We study the Dirac equation by Hormander's L2-method. By choosing appropriate weights, we get some applications in symplectic topology.

For Dirac bundles over 2-dimensional Riemannian manifolds, in compact case we give a sucient condition for the solvability of the Dirac equation in terms of a curvature integral; in noncompact case, we prove the Dirac equation is always solvable in weighted L2 space. As an application, we recover Hofer's Fredholm regular criteria of holomorphic curves in an almost complex manifold of dimension four.

On compact Riemannian manifolds, we give a new proof of Bar's theorem comparing the rst eigenvalue of the Dirac operator with that of the Yamabe type operator.

On Riemannian manifolds with cylindrical ends, we obtain solvability in L2 space with suitable exponential weights allowing mild negativity of the curvature. 

We also improve the above results when the Dirac bundle has a Z2-grading.