Large algebras of singular functions vanishing on prescribed sets

In this paper, the non-vacuousness of the family of all nowhere analytic infinitely differentiable functions on the real line vanishing on a prescribed set Z is characterized in terms of Z. In this case, large algebraic structures are found inside such family. The results obtained complete or extend a number of previous ones by several authors.Comment: 11 page

Nowhere hölderian functions and Pringsheim singular functions in the disc algebra

We prove the existence of dense linear subspaces, of infinitely generated subalgebras and of infinite dimensional Banach spaces in the disc algebra all of whose nonzero members are not α-h¨olderian at any point of the unit circle for any α > 0. This completes the recently established result of topological genericity of this kind of functions, as well as the corresponding lineability statements about functions that are nowhere differentiable at the boundary. Topological and algebraic genericity is also studied for the family of boundary-smooth holomorphic functions that are Pringsheim singular at any point of the unit circle.Plan Andaluz de Investigación (Junta de Andalucía)Ministerio de Economía y Competitividad (MINECO). Españ

Nowhere minimal CR submanifolds and Levi-flat hypersurfaces

A local uniqueness property of holomorphic functions on real-analytic nowhere minimal CR submanifolds of higher codimension is investigated. A sufficient condition called almost minimality is given and studied. A weaker necessary condition, being contained a possibly singular real-analytic Levi-flat hypersurface is studied and characterized. This question is completely resolved for algebraic submanifolds of codimension 2 and a sufficient condition for noncontainment is given for non algebraic submanifolds. As a consequence, an example of a submanifold of codimension 2, not biholomorphically equivalent to an algebraic one, is given. We also investigate the structure of singularities of Levi-flat hypersurfaces.Comment: 21 pages; conjecture 2.8 was removed in proof; to appear in J. Geom. Ana