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Locally constant functions

Joan Hart, Kenneth Kunen (1996)

Fundamenta Mathematicae

Let X be a compact Hausdorff space and M a metric space. E 0 ( X , M ) is the set of f ∈ C(X,M) such that there is a dense set of points x ∈ X with f constant on some neighborhood of x. We describe some general classes of X for which E 0 ( X , M ) is all of C(X,M). These include βℕ, any nowhere separable LOTS, and any X such that forcing with the open subsets of X does not add reals. In the case where M is a Banach space, we discuss the properties of E 0 ( X , M ) as a normed linear space. We also build three first countable Eberlein...

Locally solid topologies on spaces of vector-valued continuous functions

Marian Nowak, Aleksandra Rzepka (2002)

Commentationes Mathematicae Universitatis Carolinae

Let X be a completely regular Hausdorff space and E a real normed space. We examine the general properties of locally solid topologies on the space C b ( X , E ) of all E -valued continuous and bounded functions from X into E . The mutual relationship between locally solid topologies on C b ( X , E ) and C b ( X ) ( = ...

Łojasiewicz ideals in Denjoy-Carleman classes

Vincent Thilliez (2013)

Studia Mathematica

The classical notion of Łojasiewicz ideals of smooth functions is studied in the context of non-quasianalytic Denjoy-Carleman classes. In the case of principal ideals, we obtain a characterization of Łojasiewicz ideals in terms of properties of a generator. This characterization involves a certain type of estimates that differ from the usual Łojasiewicz inequality. We then show that basic properties of Łojasiewicz ideals in the case have a Denjoy-Carleman counterpart.

Mapping properties of integral averaging operators

H. Heinig, G. Sinnamon (1998)

Studia Mathematica

Characterizations are obtained for those pairs of weight functions u and v for which the operators T f ( x ) = ʃ a ( x ) b ( x ) f ( t ) d t with a and b certain non-negative functions are bounded from L u p ( 0 , ) to L v q ( 0 , ) , 0 < p,q < ∞, p≥ 1. Sufficient conditions are given for T to be bounded on the cones of monotone functions. The results are applied to give a weighted inequality comparing differences and derivatives as well as a weight characterization for the Steklov operator.

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