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Linear free divisors and the global logarithmic comparison theorem

Michel Granger, David Mond, Alicia Nieto-Reyes, Mathias Schulze (2009)

Annales de l’institut Fourier

A complex hypersurface D in ℂ n is a linear free divisor (LFD) if its module of logarithmic vector fields has a global basis of linear vector fields. We classify all LFDs for n at most 4 .By analogy with Grothendieck’s comparison theorem, we say that the global logarithmic comparison theorem (GLCT) holds for D if the complex of global logarithmic differential forms computes the complex cohomology of ℂ n ∖ D . We develop a general criterion for the GLCT for LFDs and prove that it is fulfilled whenever the...

Linear topological properties of the Lumer-Smirnov class of the polydisc

Marek Nawrocki (1992)

Studia Mathematica

Linear topological properties of the Lumer-Smirnov class L N ∗ ( n ) of the unit polydisc n are studied. The topological dual and the Fréchet envelope are described. It is proved that L N ∗ ( n ) has a weak basis but it is nonseparable in its original topology. Moreover, it is shown that the Orlicz-Pettis theorem fails for L N ∗ ( n ) .

Linearly invariant families of holomorphic functions in the unit polydisc

Janusz Godula, Victor Starkov (1996)

Banach Center Publications

In this paper we extend the definition of the linearly invariant family and the definition of the universal linearly invariant family to higher dimensional case. We characterize these classes and give some of their properties. We also give a relationship of these families with the Bloch space.

Liouville type theorem for solutions of linear partial differential equations with constant coefficients

Akira Kaneko (2000)

Annales Polonici Mathematici

We discuss existence of global solutions of moderate growth to a linear partial differential equation with constant coefficients whose total symbol P(ξ) has the origin as its only real zero. It is well known that for such equations, global solutions tempered in the sense of Schwartz reduce to polynomials. This is a generalization of the classical Liouville theorem in the theory of functions. In our former work we showed that for infra-exponential growth the corresponding assertion is true if and...

Liouville-type theorems for foliations with complex leaves

Giuseppe Della Sala (2010)

Annales de l’institut Fourier

In this paper we discuss various problems regarding the structure of the foliation of some foliated submanifolds S of ℂ n , in particular Levi flat ones. As a general scheme, we suppose that S is bounded along a coordinate (or a subset of coordinates), and prove that the complex leaves of its foliation are planes.

Lipschitz properties of semi-analytic sets

Adam Parusiński (1988)

Annales de l'institut Fourier

The existence of Lipschitz stratification, in the sense of Mostowski, for compact semi-analytic sets is proved. (This stratification ensures the constance of the Lipschitz type along each stratum). The proof is independent of the complex case, considered by Mostowski, and gives also some other Lipschitz properties of semi-analytic sets.

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