Estimates of some integral operators with bounded variable kernels on the hardy and weak hardy spaces over ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper, we first introduce \({L^{{\sigma _1}}}{\left( {\log L} \right)^{{\sigma _2}}}\) conditions satisfied by the variable kernels Ω(x, z) for 0 ≤ σ ₁ ≤ 1 and σ ₂ ≥ 0. Under these new smoothness conditions, we will prove the boundedness properties of singular integral operators T _Ω, fractional integrals T _Ω,α and parametric Marcinkiewicz integrals μ _Ω ^ρ with variable kernels on the Hardy spaces H ^p (R ⁿ ) and weak Hardy spaces WH ^p (R ⁿ ). Moreover, by using the interpolation arguments, we can get some corresponding results for the above integral operators with variable kernels on Hardy–Lorentz spaces H ^p,q(R ⁿ ) for all p < q < ∞.     

On the Convergence of Solutions of Variational Problems with Implicit Pointwise Constraints in Variable Domains

Results on the convergence of minimizers and minimum values of integral and more general functionals J_s: W^1,p(Ω_s) → ? on the sets U_s(h_s) = {v ∈ W^1,p(Ω_s): h_s(v) ≤ 0 a.e. in Ω_s}, where p > 1, {Ω_s} is a sequence of domains contained in a bounded domain Ω of ?ⁿ (n > 2), and {h_s} is a sequence of functions on ?, are announced.     

A note on the Schrödinger maximal function

It is shown that control of the Schrödinger maximal function sup_{0 <t<1} ?e^itΔf? for f ∈ H^s(Rⁿ) requires s ≥ n/2(n + 1).     

Variational Problems with Unilateral Pointwise Functional Constraints in Variable Domains

We consider a sequence of convex integral functionals F_s: W¹,^p(Ω_s) → ? and a sequence of weakly lower semicontinuous and generally nonintegral functionals G_s: W¹,^p(Ω_s) → ?, where {Ω_s} is a sequence of domains in ?ⁿ contained in a bounded domain Ω ? ?ⁿ (n ≥ 2) and p > 1. Along with this, we consider a sequence of closed convex sets V_s = {v ∈ W¹,^p(Ω_s): v ≥ K_s(v) a.e. in Ω_s}, where K_s is a mapping from the space W¹,^p(Ω_s) to the set of all functions defined on Ω_s. We establish conditions under which minimizers and minimum values of the functionals F_s + G_s on the sets V_s converge to a minimizer and the minimum value of a functional on the set V = {v ∈ W¹,^p(Ω): v ≥ K(v) a.e. in Ω}, where K is a mapping from the space W¹,^p(Ω) to the set of all functions defined on Ω. These conditions include, in particular, the strong connectedness of the spaces W¹,^p(Ω_s) with the space W¹,^p(Ω), the condition of exhaustion of the domain Ω by the domains Ω_s, the Γ-convergence of the sequence {F_s} to a functional F: W¹,^p(Ω) → ?, and a certain convergence of the sequence {G_s} to a functional G: W¹,^p(Ω) → ?. We also assume some conditions characterizing both the internal properties of the mappings K_s and their relation to the mapping K. In particular, these conditions admit the study of variational problems with irregular varying unilateral obstacles and with varying constraints combining the pointwise dependence and the functional dependence of the integral form.     

Distribution of the spectrum of a singular positive Sturm–Liouville operator perturbed by the Dirac delta function

We consider the Sturm–Liouville operator generated in the space L ²[0,+∞) by the expression l _a,b:= ?d ²/dx ² +x+aδ(x?b) and the boundary condition y(0) = 0. We prove that the eigenvalues λ _n of this operator satisfy the inequalities λ₁ ⁰ < λ₁ < λ₂ ⁰ and λ_n ⁰ ≤ λ_n < λ_n+1 ⁰, n = 2, 3,..., where {?λ_n ⁰} is the sequence of zeros of the Airy function Ai (λ). We find the asymptotics of λ_n as n → +∞ depending on the parameters a and b.     

Triebel-Lizorkin space boundedness of rough singular integrals associated to surfaces of revolution

We consider the boundedness of the rough singular integral operator T_(?,ψ,h) along a surface of revolution on the Triebel-Lizorkin space F~α_( p,q)(R~n) for ? ∈ H~1(~(Sn-1)) and ? ∈ Llog~+L(S~(n-1)) ∪_1q∞(B~((0,0))_q(S~(n-1))), respectively.     

On Excess-time Correlated Cumulative Processes

In this paper a class of correlated cumulative processes, B _s (t) = ∑^N(t)_i=1 H _s (X _i )X _i , is studied with excess level increments X _i ?s, where {N(t), t ?0} is the counting process generated by the renewal sequence T _n , T _n and X _n are correlated for given n, H _s (t) is the Heaviside function and s?0 is a given constant. Several useful results, for the distributions of B _s (t), and that of the number of excess (non-excess) increments on (0, t) and the corresponding means, are derived. First passage time problems are also discussed and various asymptotic properties of the processes are obtained. Transform results, by applying a flexible form for the joint distribution of correlated pairs (T _n , X _n ) are derived and inverted. The case of non-excess level increments, X _i < s, is also considered. Finally, applications to known stochastic shock and pro-rata warranty models are given.     

A note on function spaces in rough domains

This paper deals with some function spaces B_p,p^s(Ω) in rough domains Ω in Rⁿ.     

Nonlinear trigonometric approximations of multivariate function classes

Order-sharp estimates are established for the best N-term approximations of functions from Nikol’skii–Besov type classes B_pq^sm(T^k) with respect to the multiple trigonometric system T^(k) in the metric of L_r(T^k) for a number of relations between the parameters s, p, q, r, and m (s = (s₁,..., s_n) ∈ R₊ⁿ, 1 ≤ p, q, r ≤ ∞, m = (m₁,..., m_n) ∈ Nⁿ, k = m₁ +... + m_n). Constructive methods of nonlinear trigonometric approximation—variants of the so-called greedy algorithms—are used in the proofs of upper estimates.     

On boundary value problems for systems of nonlinear generalized ordinary differential equations

A general theorem (principle of a priori boundedness) on solvability of the boundary value problem dx = dA(t) · f(t, x), h(x) = 0 is established, where f: [a, b]×R ⁿ → R ⁿ is a vector-function belonging to the Carathéodory class corresponding to the matrix-function A: [a, b] → R ^n×n with bounded total variation components, and h: BV_s([a, b],R ⁿ ) → R ⁿ is a continuous operator. Basing on the mentioned principle of a priori boundedness, effective criteria are obtained for the solvability of the system under the condition x(t₁(x)) = B(x) · x(t ₂(x))+c ₀, where t _i: BV_s([a, b],R ⁿ ) → [a, b] (i = 1, 2) and B: BV_s([a, b], R ⁿ ) → R ⁿ are continuous operators, and c ₀ ∈ R ⁿ .     

Asymptotic formula for the convolution of a generalized divisor function

An asymptotic formula is obtained for the sum of terms σ _it (n)σ_-it (N - n) (t is real) over 0 < n < N with a remainder estimated by O _ε((1+|t|)^1+ε N ^3/4+ε) for any ε > 0. As a consequence, Porter’s result on a power scale for the average number of steps in the Euclidean algorithm is improved.     

Parabolic equation with a singular potential

We study the existence of a nonnegative generalized solution of an initial-boundary value problem for the heat equation with a singular potential in an arbitrary bounded domain Ω ? R ⁿ , n ≥ 3, containing the unit ball. We show that if the condition ∝ _Ω V ^n/2+s |x| ^s dx ≤ c _n is satisfied for some s ≥ 0 and c _n = c _n (n, s, Ω) > 0, then the problem in question has a nonnegative solution.     

Numerical ranges for pairs of operators,duality mappings with gauge function,and spectra of nonlinear operators

We define and study numerical ranges for pairs of nonlinear operators F and J which act between some Banach space X and its dual X*, with respect to some increasing gauge function φ. Connections with spectra for certain classes of nonlinear operators introduced recently in the literature are also established. As a sample example, we consider the case when F is the duality map of the Lebesgue space L ^p (Ω), J is the duality map of the corresponding Sobolev space W ₀ ^1,p (Ω), and φ(t)=t ^p?1 (1<p<∞). This leads to existence, uniqueness, and perturbation results for a homogeneous eigenvalue problem involving the p-Laplace operator.     

Lower bounds for polynomials of many variables

A polynomial P(ξ) = P(ξ₁,..., ξ _n ) is said to be almost hypoelliptic if all its derivatives D ^ν P(ξ) can be estimated from above by P(ξ) (see [16]). By a theorem of Seidenberg-Tarski it follows that for each polynomial P(ξ) satisfying the condition P(ξ) > 0 for all ξ ∈ R ⁿ , there exist numbers σ > 0 and T ∈ R¹ such that P(ξ) ≥ σ(1 + |ξ|) ^T for all ξ ∈ R ⁿ . The greatest of numbers T satisfying this condition, denoted by ST(P), is called Seidenberg-Tarski number of polynomial P. It is known that if, in addition, P ∈ I _n , that is, |P(ξ)| → ∞ as |ξ| → ∞, then T = T(P) > 0. In this paper, for a class of almost hypoelliptic polynomials of n (≥ 2) variables we find a sufficient condition for ST(P) ≥ 1. Moreover, in the case n = 2, we prove that ST(P) ≥ 1 for any almost hypoelliptic polynomial P ∈ I₂.     

LIFTING DIFFERENTIABLE CURVES FROM ORBIT SPACES

Let ρ: G → O(V) be a real finite dimensional orthogonal representation of a compact Lie group, let σ = (σ ₁, ?, σn): V → ? ⁿ , where σ ₁, ?, σn n form a minimal system of homogeneous generators of the G-invariant polynomials on V, and set d = max_i deg σ _i . We prove that for each C ^d?1,1-curve c in σ(V) ?? ⁿ there exits a locally Lipschitz lift over σ, i.e., a locally Lipschitz curve \( \overline{c} \) in V so that c = σ ° \( \overline{c} \), and we obtain explicit bounds for the Lipschitz constant of \( \overline{c} \) in terms of c. Moreover, we show that each C ^d -curve in σ(V) admits a C ¹-lift. For finite groups G we deduce a multivariable version and some further results.     

On low discrepancy sequences and low discrepancy ergodic transformations of the multidimensional unit cube

In this paper we describe a third class of low discrepancy sequences. Using a lattice Γ ? ? ^s , we construct Kronecker-like and van der Corput-like ergodic transformations T _1,Γ and T _2,Γ of [0, 1) ^s . We prove that for admissible lattices Γ, (T _ν,Γ ⁿ (x))_n≥0 is a low discrepancy sequence for all x ∈ [0, 1) ^s and ν ∈ {1, 2}. We also prove that for an arbitrary polyhedron P ? [0, 1) ^s , for almost all lattices Γ ∈ L _s = SL(s,?)/SL(s, ?) (in the sense of the invariant measure on L _s ), the following asymptotic formula

$\# \{ 0 \le n < N:T_{v,\Gamma }^n(x) \in P\} = NvolP + O({(\ln N)^{s + \varepsilon }}),N \to \infty$

holds with arbitrary small ? > 0, for all x ∈ [0, 1) ^s , and ν ∈ {1, 2}.

     

Global well-posedness of the fractional Klein-Gordon-Schrödinger system with rough initial data

We investigate the low regularity local and global well-posedness of the Cauchy problem for the coupled Klein-Gordon-Schrödinger system with fractional Laplacian in the Schrödinger equation in R¹⁺¹. We use Bourgain space method to study this problem and prove that this system is locally well-posed for Schrödinger data in H^s1 and wave data in H^s2 ×H^s2?1 for 3/4?α < s₁ ≤ 0 and ?1/2 < s₂ < 3/2, where α is the fractional power of Laplacian which satisfies 3/4 < α ≤ 1. Based on this local well-posedness result, we also obtain the global well-posedness of this system for s₁ = 0 and ?1/2 < s₂ < 1/2 by using the conservation law for the L² norm of u.     

Non-triviality of a product in the Adams <Emphasis Type="Italic">E</Emphasis> <Subscript>2</Subscript>-term

Let p be a prime greater than five and A the mod p Steenrod algebra. In this paper, we prove that \(h_n h_m \tilde \delta _{s + 4} \in Ext_A^{s + 6,t(s,n,m) + s} (Z/p,Z/p)\) is nontrivial in the Adams E₂-term when m ≥ n + 2 ≥ 7 and 0 ≤ s < p ? 4, and trivial in the Adams E₂-term when m ≥ n + 2 = 6 and 0 ≤ s < p ? 4, where \(\tilde \delta _{s + 4} \) stands for the fourth Greek letter element and t(s, n, m) = 2(p ? 1)[(s + 1) + (s + 2)p + (s + 3)p² + (s + 4)p³ + pⁿ + p^m].     

Curvature inequalities for operators of the Cowen–Douglas class

Let T be an operator tuple in the Cowen–Douglas class B _n (Ω) for Ω ? C ^m . The kernels Ker(T ? w) ^l , for w ∈ Ω, l = 1, 2, ···, define Hermitian vector bundles E _T ^l over Ω. We prove certain negativity of the curvature of E _T ^l . We also study the relation between certain curvature inequality and the contractive property of T when Ω is a planar domain.     

Detailed error analysis for a fractional Adams method with graded meshes

We consider a fractional Adams method for solving the nonlinear fractional differential equation \(\,^{C}_{0}D^{\alpha }_{t} y(t) = f(t, y(t)), \, \alpha >0\), equipped with the initial conditions \(y^{(k)} (0) = y_{0}^{(k)}, k=0, 1, \dots , \lceil \alpha \rceil -1\). Here, α may be an arbitrary positive number and ?α? denotes the smallest integer no less than α and the differential operator is the Caputo derivative. Under the assumption \(\,^{C}_{0}D^{\alpha }_{t} y \in C^{2}[0, T]\), Diethelm et al. (Numer. Algor. 36, 31–52, 2004) introduced a fractional Adams method with the uniform meshes t _n = T(n/N),n = 0,1,2,…,N and proved that this method has the optimal convergence order uniformly in t _n , that is O(N ^?2) if α > 1 and O(N ^?1?α ) if α ≤ 1. They also showed that if \(\,^{C}_{0}D^{\alpha }_{t} y(t) \notin C^{2}[0, T]\), the optimal convergence order of this method cannot be obtained with the uniform meshes. However, it is well-known that for y ∈ C ^m [0,T] for some \(m \in \mathbb {N}\) and 0 < α < m, the Caputo fractional derivative \(\,^{C}_{0}D^{\alpha }_{t} y(t) \) takes the form “\(\,^{C}_{0}D^{\alpha }_{t} y(t) = c t^{\lceil \alpha \rceil -\alpha } + \text {smoother terms}\)” (Diethelm et al. Numer. Algor. 36, 31–52, 2004), which implies that \(\,^{C}_{0}D^{\alpha }_{t} y \) behaves as t ^?α??α which is not in C ²[0,T]. By using the graded meshes t _n = T(n/N) ^r ,n = 0,1,2,…,N with some suitable r > 1, we show that the optimal convergence order of this method can be recovered uniformly in t _n even if \(\,^{C}_{0}D^{\alpha }_{t} y\) behaves as t ^σ ,0 < σ < 1. Numerical examples are given to show that the numerical results are consistent with the theoretical results.