Positivity 11 (2007), 399–416
2007 Birkh¨auser Verlag Basel/Switzerland
Optimal Domains for L
Via Stochastic Measures
Guillermo P. Curbera and Olvido Delgado
Abstract. We study extension of operators T : E → L
([0, 1]), where E is an
F–function space and L
([0, 1]) the space of measurable functions with the
topology of convergence in measure, to domains larger than E, and we study
the properties of such domains. The main tool is the integration of scalar
functions with respect to stochastic measures and the corresponding spaces
of integrable functions.
Mathematics Subject Classiﬁcation (2000). 46G10; 47B38; 46A16; 46E30.
Keywords. F–spaces, kernel operators, lattices, operators between function
spaces, optimal domains, space of integrable functions, space of measurable
functions, vector measures.
Let T : E → L
([0, 1]) be a linear operator, where E is a function space and
([0, 1]) the space of measurable functions with the topology of convergence in
measure. The aim of this paper is to study conditions on T and E that allow us
to extend the operator T to domains larger than E and, in that case, to study the
properties of such domains.
In the case when T : E → X with X (and also E) a Banach space, this prob-
lem has been considered in , , . For the particular case of T the operator
associated with Sobolev inequality and X a rearrangement invariant space, this
study has been done in , ; and for T a convolution operator and X = L
in . In all of these cases, the main tool has been an X–valued measure ν
canonically associated with the operator T and the corresponding space L
scalar functions which are integrable with respect to ν. The integration theory for
Banach space valued measures was developed by Bartle, Dunford and Schwartz,
, and Lewis, , . However, L
([0, 1]) with the topology of convergence in
Partially supported by D.G.I. #MTM2006-13000-C03-01 (Spain).