneous resonance is introduced and simultaneous resonant normal forms are exhibited (via conjugation with an elliptic pseudodifferential operator) under optimal simultaneous Diophantine conditions outside the resonances. In the $C^\infty$ category the results are complete, while in the Gevrey category the effect of the loss of the Gevrey regularity of the conjugating operators due to Diophantine conditions is encountered. The normal forms are used to study global hypoellipticity in $C^\infty$ and Gevrey $G^\sigma$. Finally, the exceptional sets associated with the simultaneous Diophantine conditions are studied. A generalized Hausdorff dimension is used to give precise estimates of the ‘size’ of different exceptional sets, including some inhomogeneous examples.AMS 2000 Mathematics subject classification: Primary 37C15; 11J13. Secondary 58J40; 11J20; 35H05
Publisher
Cambridge University Press (CUP)
Cited by
28 articles.
订阅此论文施引文献
订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献