Youssef Jabri
We establish an abstract critical point theorem for locally Lipschitz functionals that does not require any compactness condition of Palais-Smale type. It generalizes and unifies three other critical point theorems established in [Jabri-Moussaoui] for $C^{1}$-functionals under slightly stronger assumptions. Our approach uses continuous selections of multivalued mappings.
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The Aubin-Lions-Dubinskii theorems on compactness in Bochner spaces
本文综述了Bochner空间中紧性的Aubin-Lions-Dubinskii定理，从经典Aubin-Lions不等式及Dubinskii推广出发，并给出Chen、Jungel和Liu紧性结果的简明自洽证明。
The purpose of this paper is to provide a proof of James' weak compactness theorem that is able to be taught in a first year graduate class in functional analysis.
We show that the Arzela-Ascoli theorem and Kolmogorov compactness theorem both are consequences of a simple lemma on compactness in metric spaces. Their relation to Helly's theorem is discussed. The paper contains a detailed discussion on the historical background of the Kolmogorov compactness theorem.
We study totally bounded subsets in weighted variable exponent amalgam and Sobolev spaces. Moreover, this paper includes several detailed generalized results of some compactness criterions in these spaces.
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.
The purpose of this short note is to provide a new and very short proof of a result by Sudakov, offering an important improvement of the classical result by Kolmogorov-Riesz on compact subsets of Lebesgue spaces.
A version of Arzelà-Ascoli theorem for $X$ being $σ$-locally compact Hausdorff space is proved. The result is used in proving compactness of Fredholm, Hammerstein and Urysohn operators. Two fixed point theorems, for Hammerstein and Urysohn operator, are derived on the basis of Schauder theorem.
Every absolutely summing linear operator is weakly compact. However, for strongly summing multilinear operators and polynomials - one of the most natural extensions of the linear case to the non linear framework - weak compactness does not hold in general. We show that a subclass of the class of strongly summing multilinear operators/polynomials, sharing its main properties such as Grothendieck's Theorem, Pietsch Domination Theorem and Dvoretzky-Rogers Theorem, has even better properties like weak compactness and a natural factorization theorem.
Our main result is the following: {\it Let $E$ be a Banach space and $D$ be a weakly compact subset of $E$ with $0\notin D$. If $A$ is a bounded subset of $E$ such that every $x^*\in E^*$ with $x^*(D) >0$ attains its supremum on $A$, then $A$ is weakly relatively compact.}
On the $T1$ theorem for compactness of Calderón-Zygmund operators
本文重新表述 Calderón-Zygmund 奇异积分算子的紧性 $T1$ 定理，证明其在 $L^2(\mathbb{R}^n)$ 上紧等价于 $T1,T^*1$ 属于 CMO 且算子弱紧。
The purpose of this paper is to generalize the measure of non-compactness for the space of continuous functions over the $T_{3 \frac{1}{2}}$ space. Motivated by the generalized Arzelà-Ascoli theorem for Tichonoff space $T$ via Wallman compactifiaction $\operatorname{Wall}(T)$, we constuct a measure of non-compactness for the space $C^b(T)$. We also study some of the properties of this object and give another version of Darbo-type theorem, suitable for this particular case.
A compensated compactness theorem for pseudodifferential operators on vector bundles
本文提出在半黎曼流形上，基于伪微分算子的补偿紧致性定理，扩展了Murat-Tartar理论的应用范围。
Let X and Y be Banach spaces and F a subset of B_{Y^*}. Endow Y with the topology τ_F of pointwise convergence on F. Let T: X^* \to Y be a bounded linear operator which is (w^*, τ_F) continuous. Assume that every vector in the range of T attains its norm at an element of F. Then it is proved that T is (w^*,w) continuous.
Helly's selection theorem provides a criterion for compactness of sets of single-variable functions with bounded pointwise variation. Fra{ň}kov{á} has given a proper extension of Helly's theorem to the setting of single-variable regulated functions. We show how a similar approach yields extensions of the standard compactness criterion for multi-variable functions of bounded variation.
Spaces of distributions with Sobolev wave front in a fixed conic set: compactness, pullback by smooth maps and the compensated compactness theorem
本文为具有固定锥形集中Sobolev波前集的分布空间引入局部凸拓扑，推广拉回、紧性、伪微分算子连续性与Rellich引理，并应用于微局域缺陷测度与补偿紧性定理。
We introduce two measures of weak non-compactness $Ja_E$ and $Ja$ that quantify, via distances, the idea of boundary behind James' compactness theorem. These measures tell us, for a bounded subset $C$ of a Banach space $E$ and for given $x^*\in E^*$, how far from $E$ or $C$ one needs to go to find $x^{**}\in \bar{C}^{w^*}\subset E^{**}$ with $x^{**}(x^*)=\sup x^* (C)$. A quantitative version of James' compactness theorem is proved using $Ja_E$ and $Ja$, and in particular it yields the following result: {\it Let $C$ be a closed convex bounded subset of a Banach space $E$ and $r>0$. If there is …
The Gromoll-Meyer's generalized Morse lemma (so called splitting lemma) near degenerate critical points on Hilbert spaces, which is one of key results in infinite dimensional Morse theory, is usually stated for at least $C^2$-smooth functionals. It obstructs one using Morse theory to study most of variational problems of form $F(u)=\int_Ωf(x, u,..., D^mu)dx$ as in (\ref{e:1.1}). In this paper we establish a splitting theorem and a shifting theorem for a class of continuously directional differentiable functionals (lower than $C^1$) on a Hilbert space $H$ which have higher smoothness (but lower…
A wedge product theorem of compensated compactness theory with critical exponents on Riemannian manifolds
本文研究了黎曼流形上弱收敛微分形式楔积的极限行为，提出并证明了补偿紧致性定理，扩展了Briane等人的div-curl引理，探讨了Gauss-Codazzi-Ricci方程和等距浸入的L^p外几何弱连续性。
We present some extensions of classical results that involve elements of the dual of Banach spaces, such as Bishop-Phelp's theorem and James' compactness theorem, but restricting to sets of functionals determined by geometrical properties. The main result, which answers a question posed by F. Delbaen, is the following: Let $E$ be a Banach space such that $(B_{E^\ast}, ω^\ast)$ is convex block compact. Let $A$ and $B$ be bounded, closed and convex sets with distance $d(A,B) > 0$. If every $x^\ast \in E^\ast$ with \[ \sup(x^\ast,B) < \inf(x^\ast,A) \] attains its infimum on $A$ and its supremum …
We prove a compactness theorem for the dual Gromov-Hausdorff propinquity as a noncommutative analogue of the Gromov compactness theorem for the Gromov-Hausdorff distance. Our theorem is valid for subclasses of quasi-Leibniz compact quantum metric spaces of the closure of finite dimensional quasi-Leibniz compact quantum metric spaces for the dual propinquity. While finding characterizations of this class proves delicate, we show that all nuclear, quasi-diagonal quasi-Leibniz compact quantum metric spaces are limits of finite dimensional quasi-Leibniz compact quantum metric spaces. This result i…
Weak Compactness Criterion in $ W^{k, 1} $ with an Existence Theorem of Minimizers
针对非自反Sobolev空间变分问题极小化子存在定理未充分探索的问题，本文证明了$W^{k,1}$中的弱紧性准则（推广Dunford-Pettis定理），并将极小化子存在定理从自反空间推广至非自反空间，范畴论概念助力研究。
Riesz-Kolmogorov type compactness criteria in function spaces with applications
本文提出适用于多种函数空间的Riesz-Kolmogorov型紧性准则，并用于刻画Bergman空间上的紧Toeplitz算子及推导Hardy空间上Hankel算子的紧性。
Locally $L^0$-convex modules were introduced in [D. Filipovic, M. Kupper, N. Vogelpoth. Separation and duality in locally $L^0$-convex modules. J. Funct. Anal. 256(12), 3996-4029 (2009)] as the analytic basis for the study of conditional risk measures. Later, the algebra of conditional sets was introduced in [S. Drapeau, A. Jamneshan, M. Karliczek, M. Kupper. The algebra of conditional sets and the concepts of conditional topology and compactness. J. Math. Anal. Appl. 437(1), 561-589 (2016)]. In this paper we study locally $L^0$-convex modules, and find exactly which subclass of locally $L^0$-…
We describe a topological predual to differential forms constructed as an inductive limit of a sequence of Banach spaces. This subspace of currents has nice properties, in that Dirac chains and polyhedral chains are dense, and its operator algebra contains operators predual to exterior derivative, Hodge star, Lie derivative, and interior product. Using these operators, we establish higher order divergence theorems for net flux of k-vector fields across nonsmooth boundaries, Stokes' theorem for domains in open sets which are not necessarily regular, and a new fundamental theorem for nonsmooth d…
Boundedness and compactness kernel theorem for $α$-modulation spaces
本文建立了α-调制空间的核定理，用分布核在混合α-调制空间中的隶属关系刻画线性算子的有界性，并用闭子空间中的核刻画紧性。
The concept of convex compactness, weaker than the classical notion of compactness, is introduced and discussed. It is shown that a large class of convex subsets of topological vector spaces shares this property and that is can be used in lieu of compactness in a variety of cases. Specifically, we establish convex compactness for certain familiar classes of subsets of the set of positive random variables under the topology induced by convergence in probability. Two applications in infinite-dimensional optimization - attainment of infima and a version of the Minimax theorem - are given. Moreove…
The usual Gromoll-Meyer's generalized Morse lemma near degenerate critical points on Hilbert spaces, so called splitting lemma, is stated for at least $C^2$-smooth functionals. In this paper we establish a splitting theorem and a shifting theorem for a class of continuously directional differentiable functionals (lower than $C^1$) on a Hilbert space $H$ which have higher smoothness (but lower than $C^2$) on a densely and continuously imbedded Banach space $X\subset H$ near a critical point lying in $X$. (This splitting theorem generalize almost all previous ones to my knowledge). Moreover, a n…
Equality in Degrees of Compactness: Schauder's Theorem and s-numbers
本文推广 Schauder 定理，研究算子及其伴随算子的 s-数与紧度关系，证明紧算子逼近数相等，并给出非紧算子在自然空间条件及逼近格式下的对应结论。
Quantum Trajectories. Spectral Gap, Quasi-compactness & Limit Theorems
本文针对经受重复独立测量的量子系统演化建模的量子轨迹，利用拟紧性证明其马尔可夫算子存在谱隙并描述外围谱，通过算子扰动推导相关极限定理，为量子轨迹的统计性质分析提供理论支撑。
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