Pars 1 and 2 are devoted to study of solutions of certain differential inequalities.
Namely, in Part 1 we show that a germ of an analytic set (real or complex) admits
a Gagliardo-Nirenberg type inequality with a certain exponent s>=1. At a regular point
s=1, and the inequality becomes classical. As our examples show, s can be strictly greater than 1 even for an isolated singularity.
In Part 2 we prove the property of unique continuation for solutions of differential inequality |\Delta u|<=|Vu| for a large class of potentials V. This result can be applied to the problem of absence of positive eigenvalues for
self-adjoint Schroedinger operator -\Delta+V defined in the sense of the form sum.
The results of Part 2 are joint with Leonid Shartser.
In Parts 3 and 4 we derive the basic elements of complex function theory within
some subalgebras of holomorphic functions (including extension from submanifolds, corona type theorem, properties of divisors, approximation property). Our key instruments and results are the analogues of Cartan theorems A and B for the `coherent sheaves' on the maximal ideal spaces of these subalgebras, and of Oka-Cartan theorem on coherence of the sheaves of ideals of the corresponding complex analytic subsets.
More precisely, in Part 3 we consider the algebras of holomorphic functions on regular
coverings of complex manifolds whose restrictions to each fiber belong to a translation-invariant Banach subalgebra of bounded functions endowed with sup-norm.
The model examples of such subalgebras are Bohr's holomorphic almost periodic functions on tube domains, and all fibrewise bounded holomorphic functions on regular coverings of complex manifolds.
In Part 4 the primary object of study is the subalgebra of bounded holomorphic functions on the unit disk whose moduli can have only boundary discontinuities of the first kind.
The results of Parts 3 and 4 are joint with Alexander Brudnyi.
| Identifer | oai:union.ndltd.org:TORONTO/oai:tspace.library.utoronto.ca:1807/32309 |
| Date | 26 March 2012 |
| Creators | Kinzebulatov, Damir |
| Contributors | Milman, Pierre |
| Source Sets | University of Toronto |
| Language | en_ca |
| Detected Language | English |
| Type | Thesis |
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