We give a necessary and sufficient condition for non-local functionals on vector-valued Lebesgue spaces to be weakly sequentially lower semi-continuous. Here a non-local functional shall have the form of a double integral of a density which depends on the function values at two different points.
The characterisation we get is essentially that the density has to be convex in one variable if we integrate over the other one with an arbitrary test function in it.
Moreover, we show that this condition is in the case of non-local functionals on real-valued Lebesgue spaces (up to some equivalence in the density) equivalent to the separate convexity of the density.
We obtain formulae to calculate the asymptotic center and radius of bounded sequences in ${\cal C}_0(L)$ spaces. We also study the existence of continuous selectors for the asymptotic center map in general Banach spaces. In Hilbert spaces, even a Hölder-type estimation is given.
In this paper we extend our findings in [3] and answer further questions regarding continuity and discontinuity of seminorms on infinite-dimensional vector spaces.
Let $S$ be a seminorm on an infinite-dimensional real or complex vector space $X$. Our purpose in this note is to study the continuity and discontinuity properties of $S$ with respect to certain norm-topologies on $X$.
The main purpose of this paper is to treat semigroups properties, like norm continuity, compactness and differentiability for perturbed semigroups in Banach spaces. In particular, we investigate three large classes of perturbations, Miyadera-Voigt, Desch-Schappacher and Staffans-Weiss perturbations. Our approach is mainly based on feedback theory of Salamon-Weiss systems. Our results are applied to abstract boundary integro-differential equations in Banach spaces.
We show weak* in measures on $\barØ$/ weak-$L^1$ sequential continuity of $u\mapsto f(x,\nabla u):W^{1,p}(Ø;\R^m)\to L^1(Ø)$, where $f(x,\cdot)$ is a null Lagrangian for $x\inØ$, it is a null Lagrangian at the boundary for $x\in\partialØ$ and $|f(x,A)|\le C(1+|A|^p)$. We also give a precise characterization of null Lagrangians at the boundary in arbitrary dimensions. Our results explain, for instance, why $u\mapsto \det\nabla u:W^{1,n}(Ø;\R^n)\to L^1(Ø)$ fails to be weakly continuous. Further, we state a new weak lower semicontinuity theorem for integrands depending on null Lagrangians at the …
This paper introduces the proper notion of variational quasiconvexity associated to a group of diffeomorphisms. We prove a lower semicontinuity theorem connected to this notion. In the second part of the paper we apply this result to a class of functions introduced in [5]. Such functions are $GL(n,R)^{+}$ quasiconvex, hence they induce lower semicontinuous integrals.
We prove the strong continuity of spectral multiplier operators associated with dilations of certain functions on the general Hardy space $H^1_L$ introduced by Hofmann, Lu, Mitrea, Mitrea, Yan. Our results include the heat and Poisson semigroups as well as the group of imaginary powers.
We characterize the weak-star point of continuity property for subspaces of dual spaces with separable predual and we deduce that the weak-star point of continuity property is determined by subspaces with a Schauder basis in the natural setting of dual spaces of separable Banach spaces. As a consequence of the above characterization we get that a dual space satisfies the Radon-Nikodym property if, and only if, every seminormalized topologically weak-star null tree has a boundedly complete branch, which improves some results in \cite{DF} obtained for the separable case. Also, as a consequence o…
For a compact Hausdorff space $S$, we prove that the closed unit ball of a closed linear subalgebra of the space of real-valued continuous functions on $S$, denoted by $C(S)$, satisfies property-$(P_1)$ (the set-valued generalization of strong proximinality) for the non-empty closed bounded subsets of the bidual of $C(S)$. Various stability results related to property-$(P_1)$ and semi-continuity properties of restricted Chebyshev-center maps are also established. As a consequence, we derive that if $Y$ is a proximinal finite co-dimensional subspace of $c_0$ then the closed unit ball of $Y$ sat…
In this paper, we present a new extension of the famous Serrin's lower semicontinuity theorem for the variational functional $\int_Ωf(x,u,u')dx$,we prove its lower semicontinuity in $W_{loc}^{1,1}(Ω)$ with respect to the strong $L_{loc}^{1}$ topology assuming that the integrand $f(x,s,ξ)$ has the usual continuity on all the three variables and the convexity property on the variable $ξ$ and the local absolute continuity on the variable $x$.
We show weak lower semi-continuity of functionals assuming the new notion of a "convexly constrained" $\mathcal A$-quasiconvex integrand. We assume $\mathcal A$-quasiconvexity only for functions defined on a set $K$ which is convex. Assuming this and sufficient integrability of the sequence we show that the functional is still (sequentially) weakly lower semi-continuous along weakly convergent "convexly constrained" $\mathcal A$-free sequences. In a motivating example, the integrand is $-\det^{\frac{1}{d-1}}$ and the convex constraint is positive semi-definiteness of a matrix field.
In this paper we study continuous semigroups of positive operators on general vector lattices equipped with the relative uniform topology $τ_{ru}$. We introduce the notions of strong continuity with respect to $τ_{ru}$ and relative uniform continuity for semigroups. These notions allow us to study semigroups on non-locally convex spaces such as $L^p(\mathbb{R})$ for $0<p<1$ and non-complete spaces such as $Lip(\mathbb{R})$, $UC(\mathbb{R})$, and $C_c(\mathbb{R})$. We show that the (left) translation semigroup on the real line, the heat semigroup and some Koopman semigroups are relatively unifo…
We study continuity and Hölder continuity of $t\mapsto P_tf$, where $P_t$ is a generalized Mehler semigroup in $C_b(X)$, the space of the continuous and bounded functions from a Banach space $X$ to $R$, and $f\in C_b(X)$. The generators $L$ of such semigroups are realizations of a class of differential and pseudo-differential operators, both in finite and in infinite dimension. Examples of operators $L$ to which this theory is applicable include Ornstein-Uhlenbeck operators with fractional diffusion in finite dimension, and Ornstein-Uhlenbeck operators with associated strong-Feller semigroups,…
In this note we connect the notion of solutions of a martingale problem to the notion of a strongly continuous and locally equi-continuous semigroup on the space of bounded continuous functions equipped with the strict topology. This extends the classical connection of semigroups to Markov processes that was used successfully in the context of compact spaces to the context of Polish spaces. In addition, we consider the context of locally compact spaces and show how the transition semigroup on the space of functions vanishing at infinity can be extended to the space of bounded continuous functi…
The objects of the present study are one-parameter semigroups generated by Schrödinger operators with fairly general electromagnetic potentials. More precisely, we allow scalar potentials from the Kato class and impose on the vector potentials only local Kato-like conditions. The configuration space is supposed to be an arbitrary open subset of multi-dimensional Euclidean space; in case that it is a proper subset, the Schrödinger operator is rendered symmetric by imposing Dirichlet boundary conditions. We discuss the continuity of the image functions of the semigroup and show local-norm-contin…
Weak Continuity of the Cartan Structural System and Compensated Compactness on Semi-Riemannian Manifolds with Lower Regularity
Mathematical Institute, University of Oxford（牛津大学数学研究所）；New York University–Shanghai（纽约大学上海分校）；NYU–ECNU Institute of Mathematical Sciences（纽约大学-华东师范大学数学科学研究所）；School of Mathematical Sciences, Shanghai Jiao Tong University（上海交通大学数学科学学院）
We characterize strong continuity of general operator semigroups on some Lebesgue spaces. In particular, a characterization of strong continuity of weighted composition semigroups on classical Hardy spaces and weighted Bergman spaces with regular weights is given. As applications, our result improves the results of Siskakis, A. G. \cite{AG1} and König, W. \cite{K} and answers a question of Siskakis, A. G. proposed in \cite{AG4}. We also characterize strongly continuous semigroups of weighted composition operators on weighted Bergman spaces in terms of abelian intertwiners of multiplication ope…
We consider locally equi-continuous strongly continuous semigroups on locally convex spaces (X,tau). First, we show that if (X,tau) has the property that weak* compact sets of the dual are equi-continuous, then strong continuity of the semigroup is equivalent to weak continuity and local equi-continuity. Second, we consider locally convex spaces (X,tau) that are also equipped with a `suitable' auxiliary norm. We introduce the set N of tau continuous semi-norms that are bounded by the norm. If (X,tau) has the property that N is closed under countable convex combinations, then a number of Banach…
For a compact Hausdorff space $X$, the space $SC(X\times X)$ of separately continuous complex valued functions on $X$ can be viewed as a $C^*$-subalgebra of $C(X)^{**}\overline\otimes C(X)^{**}$, namely those elements which slice into $C(X)$. The analogous definition for a non-commutative $C^*$-algebra does not necessarily give an algebra, but we show that there is always a greatest $C^*$-subalgebra. This thus gives a non-commutative notion of separate continuity. The tools involved are multiplier algebras and row/column spaces, familiar from the theory of Operator Spaces. We make some study o…
