The Equivalence of Operator Norm between the Hardy-Littlewood Maximal Function and Truncated Maximal Function on the Heisenberg Group

In this article, we define a kind of truncated maximal function on the Heisenberg space by M γ c f x = sup 0 < r < γ 1 / m B x , r ∫ B x , r f y d y . The equivalence of operator norm between the Hardy-Littlewood maximal function and the truncated maximal function on the Heisenberg group is obtained. More specifically, when 1 < p < ∞ , the L p norm and central Morrey norm of truncated maximal function are equal to those of the Hardy-Littlewood maximal function. When p = 1 , we get the equivalence of weak norm L 1 ⟶ L 1 , ∞ and M ̇ 1 , λ ⟶ W ̇ M 1 , λ . Those results are generalization of previous work on Euclid spaces.

Conceptualizing and assessing norm strength in International Relations

What constitutes a strong or a weak norm? Scholars often refer to strong or weak, or strengthening or weakening norms, yet there are widespread inconsistencies in terminology and no agreed-upon measures. This has hindered the accumulation of knowledge and made it difficult to test competing hypotheses about norm development and contestation. To address these conceptual problems and their analytical implications, this article conceptualizes norm strength as the extent of collective expectations related to a principled idea and proposes two indicators to assess a norm’s strength: the level of international concordance with a principled idea, and the degree of international institutionalization of a principled idea. The article illustrates the applicability and utility of the proposed conceptualization by evaluating the strengths of two transitional justice norms: the norm of legal accountability and the norm of truth-seeking. In so doing, the article resolves empirical disputes over the origins and status of these norms. In particular, the analysis reveals that while legal accountability became a norm in the early 1990s and is today a strong norm, truth-seeking emerged later and remains a weak norm. More generally, the proposed framework should advance existing debates about norm contestation, localization, violation, and erosion.

Points of Weak*-Norm Continuity in the Unit Ball of the Space WC(K; X)*

For a compact Hausdorff space with a dense set of isolated points, we give a complete description of points of weak*-norm continuity in the dual unit ball of the space of Banach space valued functions that are continuous when the range has the weak topology. As an application we give a complete description of points of weak-norm continuity of the unit ball of the space of vector measures when the underlying Banach space has the Radon-Nikodym property.