No abstract uploaded!

No abstract uploaded!

The α-modulation spaces$M$^{$s$,α}_{$p$,$q$}($R$^{$d$}), α∈[0,1], form a family of spaces that contain the Besov and modulation spaces as special cases. In this paper we prove that a pseudodifferential operator σ($x$,$D$) with symbol in the Hörmander class$S$^{$b$}_{ρ,0}extends to a bounded operator σ($x$,$D$):$M$^{$s$,α}_{$p$,$q$}($R$^{$d$})→$M$^{$s$-$b$,α}_{$p$,$q$}($R$^{$d$}) provided 0≤α≤ρ≤1, and 1<$p$,$q$<∞. The result extends the well-known result that pseudodifferential operators with symbol in the class$S$^{$b$}_{1,0}maps the Besov space$B$^{$s$}_{$p$,$q$}($R$^{$d$}) into$B$^{$s$-$b$}_{$p$,$q$}($R$^{$d$}).

For smooth compact oriented Riemannian manifolds$M$of dimension 4$k$+2,$k$≥0, with or without boundary, and a vector bundle$F$on$M$with an inner product and a flat connection, we construct a modification of the Hodge star operator on the middle-dimensional (parabolic) cohomology of$M$twisted by$F$. This operator induces a canonical complex structure on the middle-dimensional cohomology space that is compatible with the natural symplectic form given by integrating the wedge product. In particular, when$k$=0 we get a canonical almost complex structure on the non-singular part of the moduli space of flat connections on a Riemann surface, with monodromies along boundary components belonging to fixed conjugacy classes when the surface has boundary, that is compatible with the standard symplectic form on the moduli space.

We prove that the rotation in time T of a trajectory of a K-Lipschitz vector ¯eld in Rn around a given point (stationary or non-stationary) is bounded by A + BKT with A;B absolute constants. In particular, trajectories of a Lipschitz vector ¯eld in ¯nite time cannot have an in¯nite rotation around a given point (while trajectories of a C1 vector ¯eld may have an in¯- nite rotation around a straight line in ¯nite time). The bound above extends to the mutual rotation of two trajectories (for the time intervals T and T0, respectively) of a K-Lipschitz vector ¯eld in R3: this rotation is bounded from above by the quantity CK min(T; T0) + DK2TT0.