Narrow Results

We study the harmonic space of line bundle valued forms over a covering manifold with a discrete group action, and obtain an asymptotic estimate for the von Neumann dimension of the space of harmonic $(n,q)$-forms with values in high tensor powers of a semipositive line bundle. In particular, we estimate the von Neumann dimension of the corresponding reduced $L^2$-Dolbeault cohomology group. The main tool is a local estimate of the pointwise norm of harmonic forms with values in semipositive line bundles over Hermitian manifolds.

In this paper, we generalize the results of¹, on theory of hyperholomorphic analysis to the case for several variables, that is, Clifford analysis. These results are not only of theoretical importance but also parctical interest. In fact, we also generalize³, the Dirac operator to . When the base i ={i₁, …,i_n} is given, the Dirac operator D is determined, while ^ψD may consider as the rotation transform on the vector ψ = {ψ¹, … , ψⁿ}. Such a generalization is similar to Descartes coordinate system to the moving coordinate system, therefore, it is very convenient to build some basic formulas on Clifford analysis and to find some analytic representations of differential equations. From this idea, we obtain a very important representation of Bergman kernel function: