On some classes of q-plurisubharmonic functions and q-pseudoconcave sets
Pawlaschyk
Thomas Patrick
Pawlaschyk, Thomas Patrick
aut
2015-11-24
2015-12-10
en
<p>We introduce real q-convex functions on open subsets of the real Euclidean space. We study their properties and develop approximation techniques by continuous ones and by those who are real q-convex with corners. In the smooth case, a function is real q-convex if and only if its real Hessian has at most q negative eigenvalues. We compare these functions to their complex relatives: the q-plurisubharmonic functions defined on open sets of the complex Euclidean space. We show that each real q-convex function is q-plurisubharmonic. On the converse, each q-plurisubharmonic function, which is invariant in the imaginary part, is real q-convex. Another example for q-plurisubharmonic functions is given by those which are plurisubharmonic on leaves of a singular foliation by analytic sets of codimension q.<br><br>
The q-plurisubharmonic functions are used to define q-pseudoconcave sets. These are closed subsets S of an open set D in the complex Euclidean space for which the negative logarithm of the distance function to S is q-plurisubharmonic on D minus S. We prove that (Hartogs) q-pseudoconcave graphs of certain continuous mappings locally admit a foliation by complex submanifolds of dimension q. We construct generalized convex hulls of compact sets with respect to subfamilies of q-plurisubharmonic functions and compare them to each other and to hulls already existing in the literature (such as Basener's hull or the rationally convex hull).<br><br>
We verify that the Shilov boundary of a compact set for certain families of upper semi-continuous functions exists and coincides with the set of all peak points for the given family. As an application, we study the Shilov boundary of a convex body for q-plurisubharmonic functions and generalize Bychkov's result from 1981. More precisely, the complement of the Shilov q-boundary of a convex body K consists of all boundary points p with the following property: there exists an open neighbourhood U of p in the boundary of K such that each point z in U lies in some open part of a complex plane of dimension at least q+1.
urn:nbn:de:hbz:468-20151210-101726-1
2015-12-10T09:17:02.567Z
2015-12-14T07:21:26.277Z
published
Diss
fbc/mathematik/diss2015/pawlaschyk