On the positive definiteness of some functions related to the Schoenberg problem

For a broad class of functions f: [0,+∞) → ?, we prove that the function f(ρ ^λ(x)) is positive definite on a nontrivial real linear space E if and only if 0 ≤ λ ≤ α(E, ρ). Here ρ is a nonnegative homogeneous function on E such that ρ(x) ? 0 and α(E, ρ) is the Schoenberg constant.     

Uniform,on the entire axis,convergence of the spectral expansion for Schrödinger operator with a potential-distribution

A uniform, on ?, estimate for the increment of the spectral function θ(λ; x, y) at x = y is proved for the self-adjoint Schrödinger operator A defined on the entire axis ? by the differential operation (?d/dx)² + q(x) with a potential-distribution q(x) that uniformly locally belongs to the space W ₂ ^?1. As a consequence, it is shown that for any function f(x) from the domain of power Aα of the operator with α > 1/4, the spectral expansion of the function that corresponds to the operator A is convergent absolutely and uniformly on the entire axis ?.     

Compactness of Riesz transform commutator associated with Bessel operators

Let λ > 0 and

$${\Delta _\lambda }: = - \frac{{{d^2}}}{{d{x^2}}} - \frac{{2\lambda }}{x}\frac{d}{{dx}}$$

be the Bessel operator on R₊:= (0,∞). We first introduce and obtain an equivalent characterization of CMO(R₊, x^2λdx). By this equivalent characterization and by establishing a new version of the Fréchet-Kolmogorov theorem in the Bessel setting, we further prove that a function b ∈ BMO(R⁺, x^2λdx) is in CMO(R⁺, x^2λdx) if and only if the Riesz transform commutator xxxx is compact on L^p(R⁺, x^2λdx) for all p ∈ (1,∞).

     

bmo _ρ(ω) Spaces and Riesz transforms associated to Schrdinger operators

We introduce the BMO-type space bmoρ(ω) and establish the duality between h_ρ~1(ω) and bmo _ρ(ω),where ω∈A_1~(ρ, ∞)(R~n) and ω's locally behave as Muckenhoupt's weights but actually include them. We also give the Fefferman-Stein type decomposition of bmoρ(ω) with respect to Riesz transforms associated to Schrdinger operator L, where L =-? + V is a Schrdinger operator on R~n(n≥3) and V is a non-negative function satisfying the reverse Hlder inequality.     

Spectral asymptotics of a nonself-adjoint elliptic differential operator with an indefinite weight function

In the present paper, we compute the leading term of the asymptotics of the angular eigenvalue distribution function of the problem Au = λω(x)u(x) in a bounded domain Ω ? R ⁿ , where A is an elliptic differential operator of order 2m with domain D(A) ? W _m ^2m (Ω). The weight function ω(x) (x ∈ Ω) is indefinite and can also take zero values on a set of positive measure.     

On the equiconvergence rate of trigonometric series expansions and eigenfunction expansions for the Sturm-Liouville operator with a distributional potential

We study the Sturm-Liouville operator L = ?d ²/dx ² + q(x) in the space L ₂[0, π] with the Dirichlet boundary conditions. We assume that the potential has the form q(x) = u′(x), u ∈ W ₂ ^θ [0, π], 0 < θ < 1/2. We consider the problem on the uniform (on the entire interval [0, π]) equiconvergence of the expansion of a function f(x) in a series in the system of root functions of the operator L with its Fourier expansion in the system of sines. We show that if the antiderivative u(x) of the potential belongs to any of the spaces W ₂ ^θ [0, π], 0 < θ < 1/2, then the equiconvergence rate can be estimated uniformly over the ball u(x) ∈ B _R = {v(x) ∈ W ₂ ^θ [0, π] | ∥v∥_{W 2} ^θ ≤ R} for any function f(x) ∈ L ₂[0, π].     

On Extraction of Smooth Solutions of a Class of Almost Hypoelliptic Equations with Constant Power

A linear differential operator P(x, D) = P(x₁,... x _n , D₁,..., D _n ) = ∑_αγ_α(x)D^α with coefficients γ_α(x) defined in E ⁿ is called formally almost hypoelliptic in E ⁿ if all the derivatives D^ν_ξP(x, ξ) can be estimated by P(x, ξ), and the operator P(x, D) has uniformly constant power in Eⁿ. In the present paper, we prove that if P(x, D) is a formally almost hypoelliptic operator, then all solutions of equation P(x, D)u = 0, which together with some of their derivatives are square integrable with a specified exponential weight, are infinitely differentiable functions.     

Bessel Property of the System of Root Functions of a Second-Order Singular Operator on an Interval

For the system of root functions of an operator defined by the differential operation ?u″ + p(x)u′ + q(x)u, x ∈ G = (0, 1), with complex-valued singular coefficients, sufficient conditions for the Bessel property in the space L₂(G) are obtained and a theorem on the unconditional basis property is proved. It is assumed that the functions p(x) and q(x) locally belong to the spaces L₂ and W₂^?1, respectively, and may have singularities at the endpoints of G such that q(x) = q_R(x) +q′S(x) and the functions q_S(x), p(x), q ₂ ^S (x)w(x), p²(x)w(x), and q_R(x)w(x) are integrable on the whole interval G, where w(x) = x(1 ? x).     

Equicontinuity of maps on a dendrite with finite branch points

Let(T, d) be a dendrite with finite branch points and f be a continuous map from T to T. Denote byω(x,f) and P(f) the ω-limit set of x under f and the set of periodic points of,respectively. Write Ω(x,f) = {y| there exist a sequence of points x_k E T and a sequence of positive integers n_1 n_2 … such that lim_(k→∞)x_k=x and lim_(k→∞)f~(n_k)(x_k) =y}. In this paper, we show that the following statements are equivalent:(1) f is equicontinuous.(2) ω(x, f) = Ω(x,f) for any x∈T.(3) ∩_(n=1)~∞f~n(T) = P(f),and ω(x,f)is a periodic orbit for every x ∈ T and map h : x→ω(x,f)(x ET)is continuous.(4) Ω(x,f) is a periodic orbit for any x∈T.     

Equiconvergence of expansions in multiple Fourier series and in fourier integrals with “lacunary sequences of partial sums”

We investigate the equiconvergence on T^N = [?π, π)^N of expansions in multiple trigonometric Fourier series and in the Fourier integrals of functions f ∈ L_p(T^N) and g ∈ L_p(R^N), p > 1, N ≥ 3, g(x) = f(x) on T^N, in the case where the “partial sums” of these expansions, i.e., S_n(x; f) and J_α(x; g), respectively, have “numbers” n ∈ Z^N and α ∈ R^N (n_j = [α_j], j = 1,..., N, [t] is the integral part of t ∈ R¹) containing N ? 1 components which are elements of “lacunary sequences.”     

On the smoothness of the conjugacy between circle maps with a break

For any α ∈ (0, 1), c ∈ ?₊ \ {1} and γ > 0 and for Lebesgue almost all irrational ρ ∈ (0, 1), any two C ^2+α -smooth circle diffeomorphisms with a break, with the same rotation number ρ and the same size of the breaks c, are conjugate to each other via a C ¹-smooth conjugacy whose derivative is uniformly continuous with modulus of continuity ω(x) = A|log x|^?γ for some A > 0.     

On multiple zeros of a partial theta function

We consider the partial theta function θ(q, x) := ∑_j=0^∞q^j(j+1)/2x^j, where x ∈ ? is a variable and q ∈ ?, 0 < |q| < 1, is a parameter. We show that, for any fixed q, if ζ is a multiple zero of the function θ(q, · ), then |ζ| ≤ 8¹¹.     

Dirac Operator with a Potential of Special Form and with the Periodic Boundary Conditions

We consider the Dirac operator on the interval [0, 1] with the periodic boundary conditions and with a continuous potential Q(x) whose diagonal is zero and which satisfies the condition Q(x) = Q^T(1?x), x ∈ [0, 1]. We establish a relationship between the spectrum of this operator and the spectra of related functional-differential operators with involution. We prove that the system of eigenfunctions of this Dirac operator has the Riesz basis property in the space L ₂ ² [0, 1].     

The Fixed Points of Contractions of <Emphasis Type="Italic">f</Emphasis>-Quasimetric Spaces

The recent articles of Arutyunov and Greshnov extend the Banach and Hadler Fixed-Point Theorems and the Arutyunov Coincidence-Point Theorem to the mappings of (q₁, q₂)-quasimetric spaces. This article addresses similar questions for f-quasimetric spaces.Given a function f: R ₊² → R₊ with f(r₁, r₂) → 0 as (r₁, r₂) → (0, 0), an f-quasimetric space is a nonempty set X with a possibly asymmetric distance function ρ: X² → R+ satisfying the f-triangle inequality: ρ(x, z) ≤ f(ρ(x, y), ρ(y, z)) for x, y, z ∈ X. We extend the Banach Contraction Mapping Principle, as well as Krasnoselskii’s and Browder’s Theorems on generalized contractions, to mappings of f-quasimetric spaces.     

An indefinite concave-convex equation under a Neumann boundary condition I

We investigate the problem (P _λ) ?Δu = λb(x)|u| ^q?2 u + a(x)|u| ^p?2 u in Ω, ?u/?n = 0 on ?Ω, where Ω is a bounded smooth domain in R ^N (N ≥ 2), 1 < q < 2 < p, λ ∈ R, and a, b ∈ \({C^\alpha }\left( {\overline \Omega } \right)\) with 0 < α < 1. Under certain indefinite type conditions on a and b, we prove the existence of two nontrivial nonnegative solutions for small |λ|. We then characterize the asymptotic profiles of these solutions as λ → 0, which in some cases implies the positivity and ordering of these solutions. In addition, this asymptotic analysis suggests the existence of a loop type component in the non-negative solutions set. We prove the existence of such a component in certain cases, via a bifurcation and a topological analysis of a regularized version of (P _λ).     

Power-Commuting Generalized Skew Derivations in Prime Rings

Let R be a non-commutative prime ring of characteristic different from 2 with extended centroid C, F ≠ 0 a generalized skew derivation of R, and n ≥ 1 such that [F(x), x] ⁿ  = 0, for all x ∈ R. Then there exists an element λ ∈ C such that F(x) = λx, for all x ∈ R.     

On perturbation theory for points of discrete spectrum of dissipative operator functions

We consider the operator function L(α, θ) = A(α) ? θR of two complex arguments, where A(α) is an analytic operator function defined in some neighborhood of a real point α ₀ ∈ ? and ranging in the space of bounded operators in a Hilbert space ?. We assume that A(α) is a dissipative operator for real α in a small neighborhood of the point α ₀ and R is a bounded positive operator; moreover, the point α ₀ is a normal eigenvalue of the operator function L(α, θ ₀) for some θ ₀ ∈ ?, and the number θ ₀ is a normal eigenvalue of the operator function L(α ₀ θ). We obtain analogs and generalizations of well-known results for self-adjoint operator functions A(α) on the decomposition of α- and θ-eigenvalues in neighborhoods of the points α ₀ and θ ₀, respectively, and on the representation of the corresponding eigenfunctions by series.     

Existence of positive solutions for a general nonlinear eigenvalue problem

Let Ω R n be a bounded domain, H = L 2 (Ω), L : D(L) H → H be an unbounded linear operator, f ∈ C(■× R, R) and λ∈ R. The paper is concerned with the existence of positive solutions for the following nonlinear eigenvalue problem Lu = λf (x, u), u ∈ D(L), which is the general form of nonlinear eigenvalue problems for differential equations. We obtain the global structure of positive solutions, then we apply the results to some nonlinear eigenvalue problems for a second-order ordinary differential equation and a fourth-order beam equation, respectively. The discussion is based on the fixed point index theory in cones.     

A mapping method in inverse Sturm-Liouville problems with singular potentials

In the space L ₂[0, π], the Sturm-Liouville operator L _D(y) = ?y″ + q(x)y with the Dirichlet boundary conditions y(0) = y(π) = 0 is analyzed. The potential q is assumed to be singular; namely, q = σ′, where σ ∈ L ₂[0, π], i.e., q ∈ W ₂ ^?1 [0, π]. The inverse problem of reconstructing the function σ from the spectrum of the operator L _D is solved in the subspace of odd real functions σ(π/2 ? x) = ?σ(π/2 + x). The existence and uniqueness of a solution to this inverse problem is proved. A method is proposed that allows one to solve this problem numerically.     

Riesz basis property of system of root functions of second-order differential operator with involution

The properties of the root functions are studied for an arbitrary operator generated in L ₂(?1, 1) by the operation with involution of the form Lu = ?u″(x)+αu″(?x)+q(x)u(x)+ qν(x)u(ν(x)), where α ∈ (?1, 1), ν(x) is an absolutely continuous involution of the segment [?1, 1] and the coefficients q(x) and qν(x) are summable functions on (?1, 1). Necessary and sufficient conditions are obtained for the unconditional basis property in L ₂(?1, 1) for the system of the root functions of the operator.