On the asymptotic behavior of the solutions of semilinear nonautonomous equations

We consider nonautonomous semilinear evolution equations of the form $$\frac{dx}{dt}= A(t)x+f(t,x) . $$ Here A(t) is a (possibly unbounded) linear operator acting on a real or complex Banach space $\mathbb{X}$ and $f: \mathbb{R}\times\mathbb {X}\to\mathbb{X}$ is a (possibly nonlinear) continuous function. We assume that the linear equation (1) is well-posed (i.e. there exists a continuous linear evolution family {U(t,s)}_(t,s)∈Δ such that for every s∈?₊ and x∈D(A(s)), the function x(t)=U(t,s)x is the uniquely determined solution of Eq. (1) satisfying x(s)=x). Then we can consider the mild solution of the semilinear equation (2) (defined on some interval [s,s+δ),δ>0) as being the solution of the integral equation $$x(t) = U(t, s)x + \int_s^t U(t, \tau)f\bigl(\tau, x(\tau)\bigr) d\tau,\quad t\geq s . $$ Furthermore, if we assume also that the nonlinear function f(t,x) is jointly continuous with respect to t and x and Lipschitz continuous with respect to x (uniformly in t∈?₊, and f(t,0)=0 for all t∈?₊) we can generate a (nonlinear) evolution family {X(t,s)}_(t,s)∈Δ , in the sense that the map $t\mapsto X(t,s)x:[s,\infty)\to\mathbb{X}$ is the unique solution of Eq. (4), for every $x\in\mathbb{X}$ and s∈?₊. Considering the Green’s operator $(\mathbb{G}{f})(t)=\int_{0}^{t} X(t,s)f(s)ds$ we prove that if the following conditions hold

• the map $\mathbb{G}{f}$ lies in $L^{q}(\mathbb{R}_{+},\mathbb{X})$ for all $f\in L^{p}(\mathbb{R}_{+},\mathbb{X})$ , and
• $\mathbb{G}:L^{p}(\mathbb{R}_{+},\mathbb{X})\to L^{q}(\mathbb {R}_{+},\mathbb{X})$ is Lipschitz continuous, i.e. there exists K>0 such that $$\|\mathbb{G} {f}-\mathbb{G} {g}\|_{q} \leq K\|f-g\|_{p} , \quad\mbox{for all}\ f,g\in L^p(\mathbb{R}_+,\mathbb{X}) , $$

then the above mild solution will have an exponential decay.     

Estimates in Beurling-Helson type theorems: Multidimensional case

We consider the spaces A _p ( $\mathbb{T}^m $ ) of functions f on the m-dimensional torus $\mathbb{T}^m $ such that the sequence of Fourier coefficients $\hat f = \{ \hat f(k),k \in \mathbb{Z}^m \} $ belongs to l ^p (? ^m ), 1 ≤ p < 2. The norm on A _p ( $\mathbb{T}^m $ ) is defined by $\left\| f \right\|_{A_p (\mathbb{T}^m )} = \left\| {\hat f} \right\|_{l^p (\mathbb{Z}^m )} $ . We study the rate of growth of the norms $\left\| {e^{i\lambda \phi } } \right\|_{A_p (\mathbb{T}^m )} $ as |λ| → ∞, λ ∈ ?, for C ¹-smooth real functions φ on $\mathbb{T}^m $ (the one-dimensional case was investigated by the author earlier). The lower estimates that we obtain have direct analogs for the spaces A _p (? ^m ).     

Completeness of the Grid Translates of Functions

If $f\in L^{p}(\mathbb{R}^{d})$ is a bounded real valued continuous function which has a unique maximum or a unique minimum at a point $x_{0}\in \mathbb{R}^{d}$ and if the inverse image of the neighborhoods of f(x ₀) shrinks regularly to x ₀, then $\mathrm{ span }\{f^{m}(x-2^{-m}\varSigma_{i=1}^{d} j_{i} e_{i})\mid m\in\mathbb{N}, j_{i}\in\mathbb{Z}\}$ is a dense subset of $L^{p}(\mathbb{R}^{d}), 1\le p<\infty$ where f ^m (x)=f(x) ^m and {e _i } is the natural basis of $\mathbb{R}^{d}$ . The result extends to all homogeneous groups, Riemannian symmetric spaces of noncompact type, Damek-Ricci spaces etc.     

On the Equivariant Cohomology of Subvarieties of a \mathfrak{B}-Regular Variety

By a $\mathfrak{B}$ -regular variety, we mean a smooth projective variety over $\mathbb{C}$ admitting an algebraic action of the upper triangular Borel subgroup $\mathfrak{B} \subset {\text{SL}}_{2} {\left( \mathbb{C} \right)}$ such that the unipotent radical in $\mathfrak{B}$ has a unique fixed point. A result of Brion and the first author [4] describes the equivariant cohomology algebra (over $\mathbb{C}$ ) of a $\mathfrak{B}$ -regular variety X as the coordinate ring of a remarkable affine curve in $X \times \mathbb{P}^{1}$ . The main result of this paper uses this fact to classify the $\mathfrak{B}$ -invariant subvarieties Y of a $\mathfrak{B}$ -regular variety X for which the restriction map i _Y : H ^*(X) → H ^*(Y) is surjective.     

On the saturation of subfields of invariants of finite groups

Every subfield $ \mathbb{K} $ (φ) of the field of rational fractions $ \mathbb{K} $ (x ₁,..., x _n ) is contained in a unique maximal subfield of the form $ \mathbb{K} $ (ω). The element ω is said to be generating for the element φ. A subfield of $ \mathbb{K} $ (x ₁,..., x _n ) is said to be saturated if, together with every its element, the subfield also contains the generating element. In the paper, the saturation property is studied for the subfields of invariants $ \mathbb{K} $ (x ₁,..., x _n ) ^G of a finite group G of automorphisms of the field $ \mathbb{K} $ (x ₁..., x _n ).     

Dirac cohomology for graded affine Hecke algebras

We define an analogue of the Casimir element for a graded affine Hecke algebra $ \mathbb{H} $ , and then introduce an approximate square-root called the Dirac element. Using it, we define the Dirac cohomology H ^D (X) of an $ \mathbb{H} $ -module X, and show that H ^D (X) carries a representation of a canonical double cover of the Weyl group $ \widetilde{W} $ . Our main result shows that the $ \widetilde{W} $ -structure on the Dirac cohomology of an irreducible $ \mathbb{H} $ -module X determines the central character of X in a precise way. This can be interpreted as p-adic analogue of a conjecture of Vogan for Harish-Chandra modules. We also apply our results to the study of unitary representations of $ \mathbb{H} $ .     

Central Limit Theorems for Super Ornstein-Uhlenbeck Processes

Suppose that X={X _t :t≥0} is a supercritical super Ornstein-Uhlenbeck process, that is, a superprocess with an Ornstein-Uhlenbeck process on $\mathbb{R}^{d}$ corresponding to $L=\frac{1}{2}\sigma^{2}\Delta-b x\cdot\nabla$ as its underlying spatial motion and with branching mechanism ψ(λ)=?αλ+βλ ²+∫_(0,+∞)(e ^?λx ?1+λx)n(dx), where α=?ψ′(0+)>0, β≥0, and n is a measure on (0,∞) such that ∫_(0,+∞) x ² n(dx)<+∞. Let $\mathbb{P} _{\mu}$ be the law of X with initial measure μ. Then the process W _t =e ^?αt ∥X _t ∥ is a positive $\mathbb{P} _{\mu}$ -martingale. Therefore there is W _∞ such that W _t →W _∞, $\mathbb{P} _{\mu}$ -a.s. as t→∞. In this paper we establish some spatial central limit theorems for X. Let $\mathcal{P}$ denote the function class $$ \mathcal{P}:=\bigl\{f\in C\bigl(\mathbb{R}^d\bigr): \mbox{there exists } k\in\mathbb{N} \mbox{ such that }|f(x)|/\|x\|^k\to 0 \mbox{ as }\|x\|\to\infty \bigr\}. $$ For each $f\in\mathcal{P}$ we define an integer γ(f) in term of the spectral decomposition of f. In the small branching rate case α<2γ(f)b, we prove that there is constant $\sigma_{f}^{2}\in (0,\infty)$ such that, conditioned on no-extinction, $$\begin{aligned} \biggl(e^{-\alpha t}\|X_t\|, ~\frac{\langle f , X_t\rangle}{\sqrt{\|X_t\|}} \biggr) \stackrel{d}{\rightarrow}\bigl(W^*,~G_1(f)\bigr), \quad t\to\infty, \end{aligned}$$ where W ^? has the same distribution as W _∞ conditioned on no-extinction and $G_{1}(f)\sim \mathcal{N}(0,\sigma_{f}^{2})$ . Moreover, W ^? and G ₁(f) are independent. In the critical rate case α=2γ(f)b, we prove that there is constant $\rho_{f}^{2}\in (0,\infty)$ such that, conditioned on no-extinction, $$\begin{aligned} \biggl(e^{-\alpha t}\|X_t\|, ~\frac{\langle f , X_t\rangle}{t^{1/2}\sqrt{\|X_t\|}} \biggr) \stackrel{d}{\rightarrow}\bigl(W^*,~G_2(f)\bigr), \quad t\to\infty, \end{aligned}$$ where W ^? has the same distribution as W _∞ conditioned on no-extinction and $G_{2}(f)\sim \mathcal{N}(0, \rho_{f}^{2})$ . Moreover W ^? and G ₂(f) are independent. We also establish two central limit theorems in the large branching rate case α>2γ(f)b. Our central limit theorems in the small and critical branching rate cases sharpen the corresponding results in the recent preprint of Mi?o? in that our limit normal random variables are non-degenerate. Our central limit theorems in the large branching rate case have no counterparts in the recent preprint of Mi?o?. The main ideas for proving the central limit theorems are inspired by the arguments in K. Athreya’s 3 papers on central limit theorems for continuous time multi-type branching processes published in the late 1960’s and early 1970’s.     

New Characterizations of Besov-Triebel-Lizorkin-Hausdorff Spaces Including Coorbits and Wavelets

In this paper, the authors establish new characterizations of the recently introduced Besov-type spaces $\dot{B}^{s,\tau}_{p,q}({\mathbb{R}}^{n})$ and Triebel-Lizorkin-type spaces $\dot{F}^{s,\tau}_{p,q}({\mathbb{R}}^{n})$ with p∈(0,∞], s∈?, τ∈[0,∞), and q∈(0,∞], as well as their preduals, the Besov-Hausdorff spaces $B\!\dot{H}^{s,\tau}_{p,q}({\mathbb{R}}^{n})$ and Triebel-Lizorkin-Hausdorff spaces $F\!\dot{H}^{s,\tau}_{p,q}({\mathbb{R}}^{n})$ , in terms of the local means, the Peetre maximal function of local means, and the tent space (the Lusin area function) in both discrete and continuous types. As applications, the authors then obtain interpretations as coorbits in the sense of Rauhut (Stud. Math. 180:237–253, 2007) and discretizations via biorthogonal wavelet bases for the full range of parameters of these function spaces. Even for some special cases of this setting such as $\dot{F}^{s}_{\infty,q}({\mathbb{R}}^{n})$ for s∈?, q∈(0,∞] (including ?BMO(? ⁿ ) when s=0 and q=2), the Q space Q _α (? ⁿ ), the Hardy-Hausdorff space HH _?α (? ⁿ ) for α∈(0,min{n/2,1}), the Morrey space ${\mathcal{M}}^{u}_{p}({\mathbb{R}}^{n})$ for 1<p≤u<∞, and the Triebel-Lizorkin-Morrey space $\dot{\mathcal{E}}^{s}_{upq}({\mathbb{R}}^{n})$ for 0<p≤u<∞, s∈? and q∈(0,∞], some of these results are new.     

Non-linear rough heat equations

This article is devoted to define and solve an evolution equation of the form dy _t ?=?Δy _t dt?+ dX _t (y _t ), where Δ stands for the Laplace operator on a space of the form ${L^p(\mathbb R^n)}$ , and X is a finite dimensional noisy nonlinearity whose typical form is given by ${X_t(\varphi)=\sum_{i=1}^N \, x^{i}_t f_i(\varphi)}$ , where each x?=?(x ⁽¹⁾, … , x ^(N)) is a γ-H?lder function generating a rough path and each f _i is a smooth enough function defined on ${L^p(\mathbb R^n)}$ . The generalization of the usual rough path theory allowing to cope with such kind of system is carefully constructed.     

Boundedness and Compactness of Pseudodifferential Operators with Non-Regular Symbols on Weighted Lebesgue Spaces

Applying the boundedness on weighted Lebesgue spaces of the maximal singular integral operator S _* related to the Carleson?CHunt theorem on almost everywhere convergence, we study the boundedness and compactness of pseudodifferential operators a(x, D) with non-regular symbols in ${L^\infty(\mathbb{R}, V(\mathbb{R})), PC(\overline{\mathbb{R}}, V(\mathbb{R}))}$ and ${\Lambda_\gamma(\mathbb{R}, V_d(\mathbb{R}))}$ on the weighted Lebesgue spaces ${L^p(\mathbb{R},w)}$ , with 1?< p <? ?? and ${w\in A_p(\mathbb{R})}$ . The Banach algebras ${L^\infty(\mathbb{R}, V(\mathbb{R}))}$ and ${PC(\overline{\mathbb{R}}, V(\mathbb{R}))}$ consist, respectively, of all bounded measurable or piecewise continuous ${V(\mathbb{R})}$ -valued functions on ${\mathbb{R}}$ where ${V(\mathbb{R})}$ is the Banach algebra of all functions on ${\mathbb{R}}$ of bounded total variation, and the Banach algebra ${\Lambda_\gamma(\mathbb{R}, V_d(\mathbb{R}))}$ consists of all Lipschitz ${V_d(\mathbb{R})}$ -valued functions of exponent ${\gamma \in (0,1]}$ on ${\mathbb{R}}$ where ${V_d(\mathbb{R})}$ is the Banach algebra of all functions on ${\mathbb{R}}$ of bounded variation on dyadic shells. Finally, for the Banach algebra ${\mathfrak{A}_{p,w}}$ generated by all pseudodifferential operators a(x, D) with symbols ${a(x, \lambda) \in PC(\overline{\mathbb{R}}, V(\mathbb{R}))}$ on the space ${L^p(\mathbb{R}, w)}$ , we construct a non-commutative Fredholm symbol calculus and give a Fredholm criterion for the operators ${A \in \mathfrak{A}_{p,w}}$ .     

Predual Spaces of Banach Completions of Orlicz-Hardy Spaces Associated with Operators

Let L be a linear operator in L ²(? ⁿ ) and generate an analytic semigroup {e ^?tL } _t??0 with kernels satisfying an upper bound of Poisson type, whose decay is measured by ??(L)??(0,??]. Let ?? on (0,??) be of upper type 1 and of critical lower type $\widetilde{p}_{0}(\omega)\in(n/(n+\theta(L)),1]$ and ??(t)=t ^?1/?? ^?1(t ^?1) for t??(0,??). In this paper, the authors first introduce the VMO-type space VMO _??,L (? ⁿ ) and the tent space $T^{\infty}_{\omega,\mathrm{v}}({\mathbb{R}}^{n+1}_{+})$ and characterize the space VMO _??,L (? ⁿ ) via the space $T^{\infty}_{\omega,\mathrm{v}}({{\mathbb{R}}}^{n+1}_{+})$ . Let $\widetilde{T}_{\omega}({{\mathbb{R}}}^{n+1}_{+})$ be the Banach completion of the tent space $T_{\omega}({\mathbb{R}}^{n+1}_{+})$ . The authors then prove that $\widetilde{T}_{\omega}({\mathbb{R}}^{n+1}_{+})$ is the dual space of $T^{\infty}_{\omega,\mathrm{v}}({\mathbb{R}}^{n+1}_{+})$ . As an application of this, the authors finally show that the dual space of $\mathrm{VMO}_{\rho,L^{\ast}}({\mathbb{R}}^{n})$ is the space B _??,L (? ⁿ ), where L ^* denotes the adjoint operator of L in L ²(? ⁿ ) and B _??,L (? ⁿ ) the Banach completion of the Orlicz-Hardy space H _??,L (? ⁿ ). These results generalize the known recent results by particularly taking ??(t)=t for t??(0,??).     

Existence and asymptotic behaviour of solutions of the very fast diffusion equation

Let n ≥ 3, 0 < m ≤ (n ? 2)/n, p > max(1, (1 ? m)n/2), and ${0 \le u_0 \in L_{loc}^p(\mathbb{R}^n)}$ satisfy ${{\rm lim \, inf}_{R\to\infty}R^{-n+\frac{2}{1-m}} \int_{|x|\le R}u_0\,dx = \infty}$ . We prove the existence of unique global classical solution of u _t = Δu ^m , u > 0, in ${\mathbb{R}^n \times (0, \infty), u(x, 0) = u_0(x)}$ in ${\mathbb{R}^n}$ . If in addition 0 < m < (n ? 2)/n and u ₀(x) ≈ A|x|^?q as |x| → ∞ for some constants A > 0, q < n/p, we prove that there exist constants α, β, such that the function v(x, t) = t ^α u(t ^β x, t) converges uniformly on every compact subset of ${\mathbb{R}^n}$ to the self-similar solution ψ(x, 1) of the equation with ψ(x, 0) = A|x|^?q as t → ∞. Note that when m = (n ? 2)/(n + 2), n ≥ 3, if ${g_{ij} = u^{\frac{4}{n+2}}\delta_{ij}}$ is a metric on ${\mathbb{R}^n}$ that evolves by the Yamabe flow ?g _ij /?t = ?Rg _ij with u(x, 0) = u ₀(x) in ${\mathbb{R}^n}$ where R is the scalar curvature, then u(x, t) is a global solution of the above fast diffusion equation.     

Geodesic distance for right invariant Sobolev metrics of fractional order on the diffeomorphism group

We study Sobolev-type metrics of fractional order s ≥ 0 on the group Diff _c (M) of compactly supported diffeomorphisms of a manifold M. We show that for the important special case M = S ¹, the geodesic distance on Diff _c (S ¹) vanishes if and only if ${s\leq\frac12}$ . For other manifolds, we obtain a partial characterization: the geodesic distance on Diff _c (M) vanishes for ${M=\mathbb{R}\times N, s < \frac12}$ and for ${M=S^1\times N, s\leq\frac12}$ , with N being a compact Riemannian manifold. On the other hand, the geodesic distance on Diff _c (M) is positive for ${{\rm dim}(M)=1, s > \frac12}$ and dim(M) ≥ 2, s ≥ 1. For ${M=\mathbb{R}^n}$ , we discuss the geodesic equations for these metrics. For n = 1, we obtain some well-known PDEs of hydrodynamics: Burgers’ equation for s = 0, the modified Constantin–Lax–Majda equation for ${s=\frac12}$ , and the Camassa–Holm equation for s = 1.     

Variable Exponent Herz Spaces

We introduce a new type of variable exponent function spaces  ? ^{p(·),q(·),α(·)}( ${\mathbb{R}^n}$ ) and H ^{p(·),q(·),α(·)}( ${\mathbb{R}^n}$ ) of Herz type, homogeneous and non-homogeneous versions, where all the three parameters are variable, and give comparison of continual and discrete approaches to their definition. Under the only assumption that the exponents p, q and α are subject to the log-decay condition at infinity, we prove that sublinear operators, satisfying the size condition known for singular integrals and bounded in L ^p(·)( ${\mathbb{R}^n}$ ), are also bounded in the nonhomogeneous version of the introduced spaces, which includes the case maximal and Calderón-Zygmund singular operators.     

Permanent Versus Determinant over a Finite Field

Let $ \mathbb{F} $ be a finite field of characteristic different from 2. We study the cardinality of sets of matrices with a given determinant or a given permanent for the set of Hermitian matrices $ {{\mathcal{H}}_n}\left( \mathbb{F} \right) $ and for the whole matrix space M _n ( $ \mathbb{F} $ ). It is known that for n = 2, there are bijective linear maps Φ on $ {{\mathcal{H}}_n}\left( \mathbb{F} \right) $ and M _n ( $ \mathbb{F} $ ) satisfying the condition per A = det Φ(A). As an application of the obtained results, we show that if n ≥ 3, then the situation is completely different and already for n = 3, there is no pair of maps (Φ, ?), where Φ is an arbitrary bijective map on matrices and ? : $ \mathbb{F} $ → $ \mathbb{F} $ is an arbitrary map such that per A = ?(det Φ(A)) for all matrices A from the spaces $ {{\mathcal{H}}_n}\left( \mathbb{F} \right) $ and M _n ( $ \mathbb{F} $ ), respectively. Moreover, for the space M _n ( $ \mathbb{F} $ ), we show that such a pair of transformations does not exist also for an arbitrary n > 3 if the field $ \mathbb{F} $ contains sufficiently many elements (depending on n). Our results are illustrated by a number of examples.     

Orbit Spaces of Free Involutions on the Product of Two Projective Spaces

Let X be a finitistic space having the mod 2 cohomology algebra of the product of two projective spaces. We study free involutions on X and determine the possible mod 2 cohomology algebra of orbit space of any free involution, using the Leray spectral sequence associated to the Borel fibration ${X \hookrightarrow X_{\mathbb{Z}_2} \longrightarrow B_{\mathbb{Z}_2}}$ . We also give an application of our result to show that if X has the mod 2 cohomology algebra of the product of two real projective spaces (respectively, complex projective spaces), then there does not exist any ${\mathbb{Z}_2}$ -equivariant map from ${\mathbb{S}^k \to X}$ for k ≥ 2 (respectively, k ≥ 3), where ${\mathbb{S}^k}$ is equipped with the antipodal involution.     

On the boundary of a special class of hyperbolic two-dimensional simplicial complexes

A class of hyperbolic two-dimensional complexes X is defined. It is shown that the limit set of the action of π ₁(X) on the universal covering ${\widetilde{X}}$ of X is equal to the visual boundary ${\partial \widetilde{X}}$ of and that ${\partial \widetilde{X}}$ is path connected and locally path connected. A kind of Sierpinski set is described which is homeomorphic to the visual boundary of certain ideal polyhedra.     

Homoclinic orbits for an infinite dimensional Hamiltonian system with periodic potential

In this paper, we study the following diffusion system where $z:(u,v):\mathbb{R}\times\mathbb{R}^{N}\rightarrow\mathbb{R}^{2}$ , $V(x)\in C(\mathbb{R}^{N},\mathbb{R})$ is a general periodic function, g(t,x,v), f(t,x,u) are periodic in t,x and superquadratic in v,u at infinity. By using much more direct methods to prove all Cerami sequences for the energy functional are bounded and establish the existence of homoclinic orbits, which are ground state solutions for above system.     

On Time-Dependent Functionals of Diffusions Corresponding to Divergence Form Operators

We consider processes of the form [s,T]?t?u(t,X _t ), where (X,P _s,x ) is a multidimensional diffusion corresponding to a uniformly elliptic divergence form operator. We show that if $u\in{\mathbb{L}}_{2}(0,T;H_{\rho }^{1})$ with $\frac{\partial u}{\partial t} \in{\mathbb{L}}_{2}(0,T;H_{\rho }^{-1})$ then there is a quasi-continuous version $\tilde{u}$ of u such that $\tilde{u}(t,X_{t})$ is a P _s,x -Dirichlet process for quasi-every (s,x)∈[0,T)×? ^d with respect to parabolic capacity, and we describe the martingale and the zero-quadratic variation parts of its decomposition. We also give conditions on u ensuring that $\tilde{u}(t,X_{t})$ is a semimartingale.     

On the fundamental group of {{\rm Hom}({\mathbb Z}^k,G)}

Let G be a compact Lie group. Consider the variety ${{\rm Hom}({\mathbb Z}^{k},G)}$ of representations of ${{\mathbb Z}^k}$ into G. We can see this as a based space by taking as base point the trivial representation . The goal of this paper is to prove that ${\pi_1({\rm Hom}({\mathbb Z}^k,G))}$ is naturally isomorphic to π ₁(G) ^k .