Microfunctions for Sheaves of Holomorphic Functions with Growth Conditions

Abstract

Mikio Sato devised microfunctions as a means of measuring the singularities of hyperfunctions. In 1970, Kawai and Sato introduced Fourier hyperfunctions in their study of partial differential operators. The class of Fourier hyperfunctions has been generalized by Saburi, Nagamachi, and Kaneko, among others, and most recently by Berenstein and Struppa. Berenstein and Struppa introduced Fourier p-hyperfunctions, where p is a plurisubharmonic function satisfying certain smoothness and growth conditions. p(z) = |z|^s, s \ge 1 are the cases studied by Sato, Kawai, Nagamachi, and Kaneko. Following the methods of Sato, Kawai and Kashiwara, this dissertation introduces Fourier p-microfunctions functorially, though under very severe conditions on p. These restrictions on p are satisfied when, for instance, p(z) = log^+ |f| where f is a product of 1 variable holomorphic functions with zeroes uniformly bounded away from the real axis. Kaneko has introduced Fourier microfunctions for p(z) = |Rez|^s, s > 0, using tubes. When s < 1 these p's are not plurisubharmonic. Thus the results here complement his.