Search results for "Pointwise"
showing 7 items of 47 documents
σ-Continuous and Co-σ-continuous Maps
2009
In this chapter we isolate the topological setting that is suitable for our study. We first present 2.1–2.3 to follow an understandable logical scheme nevertheless the main contribution are presented in 2.4–2.7 and our main tool will be Theorem 2.32. An important concept will be the σ-continuity of a map Φ from a topological space (X, T) into a metric space (Y, g). The σ-continuity property is an extension of continuity suitable to deal with countable decompositions of the domain space X as well as with pointwise cluster points of sequences of functions Φn : X → Y, n = 1,2,… When (X,T) is a subset of a locally convex linear topological space we shall refine our study to deal with σ-slicely …
The Theory of Normed Modules
2020
This chapter is devoted to the study of the so-called normed modules over metric measure spaces. These represent a tool that has been introduced by Gigli in order to build up a differential structure on nonsmooth spaces. In a few words, an \(L^2({{\mathfrak {m}}})\)-normed \(L^\infty ({{\mathfrak {m}}})\)-module is a generalisation of the concept of ‘space of 2-integrable sections of some measurable bundle’; it is an algebraic module over the commutative ring \(L^\infty ({{\mathfrak {m}}})\) that is additionally endowed with a pointwise norm operator. This notion, its basic properties and some of its technical variants constitute the topics of Sect. 3.1.
B1(1)Π state of KCs: high-resolution spectroscopy and description of low-lying energy levels.
2012
The diode laser induced B(1)(1)Π → X(1)Σ(+) fluorescence spectra of the KCs molecule were recorded by Fourier-transform spectrometer with resolution of 0.03 cm(-1). Buffer gas Ar was used to facilitate appearance of rotation relaxation lines in the spectra, thus enlarging the B(1)(1)Π dataset, allowing one to determine the Λ-splitting constants and to reveal numerous local perturbations. A dataset was systematically obtained for B(1)(1)Π vibrational levels ν(') ∈ [0; 5] nonuniformly covering rotational levels with J(') ∈ [7; 233]. For ν(') ∈ [0; 3], the less-perturbed data of f-components were included in the fit. A pointwise potential energy curve (PEC) based on the inverted perturbation a…
Self-improvement of pointwise Hardy inequality
2019
We prove the self-improvement of a pointwise p p -Hardy inequality. The proof relies on maximal function techniques and a characterization of the inequality by curves.
Self-improvement of weighted pointwise inequalities on open sets
2020
We prove a general self-improvement property for a family of weighted pointwise inequalities on open sets, including pointwise Hardy inequalities with distance weights. For this purpose we introduce and study the classes of $p$-Poincar\'e and $p$-Hardy weights for an open set $\Omega\subset X$, where $X$ is a metric measure space. We also apply the self-improvement of weighted pointwise Hardy inequalities in connection with usual integral versions of Hardy inequalities.
The annular decay property and capacity estimates for thin annuli
2016
We obtain upper and lower bounds for the nonlinear variational capacity of thin annuli in weighted $\mathbf{R}^n$ and in metric spaces, primarily under the assumptions of an annular decay property and a Poincar\'e inequality. In particular, if the measure has the $1$-annular decay property at $x_0$ and the metric space supports a pointwise $1$-Poincar\'e inequality at $x_0$, then the upper and lower bounds are comparable and we get a two-sided estimate for thin annuli centred at $x_0$, which generalizes the known estimate for the usual variational capacity in unweighted $\mathbf{R}^n$. Most of our estimates are sharp, which we show by supplying several key counterexamples. We also character…
Conditional convex orders and measurable martingale couplings
2014
Strassen's classical martingale coupling theorem states that two real-valued random variables are ordered in the convex (resp.\ increasing convex) stochastic order if and only if they admit a martingale (resp.\ submartingale) coupling. By analyzing topological properties of spaces of probability measures equipped with a Wasserstein metric and applying a measurable selection theorem, we prove a conditional version of this result for real-valued random variables conditioned on a random element taking values in a general measurable space. We also provide an analogue of the conditional martingale coupling theorem in the language of probability kernels and illustrate how this result can be appli…