Mathematics

In this note, following \cite{Chitescuetal2014}, we show that the Monge-Kantorovich norm on the vector space of countably additive measures on a compact metric space has a primal representation analogous to the Hanin norm, meaning that similarly to the Hanin norm, the Monge-Kantorovich norm can be seen as an extension of the Kantorovich-Rubinstein norm from the vector subspace of zero-charge measures, implying a number of novel results, such as the equivalence of the Monge-Kantorovich and Hanin norms.

We introduce a real-valued measure m L on non-Archimedean ordered fields (F,<) that extend the field of real numbers (R,<) . The definition of m L is inspired by the Loeb measures of hyperreal fields in the framework of Robinson's analysis with infinitesimals. The real-valued measure m L turns out to be general enough to obtain a canonical measurable representative in F for every Lebesgue measurable subset of R , moreover, the measure of the two sets is equal. In addition, m L it is more expressive than a class of non-Archimedean uniform measures. We focus on the properties of the real-valued measure in the case where F=R , the Levi-Civita field. In particular, we compare m L with the uniform non-Archimedean measure over R developed by Shamseddine and Berz, and we prove that the first is infinitesimally close to the second, whenever the latter is defined. We also define a real-valued integral for functions on the Levi-Civita field, and we prove that every real continuous function has an integrable representative in R . Recall that this result is false for the current non-Archimedean integration over R . The paper concludes with a discussion on the representation of the Dirac distribution by pointwise functions on non-Archimedean domains.

By Baillon's result, it is known that maximal regularity with respect to the space of continuous functions is rare; it implies that either the involved semigroup generator is a bounded operator or the considered space contains c 0 . We show that the latter alternative can be excluded under a refined condition resembling maximal regularity with respect to L ∞ .

Non-amenability of B(E) has been surprisingly difficult to prove for the classical Banach spaces, but is now known for E= ℓ p and E= L p for all 1≤p<∞ . However, the arguments are rather indirect: the proof for L 1 goes via non-amenability of ℓ ∞ (K( ℓ 1 )) and a transference principle developed by Daws and Runde (Studia Math., 2010). In this note, we provide a short proof that B( L 1 ) and some of its subalgebras are non-amenable, which completely bypasses all of this machinery. Our approach is based on classical properties of the ideal of representable operators on L 1 , and shows that B( L 1 ) is not even approximately amenable.

The main result is a submetric characterization of the class of Banach spaces admitting an equivalent norm with Rolewicz's property ( β ). As applications we prove that up to renorming, property ( β ) is stable under coarse Lipschitz embeddings and coarse quotients.

We survey some classical norm inequalities of Hardy, Kallman, Kato, Kolmogorov, Landau, Littlewood, and Rota of the type ?�Af ??2 X ?�C?�f ??X ????A 2 f ????X ,f?�dom( A 2 ), and recall that under exceedingly stronger hypotheses on the operator A and/or the Banach space X , the optimal constant C in these inequalities diminishes from 4 (e.g., when A is the generator of a C 0 contraction semigroup on a Banach space X ) all the way down to 1 (e.g., when A is a symmetric operator on a Hilbert space H ). We also survey some results in connection with an extension of the Hardy-Littlewood inequality involving quadratic forms as initiated by Everitt.

We prove the stability of isomorphisms between Banach spaces generated by interpolation methods introduced by Cwikel-Kalton-Milman-Rochberg which includes, as special cases, the real and complex methods up to equivalence of norms and also the so-called ± or G 1 and G 2 methods defined by Peetre and Gustavsson-Peetre. This result is used to show the existence of solution of certain operator analytic equation. A by product of these results is a more general variant of the Albrecht-Müller result which states that the interpolated isomorphisms satisfy uniqueness-of-inverses between interpolation spaces. We show applications for positive operators between Calderón function lattices. We also derive connections between the spectrum of interpolated operators.

Let K be a centrally symmetric spherical polytope, whose vertices form a 1 4n − net in the unit sphere in R n . We prove a uniform lower bound on the norms of hyperplane projections P:X→X , where X is the n -dimensional normed space with the unit ball K . The estimate is given in terms of the determinant function of vertices and faces of K . In particular, if N≥ n 4n and K is the convex hull of {± x 1 ,± x 2 ,…,± x N } , where x 1 , x 2 ,…, x N are independent random points distributed uniformly in the unit sphere, then every hyperplane projection P:X→X satisfies the inequality ||P||≥1+ c n N −8n−6 (for some explicit constant c n ), with the probability at least 1− 4 N .

Based on hierarchical partitions, we provide the construction of Haar-type tight framelets on any compact set K⊆ R d . In particular, on the unit block [0,1 ] d , such tight framelets can be built to be with adaptivity and directionality. We show that the adaptive directional Haar tight framelet systems can be used for digraph signal representations. Some examples are provided to illustrate results in this paper.

Admissible vectors lead to frames or coherent states under the action of a group by means of square integrable representations. This work shows that admissible vectors can be seen as weights with central support on the (left) group von Neumann algebra. The analysis involves spatial and cocycle derivatives, noncommutative L p -Fourier transforms and Radon-Nikodym theorems. Square integrability confine the weights in the predual of the algebra and everything may be written in terms of a (right selfdual) bounded element.