Lower semicontinuity for functionals with (p, q)-growth in Carnot groups

Let ?? be a bounded open subset of ${\mathbb{G}}$ , where ${\mathbb{G}}$ is a Carnot group, and let ${u: \Omega \rightarrow \mathbb{R}^d}$ be a vector valued function. We prove a lower semicontinuity result in the weak topology of the horizontal Sobolev space ${W^{1,p}_X(\Omega,\mathbb{R}^d)}$ , with p?>?1, of the integral functional of the calculus of variations of the type $$F(u)=\int\limits_\Omega f(Xu)\,dx$$ where f is a X-quasiconvex function satisfying a non-standard growth conditions and Xu is the horizontal gradient of u.     

Об аппроксимации свермками и баэисах иэ сдвигов функции

For 2π-periodic functions f ∈ L ^p ( $ \mathbb{T} $ ), 1 ≤ p < ∞, σ ∈ V ( $ \mathbb{T} $ ) and g ∈ L( $ \mathbb{T} $ ), we consider the convolutions $$ (f*d\sigma )_T (x) = \int_0^{2\pi } {f(x - t)d\sigma (t), } (f*g)_T (x) = \int_0^{2\pi } {f(x - t)g(t)dt.} $$ For fixed functions σ ∈ V ( $ \mathbb{T} $ ) and g ∈ L( $ \mathbb{T} $ ), necessary and sufficient conditions are obtained for the density of the ranges of these operators in L ^p . Similar result is proved for the dyadic convolution $$ (f*g)_2 (x) = \int_0^1 {f(x \oplus t)g(t)dt,} $$ where ⊕ is the operation of dyadic addition on [0, 1). Moreover, it is proved that in the spaces L ^p ( $ \mathbb{T} $ ), 1 ≤ p ∞, and C( $ \mathbb{T} $ ) there exist no bases of shifts of a function. Similar results are obtained for the spaces L ^p [0, 1]*, 1 ≤ p < ∞, and C[0, 1]* relative to dyadic shifts, where [0, 1]* is the modified segment [0, 1]. It is also proved that in the space L(?₊) there exists no basis of dyadic shifts of a function.     

Tail homogeneity of invariant measures of multidimensional stochastic recursions in a critical case

We consider the stochastic recursion ${X_{n+1} = M_{n+1}X_{n} + Q_{n+1}, (n \in \mathbb{N})}$ , where ${Q_n, X_n \in \mathbb{R}^d }$ , M _n are similarities of the Euclidean space ${ \mathbb{R}^d }$ and (Q _n , M _n ) are i.i.d. We study asymptotic properties at infinity of the invariant measure for the Markov chain X _n under assumption ${\mathbb{E}{[\log|M|]}=0}$ i.e. in the so called critical case.     

On the Behavior of Gegenbauer Polynomials in the Complex Plane

It is well-known that the squared modulus of every function f from the Laguerre–Polya class ${\mathcal{L}-\mathcal{P}}$ of entire functions obeys a MacLaurin series representation $$|f(x+i y)|^2=\sum_{k=0}^{\infty} L_k(f;x)\,y^{2k}, \quad x,y\in\mathbb{R}$$ , which reduces to a finite sum when f is a polynomial having only real zeros. The coefficients {L _k } are representable as non-linear differential operators acting on f, and by a classical result of Jensen L _k (f;x)?≥ 0 for ${f\in \mathcal{L}-\mathcal{P}}$ and ${x\in \mathbb{R}}$ . Here, we prove a conjecture formulated by the first-named author in 2005, which states that for ${f=P_n^{(\lambda)} }$ , the n-th Gegenbauer polynomial, the functions ${\{L_k(f;x)\}_{k=1}^{n}}$ are monotone decreasing on the negative semi-axis and monotone increasing on the positive semi-axis. This result pertains to certain polynomial inequalities in the spirit of the celebrated refinement of Markov’s inequality, found by R. J. Duffin and A. C. Schaeffer in 1941.     

Local and 2-Local Lie Derivations of Operator Algebras on Banach Spaces

Let X be a Banach space of dimension > 2. We show that every local Lie derivation of B(X) is a Lie derivation, and that a map of B(X) is a 2-local Lie derivation if and only if it has the form ${A \mapsto AT - TA + \psi(A)}$ A ? A T - T A + ψ ( A ) , where ${T \in B(X)}$ T ∈ B ( X ) and ψ is a homogeneous map from B(X) into ${\mathbb{F}I}$ F I satisfying ${\psi(A + B) = \psi(A)}$ ψ ( A + B ) = ψ ( A ) for ${A, B \in B(X)}$ A , B ∈ B ( X ) with B being a sum of commutators.     

On Isomorphisms of Hardy Spaces Associated with Schrödinger Operators

Let L=?Δ+V is a Schrödinger operator on $\mathbb{R}^{d}$ , d≥3, V≥0. Let $H^{1}_{L}$ denote the Hardy space associated with L. We shall prove that there is an L-harmonic function w, 0<δ≤w(x)≤C, such that the mapping $$H_L^1 \ni f\mapsto wf\in H^1\bigl(\mathbb{R}^d\bigr) $$ is an isomorphism from the Hardy space $H_{L}^{1}$ onto the classical Hardy space $H^{1}(\mathbb{R}^{d})$ if and only if $\Delta^{-1}V(x)=-c_{d}\int_{\mathbb{R}^{d}} |x-y|^{2-d} V(y) dy$ belongs to $L^{\infty}(\mathbb{R}^{d})$ .     

Difference Inequalities for the Groups and Semigroups of Operators on Banach Spaces

Let ${\mathbf{T}=\{T(t)\} _{t\in\mathbb{R}}}$ be a ??(X, F)-continuous group of isometries on a Banach space X with generator A, where ??(X, F) is an appropriate local convex topology on X induced by functionals from ${ F\subset X^{\ast}}$ . Let ?? _A (x) be the local spectrum of A at ${x\in X}$ and ${r_{A}(x):=\sup\{\vert\lambda\vert :\lambda \in \sigma_{A}(x)\},}$ the local spectral radius of A at x. It is shown that for every ${x\in X}$ and ${\tau\in\mathbb{R},}$ $$\left\Vert T(\tau) x-x\right\Vert \leq \left\vert \tau \right\vert r_{A}(x)\left\Vert x\right\Vert.$$ Moreover if ${0\leq \tau r_{A}(x)\leq \frac{\pi}{2},}$ then it holds that $$\left\Vert T(\tau) x-T(-\tau)x\right\Vert \leq 2\sin \left(\tau r_{A}(x)\right)\left\Vert x\right\Vert.$$ Asymptotic versions of these results for C ₀-semigroup of contractions are also obtained. If ${\mathbf{T}=\{T(t)\}_{t\geq 0}}$ is a C ₀-semigroup of contractions, then for every ${x\in X}$ and ????? 0, $$\underset{t\rightarrow \infty }{\lim } \left\Vert T( t+\tau) x-T(t) x\right\Vert\leq\tau\sup\left\{ \left\vert \lambda \right\vert :\lambda \in\sigma_{A}(x)\cap i \mathbb{R} \right\} \left\Vert x\right\Vert. $$ Several applications are given.     

On the Generalized Feynman–Kac Transformation for?Nearly Symmetric Markov Processes

Suppose that X is a right process which is associated with a non-symmetric Dirichlet form $(\mathcal{E},D(\mathcal{E}))$ on L ²(E;m). For $u\in D(\mathcal{E})$ , we have Fukushima??s decomposition: $\tilde{u}(X_{t})-\tilde{u}(X_{0})=M^{u}_{t}+N^{u}_{t}$ . In this paper, we investigate the strong continuity of the generalized Feynman?CKac semigroup defined by $P^{u}_{t}f(x)=E_{x}[e^{N^{u}_{t}}f(X_{t})]$ . Let $Q^{u}(f,g)=\mathcal{E}(f,g)+\mathcal{E}(u,fg)$ for $f,g\in D(\mathcal{E})_{b}$ . Denote by J ₁ the dissymmetric part of the jumping measure J of $(\mathcal{E},D(\mathcal{E}))$ . Under the assumption that J ₁ is finite, we show that $(Q^{u},D(\mathcal{E})_{b})$ is lower semi-bounded if and only if there exists a constant ?? ₀??0 such that $\|P^{u}_{t}\|_{2}\leq e^{\alpha_{0}t}$ for every t>0. If one of these conditions holds, then $(P^{u}_{t})_{t\geq0}$ is strongly continuous on L ²(E;m). If X is equipped with a differential structure, then this result also holds without assuming that J ₁ is finite.     

Quenched to annealed transition in the parabolic Anderson problem

We study limit behavior for sums of the form $\frac{1}{|\Lambda_{L|}}\sum_{x\in \Lambda_{L}}u(t,x),$ where the field $\Lambda_L=\left\{x\in {\bf{Z^d}}:|x|\le L\right\}$ is composed of solutions of the parabolic Anderson equation $$u(t,x) = 1 + \kappa \mathop{\int}_{0}^{t} \Delta u(s,x){\rm d}s + \mathop{\int}_{0}^{t}u(s,x)\partial B_{x}(s). $$ The index set is a box in Z ^d , namely $\Lambda_{L} = \left\{x\in {\bf Z}^{\bf d} : |x| \leq L\right\}$ and L = L(t) is a nondecreasing function $L : [0,\infty)\rightarrow {\bf R}^{+}. $ We identify two critical parameters $\eta(1) < \eta(2)$ such that for $\gamma > \eta(1)$ and L(t) = e^{γ t} , the sums $\frac{1}{|\Lambda_L|}\sum_{x\in \Lambda_L}u(t,x)$ satisfy a law of large numbers, or put another way, they exhibit annealed behavior. For $\gamma > \eta(2)$ and L(t) = e^{γ t} , one has $\sum_{x\in \Lambda_L}u(t,x)$ when properly normalized and centered satisfies a central limit theorem. For subexponential scales, that is when $\lim_{t \rightarrow \infty} \frac{1}{t}\ln L(t) = 0,$ quenched asymptotics occur. That means $\lim_{t\rightarrow \infty}\frac{1}{t}\ln\left (\frac{1}{|\Lambda_L|}\sum_{x\in \Lambda_L}u(t,x)\right) = \gamma(\kappa),$ where $\gamma(\kappa)$ is the almost sure Lyapunov exponent, i.e. $\lim_{t\rightarrow \infty}\frac{1}{t}\ln u(t,x)= \gamma(\kappa).$ We also examine the behavior of $\frac{1}{|\Lambda_L|}\sum_{x\in \Lambda_L}u(t,x)$ for L = e ^{γ t} with γ in the transition range $(0,\eta(1))$      

On the topology of semi-algebraic functions on closed semi-algebraic sets

We consider a closed semi-algebraic set ${X \subset \mathbb{R}^n}$ and a C ² semi-algebraic function ${f : \mathbb{R}^n \rightarrow\mathbb{R}}$ such that ${f_{\vert X}}$ has a finite number of critical points. We relate the topology of X to the topology of the sets ${X \cap \{ f * \alpha \}}$ , where ${* \in \{\le,=,\ge \}}$ and ${\alpha \in \mathbb{R}}$ , and the indices of the critical points of ${f_{\vert X}}$ and ${-f_{\vert X}}$ . We also relate the topology of X to the topology of the links at infinity of the sets ${X \cap \{ f * \alpha\}}$ and the indices of these critical points. We give applications when ${X=\mathbb{R}^n}$ and when f is a generic linear function.     

Cotauberian Operators on L 1(0, 1) Obtained by Lifting

We show that the set \({\mathcal{T}^d(L_1(0, 1))}\) of cotauberian operators acting on L ₁(0, 1) is not open, and \({T \in \mathcal{T}^d(L_1(0, 1))}\) does not imply T** cotauberian. As a consequence, we derive that the set \({\mathcal{T}(L_\infty(0, 1))}\) of tauberian operators acting on L _∞(0, 1) is not open, and that \({T \in \mathcal{T}(L_\infty(0, 1))}\) does not imply T** tauberian.     

Generalized Lax-Milgram theorem in Banach spaces and its application to the elliptic system of boundary value problems

We generalize the well-known Lax-Milgram theorem on the Hilbert space to that on the Banach space. Suppose that ${a(\cdot, \cdot)}$ is a continuous bilinear form on the product ${X\times Y}$ of Banach spaces X and Y, where Y is reflexive. If null spaces N _X and N _Y associated with ${a(\cdot, \cdot)}$ have complements in X and in Y, respectively, and if ${a(\cdot, \cdot)}$ satisfies certain variational inequalities both in X and in Y, then for every ${F \in N_Y^{\perp}}$ , i.e., ${F \in Y^{\ast}}$ with ${F(\phi) = 0}$ for all ${\phi \in N_Y}$ , there exists at least one ${u \in X}$ such that ${a(u, \varphi) = F(\varphi)}$ holds for all ${\varphi \in Y}$ with ${\|u\|_X \le C\|F\|_{Y^{\ast}}}$ . We apply our result to several existence theorems of L ^r -solutions to the elliptic system of boundary value problems appearing in the fluid mechanics.     

Discrete fractal dimensions of the ranges of random walks in {{\mathbb Z}^d} associate with random conductances

Let ${X= \{X_t, t \ge 0\}}$ be a continuous time random walk in an environment of i.i.d. random conductances ${\{\mu_e \in [1,\infty), e \in E_d\}}$ , where E _d is the set of nonoriented nearest neighbor bonds on the Euclidean lattice ${\mathbb{Z}^d}$ and d ≥ 3. Let ${{\rm R} = \{x \in \mathbb{Z}^d: X_t = x {\rm \,for\, some}\,t \ge 0\}}$ be the range of X. It is proved that, for almost every realization of the environment, dim_H R = dim_P R = 2 almost surely, where dim_H and dim_P denote, respectively, the discrete Hausdorff and packing dimension. Furthermore, given any set ${A \subseteq \mathbb{Z}^d}$ , a criterion for A to be hit by X _t for arbitrarily large t > 0 is given in terms of dim_H A. Similar results for Bouchoud’s trap model in ${\mathbb{Z}^d}$ (d ≥ 3) are also proven.     

Heavy tail phenomenon and convergence to stable laws for iterated Lipschitz maps

We consider the Markov chain ${\{X_n^x\}_{n=0}^\infty}$ on ${\mathbb{R}^d}$ defined by the stochastic recursion ${X_{n}^{x}= \psi_{\theta_{n}} (X_{n-1}^{x})}$ , starting at ${x\in\mathbb{R}^d}$ , where ?? ₁, ?? ₂, . . . are i.i.d. random variables taking their values in a metric space ${(\Theta, \mathfrak{r})}$ , and ${\psi_{\theta_{n}} :\mathbb{R}^d\mapsto\mathbb{R}^d}$ are Lipschitz maps. Assume that the Markov chain has a unique stationary measure ??. Under appropriate assumptions on ${\psi_{\theta_n}}$ , we will show that the measure ?? has a heavy tail with the exponent ???>?0 i.e. ${\nu(\{x\in\mathbb{R}^d: |x| > t\})\asymp t^{-\alpha}}$ . Using this result we show that properly normalized Birkhoff sums ${S_n^x=\sum_{k=1}^n X_k^x}$ , converge in law to an ??-stable law for ${\alpha\in(0, 2]}$ .     

A Representation of Hyperbolic Motions Including the Infinite-Dimensional Case

Let X be a real inner product space of (finite or infinite) dimension ???2, O(X) be its group of all surjective (hence bijective) orthogonal transformations of X, T(X) be the set of all hyperbolic translations of X and M(X, hyp) be the group of all hyperbolic motions of X. The following theorem will be proved in this note. Every ${\mu\in M(X,{\mbox hyp})}$ has a representation ?? = T · ?? with uniquely determined ${T\in T(X)}$ and uniquely determined ${\omega\in O(X)}$ .     

An extension of Mercer’s theory to L p

Let X be a topological space, either locally compact or first countable, endowed with a strictly positive measure ?? and ${\mathcal{K}:L^2(X,\nu)\to L^2(X,\nu)}$ an integral operator generated by a Mercer like kernel K. In this paper we extend Mercer??s theory for K and ${\mathcal{K}}$ under the assumption that the function ${x\in X\to K(x,x)}$ belongs to some L ^p/2(X, ??), p??? 1. In particular, we obtain series representations for K and some powers of ${\mathcal{K}}$ , with convergence in the p-mean, and show that the range of certain powers of ${\mathcal{K}}$ contains continuous functions only. These results are used to estimate the approximation numbers of a modified version of ${\mathcal{K}}$ acting on L ^p (X, ??).     

A Geometry for the Set of Split Operators

We study the set ${\mathcal{X}}$ of split operators acting in the Hilbert space ${\mathcal{H}}$ : $$\mathcal{X}=\{T\in \mathcal{B}(\mathcal{H}): N(T)\cap R(T)=\{0\} \ {\rm and} \ N(T)+R(T)=\mathcal{H}\}.$$ Inside ${\mathcal{X}}$ , we consider the set ${\mathcal{Y}}$ : $$\mathcal{Y}=\{T\in\mathcal{X}: N(T)\perp R(T)\}.$$ Several characterizations of these sets are given. For instance ${T\in\mathcal{X}}$ if and only if there exists an oblique projection ${Q}$ whose range is N(T) such that T + Q is invertible, if and only if T posseses a commuting (necessarilly unique) pseudo-inverse S (i.e. TS = ST, TST = T and STS = S). Analogous characterizations are given for ${\mathcal{Y}}$ . Two natural maps are considered: $${\bf q}:\mathcal{X} \to \mathbb{Q}:=\{{\rm oblique \ projections \ in} \, \mathcal{H} \}, \ {\bf q}(T)=P_{R(T)//N(T)}$$ and $${\bf p}:\mathcal{Y} \to \mathbb{P}:=\{{\rm orthogonal \ projections \ in} \ \mathcal{H} \}, \ {\bf p}(T)=P_{R(T)}, $$ where ${P_{R(T)//N(T)}}$ denotes the projection onto R(T) with nullspace N(T), and P _R(T) denotes the orthogonal projection onto R(T). These maps are in general non continuous, subsets of continuity are studied. For the map q these are: similarity orbits, and the subsets ${\mathcal{X}_{c_k}\subset \mathcal{X}}$ of operators with rank ${k<\infty}$ , and ${\mathcal{X}_{F_k}\subset\mathcal{X}}$ of Fredholm operators with nullity ${k<\infty}$ . For the map p there are analogous results. We show that the interior of ${\mathcal{X}}$ is ${\mathcal{X}_{F_0}\cup\mathcal{X}_{F_1}}$ , and that ${\mathcal{X}_{c_k}}$ and ${\mathcal{X}_{F_k}}$ are arc-wise connected differentiable manifolds.     

Absolutely continuous operators on function spaces and vector measures

Let (Ω, Σ, μ) be a finite atomless measure space, and let E be an ideal of L ⁰(μ) such that ${L^\infty(\mu) \subset E \subset L^1(\mu)}$ . We study absolutely continuous linear operators from E to a locally convex Hausdorff space ${(X, \xi)}$ . Moreover, we examine the relationships between μ-absolutely continuous vector measures m : Σ → X and the corresponding integration operators T _m : L ^∞(μ) → X. In particular, we characterize relatively compact sets ${\mathcal{M}}$ in ca _μ (Σ, X) (= the space of all μ-absolutely continuous measures m : Σ → X) for the topology ${\mathcal{T}_s}$ of simple convergence in terms of the topological properties of the corresponding set ${\{T_m : m \in \mathcal{M}\}}$ of absolutely continuous operators. We derive a generalized Vitali–Hahn–Saks type theorem for absolutely continuous operators T : L ^∞(μ) → X.