Generalization of Poincar ´e inequality in a Sobolev Space with exponent constant to the case of Sobolev space with a variable exponent

In this work, we study the Poincare inequality in Sobolev spaces with variable exponent. As a consequence of this ´ result we show the equivalent norms over such cones. The approach we adopt in this work avoids the difficulty arising from the possible lack of density of the space C∞ 0 (Ω).

Consequences, norms, and inaction: A critical analysis

Gawronski, Armstrong, Conway, Friesdorf and Hütter (2017, GACFH) presented a model of choices in utilitarian moral dilemmas, those in which following a moral principle or norm (the deontological response) leads to worse consequences than violating the principle (the utilitarian response). In standard utilitarian dilemmas, the utilitarian option involves action (which causes some harm in order to prevent greater harm), and the deontological response, omission. GACFH propose that responses in such dilemmas arise in three different ways: a psychological process leading to a deontological choice, a different process leading to a utilitarian choice, or a bias toward inaction or action. GACFH attempt to separate these three processes with new dilemmas in which action and omission are switched, and dilemmas in which the utilitarian and deontological processes lead to the same choice. They conclude that utilitarian and deontological responses are indeed separable, and that past research has missed this fact by treating them as naturally opposed. We argue that a bias toward harmful inaction is best understood as an explanation of deontological responding rather than as an alternative process. It thus should be included as an explanation of deontological responding, not an alternative response type. We also argue that GACFH's results can be largely explained in terms of subjects' unwillingness to accept the researchers' assumptions about which consequence is worse and which course of action is consistent with a moral norm. This problem is almost inherent in the attempt to switch act and omission while maintaining equivalent norms. We support this argument with data from experiments with new and old scenarios, in which we asked subjects to judge both norms and consequences. We also find that GACFH's results are not as consistent as they appear to be in the paper.pso

Fourier multipliers on anisotropic mixed-norm spaces of distributions

A new general Hörmander type condition involving anisotropies and mixed norms is introduced, and boundedness results for Fourier multipliers on anisotropic Besov and Triebel-Lizorkin spaces of distributions with mixed Lebesgue norms are obtained. As an application, the continuity of such operators is established on mixed Sobolev and Lebesgue spaces too. Some lifting properties and equivalent norms are obtained as well.

EQUIVALENT NORMS WITH AN EXTREMELY NONLINEABLE SET OF NORM ATTAINING FUNCTIONALS

We present a construction that enables one to find Banach spaces$X$whose sets$\operatorname{NA}(X)$of norm attaining functionals do not contain two-dimensional subspaces and such that, consequently,$X$does not contain proximinal subspaces of finite codimension greater than one, extending the results recently provided by Read [Banach spaces with no proximinal subspaces of codimension 2,Israel J. Math.(to appear)] and Rmoutil [Norm-attaining functionals need not contain 2-dimensional subspaces,J. Funct. Anal. 272(2017), 918–928]. Roughly speaking, we construct an equivalent renorming with the requested properties for every Banach space$X$where the set$\operatorname{NA}(X)$for the original norm is not “too large”. The construction can be applied to every Banach space containing$c_{0}$and having a countable system of norming functionals, in particular, to separable Banach spaces containing$c_{0}$. We also provide some geometric properties of the norms we have constructed.