In this paper we are going to discuss compactness in Lorentz sequence spaces. Firstly, it will be shown how to define such a space, check whether a sequence belongs to it and calculate its norm. Equipped with this knowledge, we will proceed to propose usable compactness criteria for Lorentz sequence spaces, employing the concept of seminorms.
In this paper, we apply the Hausdorff measure of noncompactness to obtain the necessary and sufficient conditions for certain matrix operators on the Fibonacci difference sequence spaces l_{p}(F) and l_{infinite}(F) to be compact, where 1<=p<infinite.
We give a new, simpler proof of a compactness result in $GSBD^p$, $p>1$, by the same authors, which is also valid in $GBD$ (the case $p=1$), and shows that bounded sequences converge a.e., after removal of a suitable sequence of piecewise infinitesimal rigid motions, subject to a fixed partition.
In this paper, we study stability of $M$-compactness for $l^p$ sum of Banach spaces for $1\leq p<\infty$. We also obtain a characterization of $M$-compact sets in terms of statistically maximizing sequence, a notion which is weaker than a maximizing sequence. Moreover, we introduce the notion of $\mathcal{I}$-$M$-compactness of a bounded subset $M$ of a normed linear space $X$ with respect to an ideal $\mathcal{I}$ and show that it is equivalent to $M$-compactness for non-trivial admissible ideals.
This article is dedicated to geometric structure of the Lorentz and Marcinkiewicz spaces in case of the pure atomic measure. We study complete criteria for order continuity, the Fatou property, strict monotonicity and strict convexity in the sequence Lorentz spaces $γ_{p,w}$. Next, we present a full characterization of extreme points of the unit ball in the sequence Lorentz space $γ_{1,w}$. We also establish a complete description with an isometry of the dual and predual spaces of the sequence Lorentz spaces $γ_{1,w}$ written in terms of the Marcinkiewicz spaces. Finally, we show a fundamental…
The aim of the paper is to introduce the spaces $\ell_{\infty}^λ(\widehat{F})$ and $\ell_{p}^λ(\widehat{F})$ derived by the composition of the two infinite matrices $Λ=(λ_{nk})$ and $\widehat{F}=\left( f_{nk} \right),$ which are the $BK$-spaces of non-absolute type and also derive some inclusion relations. Further, we determine the $α$-, $β$-, $γ$-duals of those spaces and also construct the basis for $\ell_{p}^λ(\widehat{F}).$ Additionally, we characterize some matrix classes on the spaces $\ell_{\infty}^λ(\widehat{F})$ and $\ell_{p}^λ(\widehat{F}).$ We also investigate some geometric propert…
There are two main aims of the paper. The first one is to extend the criterion for the precompactness of sets in Banach function spaces to the setting of quasi-Banach function spaces. The second one is to extend the criterion for the precompactness of sets in the Lebesgue spaces $L_p(\mathbb R^n)$, $1 \leq p < \infty$, to the so-called power quasi-Banach function spaces. These criteria are applied to establish compact embeddings of abstract Besov spaces into quasi-Banach function spaces. The results are illustrated on embeddings of Besov spaces $B^s_{p,q}(\mathbb R^n)$, $0<s<1$, $0<p,q\leq \in…
In this paper, we prove a compactness and semicontinuity result in $GSBD$ for sequences with bounded Griffith energy. This generalises classical results in $(G)SBV$ by Ambrosio and $SBD$ by Bellettini-Coscia-Dal Maso. As a result, the static problem in Francfort-Marigo's variational approach to crack growth admits (weak) solutions. Moreover, we obtain a compactness property for minimisers of suitable Ambrosio-Tortorelli's type energies, for which we have recently shown the $Γ$-convergence to Griffith energy.
Let $X$ be a Banach space and let $C$ be a closed convex bounded subset of $X$. It is proved that $C$ is weakly compact if, and only if, $C$ has the {it generic} fixed point property ($\mathcal{G}$-FPP) for the class of $L$-bi-Lipschitz affine mappings for every $L>1$. It is also proved that if $X$ has Pełczyński's property $(u)$, then either $C$ is weakly compact, contains an $\ell_1$-sequence or a $\mathrm{c}_0$-summing basic sequence. In this case, weak compactness of $C$ is equivalent to the $\mathcal{G}$-FPP for the strengthened class of affine mappings that are uniformly bi-Lipschitz. We…
Morrey (function) spaces and, in particular, smoothness spaces of Besov-Morrey or Triebel-Lizorkin-Morrey type enjoyed a lot of interest recently. Here we turn our attention to Morrey sequence spaces $m_{u,p}=m_{u,p}(\mathbb{Z}^d)$, $0<p\leq u<\infty$, which have yet been considered almost nowhere. They are defined as natural generalizations of the classical $\ell_p$ spaces. We consider some basic features, embedding properties, the pre-dual, a corresponding version of Pitt's compactness theorem, and can further characterize the compactness of embeddings of related finite-dimensional spaces.
We characterize the sequences $(w_i)_{i=1}^\infty$ of non-negative numbers for which \[ \sum_{i=1}^\infty a_i w_i \quad \text{ is of the same order as } \quad \sup_n \sum_{i=1}^n a_i w_{1+n-i} \] when $(a_i)_{i=1}^\infty$ runs over all non-increasing sequences of non-negative numbers. As a by-product of our work we settle a problem raised in [F. Albiac, Jose L. Ansorena and B. Wallis; arXiv:1703.07772[math.FA]] and prove that Garling sequences spaces have no symmetric basis.
The generalized Cesàro operators $\mathcal{C}_t$, for $t\in[0,1)$, introduced in the 1980's by Rhaly, are natural analogues of the classical Cesàro averaging operator $\mathcal{C}_1$ and act in various Banach sequence spaces $X\subseteq {\mathbb C}^{{\mathbb N}_0}$. In this paper we concentrate on a certain class of Banach lattices for the coordinate-wise order, which includes all separable, rearrangement invariant sequence spaces, various weighted $c_0$ and $\ell^p$ spaces and many others. In such Banach lattices $X$ the operators $\mathcal{C}_t$, for $t\in[0,1)$, are always compact (unlike $…
We show that every Lorentz sequence space $d(\textbf{w},p)$ admits a 1-complemented subspace $Y$ distinct from $\ell_p$ and containing no isomorph of $d(\textbf{w},p)$. In the general case, this is only the second nontrivial complemented subspace in $d(\textbf{w},p)$ yet known. We also give an explicit representation of $Y$ in the special case $\textbf{w}=(n^{-θ})_{n=1}^\infty$ ($0<θ<1$) as the $\ell_p$-sum of finite-dimensional copies of $d(\textbf{w},p)$. As an application, we find a sixth distinct element in the lattice of closed ideals of $\mathcal{L}(d(\textbf{w},p))$, of which only five …
For $0\leqslant ξ\leqslant ω_1$, we define the notion of $ξ$-weakly precompact and $ξ$-weakly compact sets in Banach spaces and prove that a set is $ξ$-weakly precompact if and only if its weak closure is $ξ$-weakly compact. We prove a quantified version of Grothendieck's compactness principle and the characterization of Schur spaces obtained by Dowling et al. For $0\leqslant ξ\leqslant ω_1$, we prove that a Banach space $X$ has the $ξ$-Schur property if and only if every $ξ$-weakly compact set is contained in the closed, convex hull of a weakly null (equivalently, norm null) sequence.
We prove a compactness result for bounded sequences $(u_j)_j$ of functions with bounded variation in metric spaces $(X,d_j)$ where the space $X$ is fixed but the metric may vary with $j$. We also provide an application to Carnot-Carathéodory spaces.
Defect of compactness, relative to an embedding of two Banach spaces E and F, is a difference between a weakly convergent sequence in E and its weak limit taken up to a remainder that vanishes in the norm of F. For Sobolev embeddings in particular, defect of compactness is expressed as a profile decomposition - a sum of terms, called elementary concentrations, with asymptotically disjoint supports. We discuss a profile decomposition for the Sobolev space of a Riemannian manifold with bounded geometry, which is a sum of elementary concentrations associated with concentration profiles defined on…
Defect of compactness, relative to an embedding of two Banach spaces E and F, is a difference between a weakly convergent sequence in E and its weak limit taken up to a remainder that vanishes in the norm of F. For Sobolev embeddings in particular, defect of compactness is expressed as a profile decomposition - a sum of terms, called elementary concentrations, with asymptotically disjoint supports. We discuss a profile decomposition for the Sobolev space of a Riemannian manifold with bounded geometry, which is a sum of elementary concentrations associated with concentration profiles defined on…
We prove that the Lorentz sequence space $\ell_{\frac{2m}{m+1},1}$ is, in a~precise sense, optimal among all symmetric Banach sequence spaces satisfying a Bohnenblust-Hille type inequality for $m$-linear forms or $m$-homogeneous polynomials on $\mathbb{C}^n$. Motivated by this result we develop methods for dealing with subtle Bohnenblust-Hille type inequalities in the setting of Lorentz spaces. Based on an interpolation approach and the Blei-Fournier inequalities involving mixed type spaces, we prove multilinear and polynomial Bohnenblust-Hille type inequalities in Lorentz spaces with subpolyn…
Concentration-compactness is used to prove compactness of maximising sequences for a variational problem governing symmetric steady vortex-pairs in a uniform planar ideal fluid flow, where the kinetic energy is to be maximised and the constraint set comprises the set of all equimeasurable rearrangements of a given function (representing vorticity) that have prescribed impulse (lnear momentum). A form of orbital stability is deduced.
We investigate conditions under which, for two sequences $(u_r)$ and $(v_r)$ weakly converging to $u$ and $v$ in $L^p(R^d;R^N)$ and $L^{q}(R^d;R^N)$, respectively, $1/p+1/q \leq 1$, a quadratic form $q(x;u_r,v_r)=\sum\limits_{j,m=1}^N q_{j m}(x)u_{j r} v_{m r}$ converges toward $q(x;u,v)$ in the sense of distributions. The conditions involve fractional derivatives and variable coefficients, and they represent a generalization of the known compensated compactness theory. The proofs are accomplished using a recently introduced $H$-distribution concept. We apply the developed techniques to a nonl…
Let $E$ be a Banach space such that $E'$ has the Radon-Nikodým property. The aim of this work is to connect relative weak compactness in the $E$-valued martingale Hardy space $H^{1}(μ,E)$ to a convex compactness criterion in a weaker topology, such as the topology of uniform convergence on compacts in measure. These results represent a dynamic version of the deep result of Diestel, Ruess, and Schachermayer on relative weak compactness in $L^{1}(μ,E)$. In the reflexive case, we obtain a Kadec-Pełczyński dichotomy for $H^{1}(μ,E)$-bounded sequences, which decomposes a subsequence into a relative…
We prove that, if dichotomy occurs when the concentration-compactness principle is used, the dichotomizing sequence can be choosen so that a nontrivial part of it concentrates. Iterating this argument leads to a profile decomposition for arbitrary sequences of bounded Borel measures. To illustrate our results we give an application to the structure of bouded sequences in the Sobolev space $ W^{1, p}(\R^N)$.
