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Fegen und Dünnheit mit Anwendungen auf die Laplace-und Wärmeleitungsgleichung

Wolfhard Hansen (1971)

Annales de l'institut Fourier

Several properties of balayage of measures in harmonic spaces are studied. In particular, characterisations of thinness of subsets are given. For the heat equation the following result is obtained: suppose that E = R m + 1 is given the presheaf of solutions of i = 1 m u x i = u x m + 1 and B is a subset of R m × [ - , 0 ] satisfying { ( α x , α 2 t ) : ( x , t ) B , x R m , t R } B for α > 0 arbitrarily small. Then B is thin at 0 if and only if B is polar. Similar result for the Laplace equation. At last the reduced of measures is defined and several approximation theorems on reducing and balayage...

Filters and sequences

Sławomir Solecki (2000)

Fundamenta Mathematicae

We consider two situations which relate properties of filters with properties of the limit operators with respect to these filters. In the first one, we show that the space of sequences having limits with respect to a Π 3 0 filter is itself Π 3 0 and therefore, by a result of Dobrowolski and Marciszewski, such spaces are topologically indistinguishable. This answers a question of Dobrowolski and Marciszewski. In the second one, we characterize universally measurable filters which fulfill Fatou’s lemma.

Fine behavior of functions whose gradients are in an Orlicz space

Jan Malý, David Swanson, William P. Ziemer (2009)

Studia Mathematica

For functions whose derivatives belong to an Orlicz space, we develop their "fine" properties as a generalization of the treatment found in [MZ] for Sobolev functions. Of particular importance is Theorem 8.8, which is used in the development in [MSZ] of the coarea formula for such functions.

Finite codimensional linear isometries on spaces of differentiable and Lipschitz functions

Hironao Koshimizu (2011)

Open Mathematics

We characterize finite codimensional linear isometries on two spaces, C (n)[0; 1] and Lip [0; 1], where C (n)[0; 1] is the Banach space of n-times continuously differentiable functions on [0; 1] and Lip [0; 1] is the Banach space of Lipschitz continuous functions on [0; 1]. We will see they are exactly surjective isometries. Also, we show that C (n)[0; 1] and Lip [0; 1] admit neither isometric shifts nor backward shifts.

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