Bounds for singular integrals associated with a class of hypersurfaces

For the hypersurface Γ=(y,γ(y)), the singular integral operator along Γ is defined by. $$Tf(x,x_n ) = P.V.\int_{\mathbb{R}^n } {, f(x - y,x_n ) - } \gamma (y))_{\left| y \right|^{n - 1} }^{\Omega (v)} dy$$ where Σ is homogeneous of order 0, $ \int_{\Sigma _{n \lambda } } {\Omega (y')dy'} = 0 $ . For a certain class of hypersurfaces, T is shown to be bounded on L^p(Rⁿ) provided Ω∈L _α ¹ (Σ _n?2),P>1.     

Asymptotic representations of solutions of second-order differential equations

We establish asymptotic representations as t → ω (ω ≤ + ∞) of a class of monotone solutions of the second-order differential equation y″ = f(t, y, y′), where f:[a,ω[× Δ _Y0 × Δ _Y1 is a continuous function asymptotically close on the considered class of solutions to a function of the form ±p(t)φ ₀(y)φ ₁(y′) with functions φ ₀ and φ ₁ regularly varying as y → Y ₀ and y′ → Y ₁. Here Δ _Yi , i ∈ {0, 1}, is a one-sided neighborhood of Y _i , and Y _i is either zero or ±∞.     

Generalized weak peripheral multiplicativity in algebras of Lipschitz functions

Let (X, d _X ) and (Y,d _Y ) be pointed compact metric spaces with distinguished base points e _X and e _Y . The Banach algebra of all $\mathbb{K}$ -valued Lipschitz functions on X — where $\mathbb{K}$ is either?or ? — that map the base point e _X to 0 is denoted by Lip₀(X). The peripheral range of a function f ∈ Lip₀(X) is the set Ran_µ(f) = {f(x): |f(x)| = ‖f‖_∞} of range values of maximum modulus. We prove that if T ₁, T ₂: Lip₀(X) → Lip₀(Y) and S ₁, S ₂: Lip₀(X) → Lip₀(X) are surjective mappings such that $Ran_\pi (T_1 (f)T_2 (g)) \cap Ran_\pi (S_1 (f)S_2 (g)) \ne \emptyset $ for all f, g ∈ Lip₀(X), then there are mappings φ₁φ₂: Y → $\mathbb{K}$ with φ₁(y)φ₂(y) = 1 for all y ∈ Y and a base point-preserving Lipschitz homeomorphism ψ: Y → X such that T _j (f)(y) = φ _j (y)S _j (f)(ψ(y)) for all f ∈ Lip₀(X), y ∈ Y, and j = 1, 2. In particular, if S ₁ and S ₂ are identity functions, then T ₁ and T ₂ are weighted composition operators.     

Schrödinger operators with highly singular oscillating potentials

We investigate the Feynman-Kac semigroupP _t ^V and its densityp ^V(t,.,.),t>0, associated with the Schrödinger operator ?1/2Δ+V on ?^d\{0}.V will be a highly singular, oscillating potential like $V\left( x \right) = k \cdot \left\| x \right\|^{ - 1} \cdot \sin \left( {\left\| x \right\|^{ - m} } \right)$ with arbitraryk, l, m>0. We derive conditions (onk,l,m) which are sufficientand necessary for the existence of constants α, β, γ, ∈ ? such that for allt, x, y p ^V(t, x, y)≤γ·p(βt, x, y)·e^at. On the other hand, also conditions are derived which imply thatp ^V (t, x, y)≡∞ for allt, x, y. The aim is to see to which extent quick oscillations can lead to annihilations of the singularities ofV. For this purpose, we analyse the above example in great detail. Note that forl≥2 the potential is so singular that none of the usual perturbation techniques applies.     

Real parabolic vector bundles over a real curve

We define real parabolic structures on real vector bundles over a real curve. Let (X, σ _X ) be a real curve, and let S???X be a non-empty finite subset of X such that σ _X (S)?=?S. Let N?≥?2 be an integer. We construct an N-fold cyclic cover p : Y→ X in the category of real curves, ramified precisely over each point of S, and with the property that for any element g of the Galois group Γ, and any y?∈?Y, one has $\sigma_Y(gy) = g^{-1}\sigma_Y(y)$ . We established an equivalence between the category of real parabolic vector bundles on (X, σ _X ) with real parabolic structure over S, all of whose weights are integral multiples of 1/N, and the category of real Γ-equivariant vector bundles on (Y, σ _Y ).     

On Parameter Depending Differential Systems Under Carathéodory Conditions

For every λ in a complex domain G, consider on some interval I the initial value problem y′(λ,x) = A(λ,x)y(λ,x) + b(λ,x), y(λ,x₀) - y₀. If this problem satisfies the Carathéodory conditions for every A, then there exist locally absolutely continuous and almost everywhere differentiable solutions y(λ,· ) of the initial value problem. In general, the union N of the exceptional sets N _λ ? I where y(λ, ·) is not differentiate or does not fulfill the differential equation, is not of Lebesgue measure zero. It will be shown that N is of Lebesgue measure zero provided that A and b are holomorphic with respect to λ and their integrals with respect to x are locally bounded on G × I.     

Ruin problems with compounding assets

We consider a generalization of the classical model of collective risk theory. It is assumed that the cumulative income of a firm is given by a process X with stationary independent increments, and that interest is earned continuously on the firm's assets. Then Y(t), the assets of the firm at time t, can be represented by a simple path-wise integral with respect to the income process X. A general characterization is obtained for the probability r(y) that assets will ever fall to zero when the initial asset level is y (the probability of ruin). From this we obtain a general upper bound for r(y), a general solution for the case where X has no negative jumps, and explicit formulas for three particular examples.In addition, an approximation theorem is proved using the weak convergence theory for stochastic processes. This shows that if the income process is well approximated by Brownian motion with drift, then the assets process Y is well approximated by a certain diffusion process Y₁, and r(y) is well approximated by a corresponding first passage probability r₁(y). The diffusion Y₁, which we call compounding Brownian motion, is closely related to the classical Ornstein-Uhlenbeck process.     

An integral operator connected with the Helmholtz equation

Let Lu be the integral operator defined by $\text{(L}{}_{\mathrm{k?}}\text{)(x, y) = ∝ s ∝ ?(x′, y′)(}\text{e}{}^{\mathrm{ik?}}\text{?}\text{) dx′ dy′, (x, y) ? S}$ where S is the interior of a smooth, closed Jordan curve in the plane, k is a complex number with Re k ? 0, Im k ? 0, and ?² = (x ?x′)² + (y ? y′)². We define $\text{q(x, y) = [}\text{dist}\text{((x, y), ?S)]}\text{1}\text{2}\text{, (x, y) ? S; L}{}_2\text{(q, S) = {? : ∝ s ∝ ¦ ?(x, y)¦}{}^2\text{q(x, y) dx dy < ∞}; W}{}_2{}^1\text{(q, S) = {? : ? ? L}{}_2\text{(q, S),}\text{??}\text{?x}\text{,}\text{?f}\text{?y}\text{? L}{}_2\text{(q, S)}}$, where in the definition of W₂¹(q, S) the derivatives are taken in the sense of distributions. We prove that L_k is a continuous 1-l mapping of L₂(q, S) onto W₂¹(q, S).     

Krein’s Theorems for a Dissipative Boundary Value Transmission Problem

Let us consider the differential expression $$\ell (y)=-y^{\prime \prime }+q(x)y,\quad x\in I:=[0,c)\cup (c,\infty ),$$ where c is a transmission point and is regular for the differential expression ?(y). We assume that Weyl’s limit-circle case holds for the differential expression ?(y) on I. In this paper, using Krein’s theorems, we investigate the completeness of the root vectors of a singular dissipative boundary value transmission problem generated by ?(y).     

On solvability of derivative dependent doubly singular boundary value problems

In this paper we establish existence of solutions of singular boundary value problem ?(p(x)y ^′(x))^′=q(x)f(x,y,py′) for 0<x≤b and $\lim_{x\rightarrow0^{+}}p(x)y^{\prime}(x)=0$ , α ₁ y(b)+β ₁ p(b)y ^′(b)=γ ₁ with p(0)=0 and q(x) is allowed to have integrable discontinuity at x=0. So the problem may be doubly singular. Here we consider $\lim_{x\rightarrow0^{+}}\frac{q(x)}{p'(x)}\neq0$ therefore $\lim_{x\rightarrow0^{+}}p(x)y'(x)=0$ does not imply y′(0)=0 unless $\lim_{x\rightarrow0^{+}}f(x,y(x),p(x)y'(x))=0$ .     

Uniform Estimates of Solutions of a Nonlinear Third-Order Differential Equation

One considers the differential equation $$y{\prime\prime\prime}(x) + p\bigl(x, y(x), y{\prime}(x), y{\prime\prime}(x)\bigr) |y(x)|^{k-1} y(x) = 0,$$ where k?>?1, the function p(x, y ₀ , y ₁ , y ₂) is continuous and satisfies the inequalities $$ 0 < p_* \le p(x, y_0, y_1, y_2) \le p^* < \infty,$$ as well as the Lipschitz condition with respect to the last three arguments. Uniform estimates are obtained for the moduli of the solutions with a common domain.     

Analytic integrability and characterization of centers for nilpotent singular points

A method which provides necessary conditions to obtain a local analytic first integral in a neighborhood of a nilpotent singular point is developed. As an application we provide sufficient conditions in order that systems of the form where P_n and Q_n are homogeneous polynomials of degree n = 2, 3, 4, 5 have a local analytic first integral of the form H=y²+F(x, y), where F starts with terms of order higher than 2. We remark that, in general, the existence of such integral is only guaranteed when the singular point is a nilpotent center and the system has a formal first integral, see [6]. Therefore, we characterize the nilpotent centers of systems which have a local analytic first integral.     

Holomorphic Solutions of a Functional Equation Related to Nonlinear Difference Systems

Here we consider the following functional equation, $$\Psi(X(x,\Psi(x)))=Y(x, \Psi(x)),$$ where X(x, y) and Y(x, y) are holomorphic functions in |x| < δ ₁, |y| < δ ₁. When we consider a nonlinear simultaneous system of two variables difference equations, we can reduce it to a single difference equation of first order by a solution Ψ of the above functional equation. We obtain a matrix by the linear terms of functions X and Y. When the all eigenvalues of the matrix are equal to 1, it is difficult to have a solution of the above functional equation. In the present paper, we derive a formal solution of the above functional equation under the condition. Further we prove the existence of a solution which is holomorphic and have an asymptotically expansion of the formal solution. Moreover, we will show an example of nonlinear difference system such that our results are applicable.     

Weakly Regular Semigroups of Isotone Transformations

Let X be a partially ordered set and O(X) be the semigroup of all mappings X → X that preserve the order, i.e., x ≤ y ? xα ≤ yα for all x, y ∈ X. It is proved that the semigroup O(X) is weakly regular in the wide sense if and only if at least one of the following conditions holds: (1) X is a quasi-complete chain; (2) the elements of X are not comparable pairwise; (3) X = Y ∪ Z, where y < z for y ∈ Y, z ∈ Z; (4) X = Y ∪ Z, where y ₀ ∈ Y, z ₀ ∈ Z, and y ₀ < z for z ∈ Z, y < z0 for y ∈ Y; (5) X = {a, c} ∪ B, where a < b < c for b ∈ B; (6) X = {1, 2, 3, 4, 5, 6}, where 1 < 4, 1 < 5, 2 < 5, 2 < 6, 3 < 4, 3 < 6. Moreover, if X is a quasi-ordered set but not partially ordered, then the semigroup O(X) is weakly regular in the wide sense if and only if x ≤ y for all x, y ∈ X.     

A generalization of the Buchstab equation

Letf be a multiplicative function and letΨ _f (x, y) denote the incomplete multiplicative sum Σ _n≤x,P(n)≤y f(n), whereP(n) denotes the greatest prime factor ofn. A Buchstab- and a Hildebrand equation forΨ _f (x, y) are derived.     

Sharp estimates for the convergence rate of “hyperbolic” partial sums of double fourier series in orthogonal polynomials

Two-variable functions f(x, y) from the class L ₂ = L ₂((a, b) × (c, d); p(x)q(y)) with the weight p(x)q(y) and the norm $$\left\| f \right\| = \sqrt {\int\limits_a^b {\int\limits_c^d {p(x)q(x)f^2 (x,y)dxdy} } }$$ are approximated by an orthonormal system of orthogonal P _n (x)Q _n (y), n, m = 0, 1, ..., with weights p(x) and q(y). Let $$E_N (f) = \mathop {\inf }\limits_{P_N } \left\| {f - P_N } \right\|$$ denote the best approximation of f ?? L ₂ by algebraic polynomials of the form $$\begin{array}{*{20}c} {P_N (x,y) = \sum\limits_{0 < n,m < N} {a_{m,n} x^n y^m ,} } \\ {P_1 (x,y) = const.} \\ \end{array}$$ . Consider a double Fourier series of f ?? L ₂ in the polynomials P _n (x)Q _m (y), n, m = 0, 1, ..., and its ??hyperbolic?? partial sums $$\begin{array}{*{20}c} {S_1 (f;x,y) = c_{0,0} (f)P_o (x)Q_o (y),} \\ {S_N (f;x,y) = \sum\limits_{0 < n,m < N} {c_{n,m} (f)P_n (x)Q_m (y), N = 2,3, \ldots .} } \\ \end{array}$$ A generalized shift operator Fh and a kth-order generalized modulus of continuity ?? _k (A, h) of a function f ?? L ₂ are used to prove the following sharp estimate for the convergence rate of the approximation: $\begin{gathered} E_N (f) \leqslant (1 - (1 - h)^{2\sqrt N } )^{ - k} \Omega _k (f;h),h \in (0,1), \hfill \\ N = 4,5,...;k = 1,2,... \hfill \\ \end{gathered} $ . Moreover, for every fixed N = 4, 9, 16, ..., the constant on the right-hand side of this inequality is cannot be reduced.     

On the diophantine equation x(x + 1)(x + 2)…(x + (m − 1)) =g(y)

Let g(y) ? Q[Y] be an irreducible polynomial of degree n ≥ 3. We prove that there are only finitely many rational numbers x, y with bounded denominator and an integer m ≥ 3 satisfying the equation x(x + 1) (x + 2)…(x + (m − 1) ) = g(y). We also obtain certain finiteness results when g(y) is not an irreducible polynomial.     

On Heuvers’ Logarithmic Functional Equation

?(x + y) - ?(x) - ?(y) = ?(x ^?1 + y ^?l) are identical to those of the Cauchy equation ?(xy) = ?(x) + ?(y) when ? is a function from the positive real numbers into the reals. In the present article, we prove this equivalence for functions mapping the set of nonzero elements of a field (excluding ?₂) .     

Global center manifolds in singular systems

The problem of existence of aglobal center manifold for a system of O.D.E. like (*) $$\left\{ {\begin{array}{*{20}c} {\dot x = A(y)x + F(x,y)} \\ {\dot y = G(x,y), (x,y) \in \mathbb{R}^n \times \mathbb{R}^m ,} \\ \end{array} } \right.$$ is considered. We give conditions onA(y), F(x, y), G(x, y) in order that a functionH: ? ^m →? ⁿ , with the same smoothness asA(y), F(x, y), G(x, y), exists and is such that the manifoldC={(x,y)∈? ⁿ ×? ^m ∣x=H(y),y∈? ^m } is an invariant manifold for (*), and there exists ρ>0 such that any solution of (*) satisfying sup _t∈?∣x(t)∣ <ρ must belong toC. This is why we callC global center manifold. Applications are given to the problem of existence of heteroclinic orbits in singular systems.     

Backward stochastic differential equations and applications to optimal control

We study the existence and uniqueness of the following kind of backward stochastic differential equation, $$x(t) + \int_t^T {f(x(s),y(s),s)ds + \int_t^T {y(s)dW(s) = X,} }$$ under local Lipschitz condition, where (Ω, ?,P, W(·), ?_t) is a standard Wiener process, for any given (x, y),f(x, y, ·) is an ?_t-adapted process, andX is ?_t-measurable. The problem is to look for an adapted pair (x(·),y(·)) that solves the above equation. A generalized matrix Riccati equation of that type is also investigated. A new form of stochastic maximum principle is obtained.