Elliptic operators on refined Sobolev scales on vector bundles

We introduce a refined Sobolev scale on a vector bundle over a closed infinitely smooth manifold. This scale consists of inner product Hörmander spaces parametrized with a real number and a function varying slowly at infinity in the sense of Karamata. We prove that these spaces are obtained by the interpolation with a function parameter between inner product Sobolev spaces. An arbitrary classical elliptic pseudodifferential operator acting between vector bundles of the same rank is investigated on this scale. We prove that this operator is bounded and Fredholm on pairs of appropriate Hörmander spaces. We also prove that the solutions to the corresponding elliptic equation satisfy a certain a priori estimate on these spaces. The local regularity of these solutions is investigated on the refined Sobolev scale. We find new sufficient conditions for the solutions to have continuous derivatives of a given order.

The sectorial projection defined from logarithms

For a classical elliptic pseudodifferential operator $P$ of order $>0$ on a closed manifold $X$, such that the eigenvalues of the principal symbol $p_m(x,\xi)$ have arguments in $]\theta ,\varphi [$ and $]\varphi ,\theta +2\pi [$ ($\theta <\varphi <\theta +2\pi$), the sectorial projection $\Pi_{\theta,\varphi}(P)$ is defined essentially as the integral of the resolvent along $ e^{i\varphi}\overline{\mathrm R}_{+}\cup e^{i\theta}\overline{\mathrm R}_{+}$. In a recent paper, Booss-Bavnbek, Chen, Lesch and Zhu have pointed out that there is a flaw in several published proofs that $\Pi_{\theta,\varphi}(P)$ is a $\psi$do of order 0; namely that $p_m(x,\xi)$ cannot in general be modified to allow integration of $(p_m(x,\xi )-\lambda)^{-1}$ along $ e^{i\varphi}\overline{\mathrm R}_{+}\cup e^{i\theta}\overline{\mathrm R}_{+}$ simultaneously for all $\xi$. We show that the structure of $\Pi_{\theta,\varphi}(P)$ as a $\psi$do of order 0 can be deduced from the formula $\Pi_{\theta,\varphi}(P)=\frac {i}{2\pi}(\log_{\theta} P - \log_{\varphi} P)$ proved in an earlier work (coauthored with Gaarde). In the analysis of $\log_{\theta} P$ one need only modify $p_m(x,\xi)$ in a neighborhood of $e^{i\theta}\overline{\mathrm R}_{+}$ this is known to be possible from Seeley's 1967 work on complex powers.

PERTURBATIONS OF VECTOR FIELDS ON TORI: RESONANT NORMAL FORMS AND DIOPHANTINE PHENOMENA

This 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