Homoclinic Solutions for <Emphasis Type="Italic">p</Emphasis>(<Emphasis Type="Italic">t</Emphasis>)-Laplacian–Hamiltonian Systems Without Coercive Conditions

In this paper, we study the existence and multiplicity of homoclinic solutions for the following second-order p(t)-Laplacian–Hamiltonian systems

$$\frac{{\rm d}}{{\rm d}t}(|\dot{u}(t)|^{p(t)-2}\dot{u}(t))-a(t)|u(t)|^{p(t)-2}u(t)+\nabla W(t,u(t))=0,$$

where \({t \in \mathbb{R}}\), \({u \in \mathbb{R}^n}\), \({p \in C(\mathbb{R},\mathbb{R})}\) with p(t) > 1, \({a \in C(\mathbb{R},\mathbb{R})}\), \({W\in C^1(\mathbb{R}\times\mathbb{R}^n,\mathbb{R})}\) and \({\nabla W(t,u)}\) is the gradient of W(t, u) in u. The point is that, assuming that a(t) is bounded in the sense that there are constants \({0<\tau_1<\tau_2<\infty}\) such that \({\tau_1\leq a(t)\leq \tau_2 }\) for all \({t \in \mathbb{R}}\) and W(t, u) is of super-p(t) growth or sub-p(t) growth as \({|u|\rightarrow \infty}\), we provide two new criteria to ensure the existence and multiplicity of homoclinic solutions, respectively. Recent results in the literature are extended and significantly improved.

     

Some estimates for the Schrödinger type operators with nonnegative potentials

In this paper we consider the Schrödinger operator ?Δ + V on \({\mathbb R^d}\), where the nonnegative potential V belongs to the reverse Hölder class \({B_{q_{_1}}}\) for some \({q_{_1}\geq \frac{d}{2}}\) with d ≥ 3. Let \({H^1_L(\mathbb R^d)}\) denote the Hardy space related to the Schrödinger operator L = ?Δ + V and \({BMO_L(\mathbb R^d)}\) be the dual space of \({H^1_L(\mathbb R^d)}\). We show that the Schrödinger type operator \({\nabla(-\Delta +V)^{-\beta}}\) is bounded from \({H^1_L(\mathbb R^d)}\) into \({L^p(\mathbb R^d)}\) for \({p=\frac{d}{d-(2\beta-1)}}\) with \({ \frac{1}{2}<\beta<\frac{3}{2} }\) and that it is also bounded from \({L^p(\mathbb R^d)}\) into \({BMO_L(\mathbb R^d)}\) for \({p=\frac{d}{2\beta-1}}\) with \({ \frac{1}{2}<\beta< 2}\).     

Isoperimetric inequality for the polydisk

Let \({n\in\mathbb{N}}\). For \({k\in\{1,\dots,n\}}\) let \({\Omega_k\subset \mathbb{C}}\) be a simply connected domain with a rectifiable boundary. Let \({\Omega^n=\prod_{k=1}^n\Omega_k\subset \mathbb{C}^n}\) be a generalized polydisk with distinguished boundary \({\partial\Omega^n=\prod_{k=1}^n\partial\Omega_k}\). Let E ^r (Ω ⁿ ) be the holomorphic Smirnov class on Ω ⁿ with index r. We show that the generalized isoperimetric inequality

$ \int\limits_{\Omega^n} |f_1|^p|f_2|^qdV\le \frac{1}{(4\pi)^n}\int\limits_{\partial \Omega^n}|f_1|^pdS \int\limits_{\partial \Omega^n} |f_2|^qdS, $

holds for arbitrary \({f_1\in E^p(\Omega^n)}\) and \({f_2\in E^q(\Omega^n)}\), where 0 < p, q < ∞. We also determine necessary and sufficient conditions for equality.

     

On estimation of delay location

Assume that we observe a stationary Gaussian process X(t), \({t \in [-r, T]}\) , which satisfies the affine stochastic delay differential equation

$d X(t) = \int\limits_{[-r,0]}X(t+u)\, a_\vartheta (du)\,dt +dW(t), \quad t\ge 0,$

where W(t), t ≥ 0, is a standard Wiener process independent of X(t), \({t\in [-r, 0]}\) , and \({a_\vartheta}\) is a finite signed measure on [?r, 0], \({\vartheta\in\Theta}\) . The parameter \({\vartheta}\) is unknown and has to be estimated based on the observation. In this paper we consider the case where \({\Theta=(\vartheta_0,\vartheta_1)}\) , \({-\infty\,<\,\vartheta_0 <0 \,<\,\vartheta_1\,<\,\infty}\) , and the measures \({a_\vartheta}\) are of the form

$a_\vartheta = a+b_\vartheta-b,$

where a and b are finite signed measure on [?r, 0] and \({b_\vartheta}\) is the translate of b by \({\vartheta}\) . We study the limit behaviour of the normalized likelihoods

$Z_{T,\vartheta}(u) = \frac{dP_T^{\vartheta+\delta_T u}}{dP_T^\vartheta}$

as T→ ∞, where \({P_T^\vartheta}\) is the distribution of the observation if the true value of the parameter is \({\vartheta}\) . A necessary and sufficient condition for the existence of a rescaling function δ _T such that \({Z_{T,\vartheta}(u)}\) converges in distribution to an appropriate nondegenerate limiting function \({Z_{\vartheta}(u)}\) is found. It turns out that then the limiting function \({Z_{\vartheta}(u)}\) is of the form

$Z_\vartheta(u)=\exp\left(B^H(u) - E[B^H(u)]^2/2\right),$

where \({H\in[1/2,1]}\) and B ^H (u), \({u\in\mathbb{R}}\) , is a fractional Brownian motion with index H, and δ _T  = T ^?1/(2H) ?(T) with a slowly varying function ?. Every \({H\in[1/2,1]}\) may occur in this framework. As a consequence, the asymptotic behaviour of maximum likelihood and Bayes estimators is found.

     

Asymptotic behavior of the p-torsion functions as p goes to 1

Let \({\Omega}\) be a Lipschitz bounded domain of \({\mathbb{R}^N}\), \({N\geq2}\), and let \({u_p\in W_0^{1,p}(\Omega)}\) denote the p-torsion function of \({\Omega}\), p > 1. It is observed that the value 1 for the Cheeger constant \({h(\Omega)}\) is threshold with respect to the asymptotic behavior of u_p, as \({p\rightarrow 1^+}\), in the following sense: when \({h(\Omega) > 1}\), one has \({\lim_{p\rightarrow 1^+}\left\|u_{p}\right\| _{L^\infty(\Omega)}=0}\), and when \({h(\Omega) < 1}\), one has \({\lim_{p\rightarrow 1^+}\left\|u_p\right\| _{L^\infty(\Omega)}=\infty}\). In the case \({h(\Omega)=1}\), it is proved that \({\limsup_{p\rightarrow1^+}\left\|u_p\right\|_{L^\infty(\Omega)}<\infty}\). For a radial annulus \({\Omega_{a,b}}\), with inner radius a and outer radius b, it is proved that \({\lim_{p\rightarrow 1^+}\left\|u_p\right\| _{L^\infty(\Omega_{a,b})}=0}\) when \({h(\Omega_{a,b})=1}\).     

Lifting Fixed Points of Completely Positive Semigroups

Let \({\{\phi_s\}_{s\in S}}\) be a commutative semigroup of completely positive, contractive, and weak*-continuous linear maps acting on a von Neumann algebra N. Assume there exists a semigroup \({\{\alpha_s\}_{s\in S}}\) of weak*-continuous *-endomorphisms of some larger von Neumann algebra \({M\supset N}\) and a projection \({p\in M}\) with N = pMp such that α _s (1 ? p) ≤ 1 ? p for every \({s\in S}\) and \({\phi_s(y)=p\alpha_s(y)p}\) for all \({y\in N}\). If \({\inf_{s \in S}\alpha_s(1-p)=0}\) then we show that the map \({E:M\to N}\) defined by E(x) = pxp for \({x\in M}\) induces a complete isometry between the fixed point spaces of \({\{\alpha_s\}_{s\in S}}\) and \({\{\phi_s\}_{s\in S}}\).     

Schatten-Herz Operators,Berezin Transform and Mixed Norm Spaces

For p, q > 0 we study operators T on the Bergman space \({A_{2}(\mathbb{D)}}\) in the disk such that \({\left(\sum_{j}\Vert T\Delta_{j}\Vert_{p}^{q}\right)^{1/q}<\infty,}\) where the norms \({\Vert\cdot\Vert_{p}}\) are in the Schatten class S _p (A ₂), the projection \({\Delta_{j}f=\sum_{n\in I_{j}}a_{n}z^{n}}\) for \({f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}}\) and \({I_{j}=[2^{j}-1,2^{j+1} )\cap(\mathbb{N}\cup\{0\})}\) for \({j\in\mathbb{N}\cup\{0\}.}\) We consider the relation of this property with mixed norms of the Berezin transform of T and of the related function \({f_{T}(z)={\Vert}T(k_{z})\Vert}\) where k _z is the normalized Bergman kernel. These classes of operators denoted by S(p, q) are closely related when assumed to be positive with other sets of operators, like the class of positive operators on A ₂ for which \({\left(\sum_{j\geq0}(\sum_{n\in I_{j}}|\left\langle T^pe_{n},e_{n}\right\rangle |)^{q/p}\right)^{1/q}<\infty}\) , where \({\{e_{n}\}_{n\geq0}}\) is the canonical basis of A ₂; also we study the relation of Toeplitz operators in S(p, q) with the Schatten-Herz classes, where the decomposition is through dyadic annuli of the domain \({\mathbb{D}}\) .     

Boundedness of Sublinear Operators and Commutators on Generalized Morrey Spaces

In this paper the authors study the boundedness for a large class of sublinear operators \({T_{\alpha}, \alpha \in [0,n)}\) generated by Calderón–Zygmund operators (α = 0) and generated by Riesz potential operator (α > 0) on generalized Morrey spaces \({M_{p,\varphi}}\) . As an application of the above result, the boundeness of the commutator of sublinear operators \({T_{b,\alpha}, \alpha \in [0,n)}\) on generalized Morrey spaces is also obtained. In the case \({b \in BMO}\) and T _b,α is a sublinear operator, we find the sufficient conditions on the pair \({(\varphi_1,\varphi_2)}\) which ensures the boundedness of the operators \({T_{b,\alpha}, \alpha \in [0,n)}\) from one generalized Morrey space \({M_{p,\varphi_1}}\) to another \({M_{q,\varphi_2}}\) with 1/p ? 1/q = α/n. In all the cases the conditions for the boundedness are given in terms of Zygmund-type integral inequalities on \({(\varphi_1,\varphi_2)}\) , which do not assume any assumption on monotonicity of \({\varphi_1, \, \varphi_2}\) in r. Conditions of these theorems are satisfied by many important operators in analysis, in particular, Littlewood–Paley operator, Marcinkiewicz operator and Bochner–Riesz operator.     

On complete sequences

A sequence A of nonnegative integers is called complete if all sufficiently large integers can be represented as the sum of distinct terms taken form A. For a sequence \({S=\{s_{1}, s_{2}, \dots\}}\) of positive integers and a positive real number α, let S _α denote the sequence \({\{\lfloor\alpha s_{1}\rfloor, \lfloor\alpha s_{2}\rfloor, \dots\}}\), where \({\lfloor x \rfloor}\) denotes the greatest integer not greater than x. Let \({{U_S = \{\alpha \mid S_\alpha} \, is complete\}}\). Hegyvári [6] proved that if \({\lim_{n\to\infty} (s_{n+1}-s_{n})=+ \infty}\), \({s_{n+1} < \gamma s_{n}}\) for all integers \({n \geqq n_{0}}\), where \({1 < \gamma < 2}\), and \({U_{S}\ne\emptyset}\), then \({\mu(U_{S}) > 0}\), where \({\mu(U_{S})}\) is the Lebesgue measure of U _S . Yong-Gao Chen and the first author [4] proved that, if \({s_{n+1} < \gamma s_{n}}\) for all integers \({n \geqq n_{0}}\), where \({1 < \gamma \leqq 7/4=1.75}\), then \({\mu(U_{S}) > 0}\). In this paper, we prove that the conclusion holds for \({1 < \gamma \leqq \sqrt[4]{13}=1.898\dots\;}\).     

Number fields in fibers: the geometrically abelian case with rational critical values

Let X be an algebraic curve over \({\mathbb {Q}}\) and \({t\in {\mathbb {Q}}(X)}\) a non-constant rational function such that \({{\mathbb {Q}}(X)\ne {\mathbb {Q}}(t)}\). For every \({ n \in {\mathbb {Z}}}\) pick \({P_ n \in X(\bar{{\mathbb {Q}}})}\) such that \({t(P_n)=n}\). We conjecture that, for large N, among the number fields \({\mathbb {Q}}(P_1), \ldots , {\mathbb {Q}}(P_N)\) there are at least cN distinct. We prove this conjecture in the special case when \(\bar{{\mathbb {Q}}}(X)/\bar{{\mathbb {Q}}}(t)\) is an abelian field extension and the critical values of t are all rational. This implies, in particular, that our conjecture follows from a more famous conjecture of Schinzel.     

Geometric and probabilistic analysis of convex bodies with unconditional structures,and associated spaces of operators

Let \({{\|\cdot\|}}\) be a norm on \({\mathbb{R}^n}\) and \({\|.\|_*}\) be the dual norm. If \({\|\cdot\|}\) has a normalized 1-symmetric basis \({\{e_i\}_{i=1}^n}\) then the following inequalities hold: for all \({x,y\in \mathbb{R}^n}\), \({\|x\|\cdot\|y\|_*\le \max(\|x\|_1\cdot\|y\|_\infty,\|x\|_\infty\cdot\|y\|_1)}\) and if the basis is only 1-unconditional and normalized then for all \({x \in \mathbb{R}^n}\) , \({\|x\|+\|x\|_{*}\leq \|x\|_1+\|x\|_\infty}\) . We consider other geometric generalizations and apply these results to get, as a special case, estimates on best random embeddings of k-dimensional Hilbert spaces in the spaces of nuclear operators \({{\mathcal N}(K,K)}\) of dimension n ², for all k = [λn ²] and 0 < λ < 1. We obtain universal upper bounds independent on the 1-symmetric norm \({\|.\|}\) for the products of pth moments

$\left( {\mathbb{E}} \left\|\sum_{i=1}^n f_i(\omega)\,e_i\right\|^p\cdot\, \mathbb {E} \left\|\sum_{i=1}^n f_i(\omega)\,e_i\right\|_*^p\right)^{1/p}$

for independent random variables {f _i (ω)}, and 1 ≤ p < ∞.

     

The Katchalski–Lewis transversal problem for regular polygons

If every k-membered subfamily of a family of plane convex bodies has a line transversal, then we say that this family has property T(k). We say that a family \({\mathcal{F}}\) has property \({T-m}\), if there exists a subfamily \({\mathcal{G} \subset \mathcal{F}}\) with \({|\mathcal{F} - \mathcal{G}| \le m}\) admitting a line transversal. Heppes [7] posed the problem whether there exists a convex body K in the plane such that if \({\mathcal{F}}\) is a finite T(3)-family of disjoint translates of K, then m = 3 is the smallest value for which \({\mathcal{F}}\) has property \({T-m}\). In this paper, we study this open problem in terms of finite T(3)-families of pairwise disjoint translates of a regular 2n-gon \({(n \ge 5)}\). We find out that, for \({5 \le n \le 34}\), the family has property \({T - 3}\) ; for \({n \ge 35}\), the family has property \({T - 2}\).     

Interlacing of zeros of Gegenbauer polynomials of non-consecutive degree from different sequences

A theorem due to Stieltjes’ states that if \({\{p_n\}_{n=0}^\infty}\) is any orthogonal sequence then, between any two consecutive zeros of p _k , there is at least one zero of p _n whenever k < n, a property called Stieltjes interlacing. We show that Stieltjes interlacing extends to the zeros of Gegenbauer polynomials \({C_{n+1}^{\lambda}}\) and \({C_{n-1}^{\lambda+t}}\), \({\lambda > -\frac 12}\), if 0 < t ≤ k + 1, and also to the zeros of \({C_{n+1}^{\lambda}}\) and \({C_{n-2}^{\lambda +k}}\) if \({k\in\{1,2,3\}}\). More generally, we prove that Stieltjes interlacing holds between the zeros of the kth derivative of \({C_{n}^{\lambda}}\) and the zeros of \({C_{n+1}^{\lambda}}\), \({k\in\{1,2,\dots,n-1\}}\) and we derive associated polynomials that play an analogous role to the de Boor–Saff polynomials in completing the interlacing process of the zeros.     

Spaces with normal weights and Hadamard gap series

We give a sufficient and necessary condition for an analytic function f(z) on the unit disc \({\mathbb{D}}\) with Hadamard gaps, that is, for \({f(z)=\sum_{k=1}^{\infty}a_kz^{n_k}}\) where \({n_{k+1}/n_k\geq\lambda >1 }\) for all \({k\in \mathbb{N}}\), to belong to the weighted-type space \({ H_\mu^{\infty}}\), under some condition posed on the weight function μ. We can define the corresponding little weighted-type space \({H_{\mu,0}^{\infty}}\) and give a criterion for functions to belong to it.     

On generalized Stanley sequences

Let \({\mathbb{N}}\) denote the set of all nonnegative integers. Let \({k \ge 3}\) be an integer and \({A_{0} = \{a_{1}, \dots, a_{t}\} (a_{1} < \cdots < a_{t})}\) be a nonnegative set which does not contain an arithmetic progression of length k. We denote \({A = \{a_{1}, a_{2}, \ldots{}\}}\) defined by the following greedy algorithm: if \({l \ge t}\) and \({a_{1}, \dots{}, a_{l}}\) have already been defined, then \({a_{l+1}}\) is the smallest integer \({a > a_{l}}\) such that \({\{a_{1}, \dots, a_{l}\} \cup \{a\}}\) also does not contain a k-term arithmetic progression. This sequence A is called the Stanley sequence of order k generated by A₀. We prove some results about various generalizations of the Stanley sequence.     

Local Inverse Scattering at a Fixed Energy for Radial Schrödinger Operators and Localization of the Regge Poles

We study inverse scattering problems at a fixed energy for radial Schrödinger operators on \({\mathbb{R}^n}\), \({n \geq 2}\). First, we consider the class \({\mathcal{A}}\) of potentials q(r) which can be extended analytically in \({\Re z \geq 0}\) such that \({\mid q(z)\mid \leq C \ (1+ \mid z \mid )^{-\rho}}\), \({\rho > \frac{3}{2}}\). If q and \({\tilde{q}}\) are two such potentials and if the corresponding phase shifts \({\delta_l}\) and \({\tilde{\delta}_l}\) are super-exponentially close, then \({q=\tilde{q}}\). Second, we study the class of potentials q(r) which can be split into q(r) = q ₁(r) + q ₂(r) such that q ₁(r) has compact support and \({q_2 (r) \in \mathcal{A}}\). If q and \({\tilde{q}}\) are two such potentials, we show that for any fixed \({a>0, {\delta_l - \tilde{\delta}_l \ = \ o \left(\frac{1}{l^{n-3}}\ \left({\frac{ae}{2l}}\right)^{2l}\right)}}\) when \({l \rightarrow +\infty}\) if and only if \({q(r)=\tilde{q}(r)}\) for almost all \({r \geq a}\). The proofs are close in spirit with the celebrated Borg–Marchenko uniqueness theorem, and rely heavily on the localization of the Regge poles that could be defined as the resonances in the complexified angular momentum plane. We show that for a non-zero super-exponentially decreasing potential, the number of Regge poles is always infinite and moreover, the Regge poles are not contained in any vertical strip in the right-half plane. For potentials with compact support, we are able to give explicitly their asymptotics. At last, for potentials which can be extended analytically in \({\Re z \geq 0}\) with \({\mid q(z)\mid \leq C (1+ \mid z \mid)^{-\rho}}\), \({\rho >1}\), we show that the Regge poles are confined in a vertical strip in the complex plane.     

The Representation of a C0-Semigroup of Linear Operators in a Banach Space on the Set of Entire Vectors of its Generator

For a C₀-semigroup \({\{U(t)\}_{t \geq 0}}\) of linear operators in a Banach space \({{\mathfrak{B}}}\) with generator A, we describe the set of elements \({x \in {\mathfrak{B}}}\) whose orbits U(t)x can be extended to entire \({{\mathfrak{B}}}\)-valued functions of a finite order and a finite type, and establish the conditions under which this set is dense in \({{\mathfrak{B}}}\). The Hille problem of finding vectors \({x \in {\mathfrak{B}}}\) such that there exists the limit \({\lim\limits_{n \to \infty}\left(I + \frac{tA}{n}\right)^{n}x}\) is also solved in the paper. We prove that this limit exists if and only if x is an entire vector of the operator A, and if this is the case, then it coincides with U(t)x.     

Strichartz estimates via the Schrödinger maximal operator

We consider the Schrödinger operator \({e^{it\Delta}}\) acting on initial data f in \({\dot{H}^s}\). We show that an affirmative answer to a question of Carleson, concerning the sharp range of s for which \({\lim_{t\to 0}e^{it\Delta}f(x)=f(x)}\) a.e. \({x\in \mathbb {R}^n}\), would imply an affirmative answer to a question of Planchon, concerning the sharp range of q and r for which \({e^{it\Delta}}\) is bounded in \({L_x^q(\mathbb {R}^n,L^r_t(\mathbb {R}))}\). When n  =  2, we unconditionally improve the range for which the mixed norm estimates hold.     

Disposition p-groups

For any prime p and positive integers c, d there is up to isomorphism a unique p-group \({G_{d}^{c}(p)}\) of least order having any (finite) p-group G with rank \({d(G) \le d}\) and Frattini class \({c_{p}(G) \le c}\) as epimorphic image. Here \({c_{p}(G) = n}\) is the least positive integer such that G has a central series of length n with all factors being elementary. This “disposition” p-group \({G_{d}^{c}(p)}\) has been examined quite intensively in the literature, sometimes controversially. The objective of this paper is to present a summary of the known facts, and to add some new results. For instance we show that for \({G = G_{d}^{c}(p)}\) the centralizer \({C_{G}(x) = \langle Z(G), x \rangle}\) whenever \({x \in G}\) is outside the Frattini subgroup, and that for odd p and \({d \ge 2}\) the group \({E = G_{d}^{c+1}(p)/(G_{d}^{c+1}(p))^{p^{c}}}\) is a distinguished Schur cover of G with \({E/Z(E) \cong G}\). We also have a fibre product construction of \({G_{d}^{c+1}(p)}\) in terms of \({G = G_{d}^{c}(p)}\) which might be of interest for Galois theory.     

Min-max and robust polynomial optimization

We consider the robust (or min-max) optimization problem

$J^*:=\max_{\mathbf{y}\in{\Omega}}\min_{\mathbf{x}}\{f(\mathbf{x},\mathbf{y}): (\mathbf{x},\mathbf{y})\in\mathbf{\Delta}\}$

where f is a polynomial and \({\mathbf{\Delta}\subset\mathbb{R}^n\times\mathbb{R}^p}\) as well as \({{\Omega}\subset\mathbb{R}^p}\) are compact basic semi-algebraic sets. We first provide a sequence of polynomial lower approximations \({(J_i)\subset\mathbb{R}[\mathbf{y}]}\) of the optimal value function \({J(\mathbf{y}):=\min_\mathbf{x}\{f(\mathbf{x},\mathbf{y}): (\mathbf{x},\mathbf{y})\in \mathbf{\Delta}\}}\). The polynomial \({J_i\in\mathbb{R}[\mathbf{y}]}\) is obtained from an optimal (or nearly optimal) solution of a semidefinite program, the ith in the “joint + marginal” hierarchy of semidefinite relaxations associated with the parametric optimization problem \({\mathbf{y}\mapsto J(\mathbf{y})}\), recently proposed in Lasserre (SIAM J Optim 20, 1995-2022, 2010). Then for fixed i, we consider the polynomial optimization problem \({J^*_i:=\max\nolimits_{\mathbf{y}}\{J_i(\mathbf{y}):\mathbf{y}\in{\Omega}\}}\) and prove that \({\hat{J}^*_i(:=\displaystyle\max\nolimits_{\ell=1,\ldots,i}J^*_\ell)}\) converges to J* as i → ∞. Finally, for fixed ? ≤ i, each \({J^*_\ell}\) (and hence \({\hat{J}^*_i}\)) can be approximated by solving a hierarchy of semidefinite relaxations as already described in Lasserre (SIAM J Optim 11, 796–817, 2001; Moments, Positive Polynomials and Their Applications. Imperial College Press, London 2009).