| Co-Investigator(Kenkyū-buntansha) |
TAKAMURA Hiroyuki Univ.of Tsukuba, Inst.of Math., Assistant, 数学系, 助手 (40241781)
KAKEHI Tomoyuki Univ.of Tsukuba, Inst.of Math., Lecturer, 数学系, 講師 (70231248)
YAMAZAKI Mitsuru Univ.of Tsukuba, Inst.of Math., Associate Professor, 数学系, 助教授 (30240732)
SHIBATA Yoshihiro Univ.of Tsukuba, Inst.of Math., Associate Professor, 数学系, 助教授 (50114088)
KAJITANI Kunihiko Univ.of Tsukuba, Inst.of Math., Professor, 数学系, 教授 (00026262)
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| Research Abstract |
First, we investigated the theory of hyperfunctions from a viewpoint of (classical) analysis, and proved fundamental results from this viewpoint. In doing so, we considered the (inverse) Fourier transform S'_<epsilon> of {exp [epsilon <xi>] u (xi) ; u*S'}, and regarded the space of hyperfunctions as the local space of *_<epsilon>0>S'_<epsilon>. Here S denotes the Schwartz space. Furthermore, we established the (classical) analytical theory of pseudodifferential operators and microlocal analysis for hyper-functions, generalizing calculus of pseudodifferential operators and Fourier integral operators in S' to calculus in S'_<epsilon>. In the studies of partial differential operators (and pseudodifferential operators), we could apply the same arguments to S'_<epsilon>, especially the space of hyperfunctions, as used in the category of distributions. And we made it possible to investigate, with unified treatments, partial differential operators in the spaces of distributions, ultradistributions (and Gevrey), and hyperfunctions (and analytic functions). For example, we treated the problems, deriving a priori (energy) estimates. In particular, we obtained results on propagation of analytic singularities and analytic hypoellipticity for analytic pseudodifferential operators from a priori estimates. We also proved that the relation between hypoellipticity and local solvability in the category of hyperfunctions is the same as in the category of distributions. And we obtained several results on local solvability in the space of hyperfunctions from a priori estimates.
We studied partial differential operators from a viewpoint of derivation of a priori estimates. And we obtained a priori estimates for various problems. The related problems were studied by the investigators of this project. We believe that the results obtained here are of great use for the studies on partial differential operators.
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