Integral representation formulas and applications

Khalid Filali Adib, Purdue University

Abstract

In this work, we consider the integral representations formulas constructed to study the Cauchy-Riemann equation and the tangential Cauchy-Riemann equation. We extend some of the classical results already known in the case of a strictly pseudo-convex domain to a class of convex domains. Among the results we obtained is the integral formula for the $\bar\partial\sb{b}$-equation on some convex domains. This result is proved in chapter 2. The last result we give in this chapter concerns a problem of division. By applying the same method as in the previous paragraphs, we prove the regularity of the solutions of the division problem and show that the solutions are regular. This result was obtained in the case of a convex domain satisfying some technical conditions. In chapter 3, we study the regularity of the Bergman kernel on strictly pseudo-convex domains in $\doubc\sp{n}.$ We obtain representation of the Bergman kernel on domains which are intersection of two transverse strictly pseudo-convex domains. The same study allows us to have Holder estimates for the Cauchy-Riemann equation on some convex domains.

Degree

Ph.D.

Advisors

Bell, Purdue University.

Subject Area

Mathematics

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