No abstract uploaded!

No abstract uploaded!

In this paper, we consider the continuity property of pseudo-differential operators with symbols whose Fourier transforms have compact support. As applications, we obtain the$L$^{$p$}-boundedness for symbols in Besov spaces and in modulation spaces.

We introduce tropical vector bundles, morphisms and rational sections of these bundles and define the pull-back of a tropical vector bundle and of a rational section along a morphism. Most of the definitions presented here for tropical vector bundles will be contained in$Torchiani, C.$,$Line Bundles on Tropical Varieties$, Diploma thesis, Technische Universität Kaiserslautern, Kaiserslautern,2010, for the case of line bundles. Afterwards we use the bounded rational sections of a tropical vector bundle to define the Chern classes of this bundle and prove some basic properties of Chern classes. Finally we give a complete classification of all vector bundles on an elliptic curve up to isomorphisms.

In this paper, we present two kinds of total Chern forms c(E,G) and c(E,G) as well as a total Segre form c(E,G) of a holomorphic Finsler vector bundle c(E,G) expressed by the Finsler metric c(E,G), which answers a question of Faran [The equivalence problem for complex Finsler Hamiltonians, in <i>Finsler Geometry</i>, Contemporary Mathematics, Vol. 196 (American Mathematical Society, Providence, RI, 1996), pp. 133144] to some extent. As some applications, we show that the signed Segre forms c(E,G) are positive c(E,G)-forms on c(E,G) when c(E,G) is of positive Kobayashi curvature; we prove, under an extra assumption, that a FinslerEinstein vector bundle in the sense of Kobayashi is semi-stable; we introduce a new definition of a flat Finsler metric, which is weaker than Aikous one [Finsler geometry on complex vector bundles, in <i>A Sampler of RiemannFinsler Geometry</i>, MSRI Publications, Vol. 50 (Cambridge