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Extreme cases of weak type interpolation.

Evgeniy Pustylnik (2005)

Revista Matemática Iberoamericana

We consider quasilinear operators T of joint weak type (a, b; p, q) (in the sense of [2]) and study their properties on spaces Lφ,E with the norm||φ(t) f*(t)||Ê, where Ê is arbitrary rearrangement-invariant space with respect to the measure dt/t. A space Lφ,E is said to be "close" to one of the endpoints of interpolation if the corresponding Boyd index of this space is equal to 1/a or to 1/p. For all possible kinds of such "closeness", we give sharp estimates for the function ψ(t) so as to obtain...

Extreme compact operators from Orlicz spaces to C ( Ω )

Shutao Chen, Marek Wisła (1993)

Commentationes Mathematicae Universitatis Carolinae

Let E ϕ ( μ ) be the subspace of finite elements of an Orlicz space endowed with the Luxemburg norm. The main theorem says that a compact linear operator T : E ϕ ( μ ) → C ( Ω ) is extreme if and only if T * ω ∈ Ext B ( ( E ϕ ( μ ) ) * ) on a dense subset of Ω , where Ω is a compact Hausdorff topological space and 〈 T * ω , x 〉 = ( T x ) ( ω ) . This is done via the description of the extreme points of the space of continuous functions C ( Ω , L ϕ ( μ ) ) , L ϕ ( μ ) being an Orlicz space equipped with the Orlicz norm (conjugate to the Luxemburg one). There is also given a theorem on closedness of the set of extreme...

Extreme topological measures

S. V. Butler (2006)

Fundamenta Mathematicae

It has been an open question since 1997 whether, and under what assumptions on the underlying space, extreme topological measures are dense in the set of all topological measures on the space. The present paper answers this question. The main result implies that extreme topological measures are dense on a variety of spaces, including spheres, balls and projective planes.

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