Perturbations of Vector Fields on Tori: Resonant Normal Forms and Diophantine Phenomena

Abstract

This paper concerns perturbations of smooth vector fields on T n (constant if n 3) with zeroth-order C ∞ and Gevrey G σ , σ 1, 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 ∞ 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 ∞ and Gevrey G σ . 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.

Existence and Regularity of Periodic Solutions to Certain First-Order Partial Differential Equations

“…The Diophantine condition (DC1) (respectively, (DC2)) is related to the notion of nonresonant (respectively, resonant) condition that appears in [7].…”

On Gevrey solvability and regularity

“…Local and global G s , 1 ≤ s ≤ ∞, hypoellipticity has been studied by many authors, including Albanese, Corli, and Rodino [1], Amano [3], Baouendi and Goulaouic [4], Bell and Mohammed [5], Bergamasco, Cordaro, and Malagutti [6], Bove and Tartakoff [8], [9], Christ [12]- [14], Cordaro and Himonas [15], [16], Dickinson, Gramchev, and Yoshino [17], Fedii [18], Fujiwara and Omori [19], Gramchev,Popivanov,and Yoshino [20]- [22], Greenfield and Wallach [24], Hanges and Himonas [25], [26], Helffer [27], Himonas [28], [29], Himonas and Petronilho [30]- [33], Himonas, Petronilho, and dos Santos [34], Hörmander [35], [36], Kohn [38], Pham The Lai and Robert [39], Metivier [40], Oleinik and Radkevic [41], Omori and Kobayashi [42], Rodino [47], Rothschild and Stein [48], Tartakoff [49].…”

Simultaneous reduction to normal forms of commuting singular vector fields with linear parts having Jordan blocks

“…In view of (3.9), the equation 13) where M N is given by 14) and M 0 = 0. We note that, for the equation…”