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Manifold-valued generalized functions in full Colombeau spaces

Michael Kunzinger, Eduard Nigsch (2011)

Commentationes Mathematicae Universitatis Carolinae

We introduce the notion of generalized function taking values in a smooth manifold into the setting of full Colombeau algebras. After deriving a number of characterization results we also introduce a corresponding concept of generalized vector bundle homomorphisms and, based on this, provide a definition of tangent map for such generalized functions.

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.

Marcinkiewicz multipliers of higher variation and summability of operator-valued Fourier series

Earl Berkson (2014)

Studia Mathematica

Let f V r ( ) r ( ) , where, for 1 ≤ r < ∞, V r ( ) (resp., r ( ) ) denotes the class of functions (resp., bounded functions) g: → ℂ such that g has bounded r-variation (resp., uniformly bounded r-variations) on (resp., on the dyadic arcs of ). In the author’s recent article [New York J. Math. 17 (2011)] it was shown that if is a super-reflexive space, and E(·): ℝ → () is the spectral decomposition of a trigonometrically well-bounded operator U ∈ (), then over a suitable non-void open interval of r-values, the condition...

Marczewski-Burstin-like characterizations of σ-algebras, ideals, and measurable functions

Jack Brown, Hussain Elalaoui-Talibi (1999)

Colloquium Mathematicae

ℒ denotes the Lebesgue measurable subsets of ℝ and 0 denotes the sets of Lebesgue measure 0. In 1914 Burstin showed that a set M ⊆ ℝ belongs to ℒ if and only if every perfect P ∈ ℒ$ℒ0 h a s a p e r f e c t s u b s e t Q $ 0 which is a subset of or misses M (a similar statement omitting “is a subset of or” characterizes 0 ). In 1935, Marczewski used similar language to define the σ-algebra (s) which we now call the “Marczewski measurable sets” and the σ-ideal ( s 0 ) which we call the “Marczewski null sets”. M ∈ (s) if every perfect set P has...

Markov operators acting on Polish spaces

Tomasz Szarek (1997)

Annales Polonici Mathematici

We prove a new sufficient condition for the asymptotic stability of Markov operators acting on measures. This criterion is applied to iterated function systems.

Mass transport problem and derivation

Nacereddine Belili, Henri Heinich (1999)

Applicationes Mathematicae

A characterization of the transport property is given. New properties for strongly nonatomic probabilities are established. We study the relationship between the nondifferentiability of a real function f and the fact that the probability measure λ f * : = λ ( f * ) - 1 , where f*(x):=(x,f(x)) and λ is the Lebesgue measure, has the transport property.

Matchings and the variance of Lipschitz functions

Franck Barthe, Neil O'Connell (2009)

ESAIM: Probability and Statistics

We are interested in the rate function of the moderate deviation principle for the two-sample matching problem. This is related to the determination of 1-Lipschitz functions with maximal variance. We give an exact solution for random variables which have normal law, or are uniformly distributed on the Euclidean ball.

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