Spherical means and the restriction phenomenon

Let Γ be a smooth compact convex planar curve with arc length dm and let dσ=ψ dm where ψ is a cutoff function. For Θ∈SO (2) set σ_Θ(E) = σ(ΘE) for any measurable planar set E. Then, for suitable functions f in ℝ², the inequality.

On the weighted elliptic problems involving multi-singular potentials and multi-critical exponents

Suppose Ω belong to R^N（N≥3） is a smooth bounded domain,ξi∈Ω,0〈ai〈√μ,μ：=（（N-1）/2）^2,0≤μi〈（√μ-ai）^2,ai〈bi〈ai＋1 and pi：=2N/N-2（1＋ai-bi）are the weighted critical Hardy-Sobolev exponents, i = 1, 2,..., k, k ≥ 2. We deal with the conditions that ensure the existence of positive solutions to the multi-singular and multi-critical elliptic problem ∑i=1^k（-div（｜x-ξi｜^-2ai△↓u）-μiu/｜x-ξi｜^2（1＋ai）-u^pi-1/｜x-ξi｜^bipi）=0with Dirichlet boundary condition, which involves the weighted Hardy inequality and the weighted Hardy-Sobolev inequality. The results depend crucially on the parameters ai, bi and #i, i -- 1, 2,..., k.     

About entire functions with specialL 2-properties on one-dimensional subspaces of ℂ n

In this paper we consider special elements of the Fock space #x2131; _n . That is the space of entire functionsf:ℂ: ⁿ →ℂ, such that the followingL ²- condition is satisfied: . Here we show that there exists an entire functiong:ℂ ⁿ →ℂ such that for every one-dimensional subspace Π⊂ℂ ⁿ and for all 0<∈<2 we have , but in the limit case ∈=0 we have . This result is analogue to a result from [1]. There holomorphic functions on the unit-ball are investigated. Furthermore the proof — as the one in [1] — uses a theorem from [2]. Therefore we give another application of the results from [2] — namely for spaces of entire functions.     

Solutions to Some Open Problems in Fluid Dynamics

Let u=u（x,t,uo）represent the global solution of the initial value problem for the one-dimensional fluid dynamics equation ut-εuxxt＋δux＋γHuxx＋βuxxx＋f（u）x=αuxx,u（x,0）=uo（x）, whereα〉0,β〉0,γ〉0,δ〉0 andε〉0 are constants.This equation may be viewed as a one-dimensional reduction of n-dimensional incompressible Navier-Stokes equations. The nonlinear function satisfies the conditions f（0）=0,｜f（u）｜→∞as ｜u｜→∞,and f∈C^1（R）,and there exist the following limits Lo=lim sup/u→o f（u）/u^3 and L∞=lim sup/u→∞ f（u）/u^5 Suppose that the initial function u0∈L^I（R）∩H^2（R）.By using energy estimates,Fourier transform,Plancherel＇s identity,upper limit estimate,lower limit estimate and the results of the linear problem vt-εv（xxt）＋δvx＋γHv（xx）＋βv（xxx）=αv（xx）,v（x,0）=vo（x）, the author justifies the following limits（with sharp rates of decay） lim t→∞[（1＋t）^（m＋1/2）∫｜uxm（x,t）｜^2dx]=1/2π（π/2α）^（1/2）m!!/（4α）^m[∫R uo（x）dx]^2, if∫R uo（x）dx≠0, where 0!!=1,1!!=1 and m!!=1·3…（2m-3）…（2m-1）.Moreover lim t→∞[（1＋t）^（m＋3/2）∫R｜uxm（x,t）｜^2dx]=1/2π（x/2α）^（1/2）（m＋1）!!/（4α）^（m＋1）[∫Rρo（x）dx]^2, if the initial function uo（x）=ρo′（x）,for some functionρo∈C^1（R）∩L^1（R）and∫Rρo（x）dx≠0.     

Trigonometric series of classesL^p (\mathbb{T}), p \in ]1;\infty [ and their conservative meansand their conservative means

Suppose that a lower triangular matrix μ:[μ _m ⁽ⁿ⁾ ] defines a conservative summation method for series, i.e.,

Growth Estimates for Cauchy Integrals of Measures and Rectifiability

We show that if μ is a finite Borel measure on the complex plane such that

Exponents for three-dimensional simultaneous Diophantine approximations

Let Θ = (θ ₁,θ ₂,θ ₃) ∈ ℝ³. Suppose that 1, θ ₁, θ ₂, θ ₃ are linearly independent over ℤ. For Diophantine exponents

Positive Solution of Multi-Point Boundary Value Problems for the One-Dimensional p-Laplacian with Singularities

In this paper, we obtain positive solution to the following multi-point singular boundary value problem with p-Laplacian operator,{（ φp（u＇））＇＋q（t）f（t,u,u＇）=0,0〈t〈1,u（0）=∑i=1^nαiu（ξi）,u＇（1）=∑i=1^nβiu＇（ξi）,whereφp（s）=｜s｜^p-2s,p≥2;ξi∈（0,1）（i=1,2,…,n）,0≤αi,βi〈1（i=1,2,…n）,0≤∑i=1^nαi,∑i=1^nβi〈1,and q（t） may be singular at t=0,1,f（t,u,u＇）may be singular at u＇=0     

Central polynomials and matrix invariants

LetK be a field, charK=0 andM _n (K) the algebra ofn×n matrices overK. If λ=(λ₁,…,λ _m ) andμ=(μ ₁,…,μ _m ) are partitions ofn ² let wherex ₁,…,x _n ²,y ₁,…,y _n ² are noncommuting indeterminates andS _n ² is the symmetric group of degreen ². The polynomialsF ^{λ, μ} , when evaluated inM _n (K), take central values and we study the problem of classifying those partitions λ,μ for whichF ^{λ, μ} is a central polynomial (not a polynomial identity) forM _n (K). We give a formula that allows us to evaluateF ^{λ, μ} inM(K) in general and we prove that if λ andμ are not both derived in a suitable way from the partition δ=(1, 3,…, 2n−3, 2n−1), thenF ^{λ, μ} is a polynomial identity forM _n (K). As an application, we exhibit a new class of central polynomials forM _n (K). In memory of Shimshon Amitsur Research supported by a grant from MURST of Italy.     

The Cauchy-Dirichlet problem for the heat equation in Besov spaces

The solvability in anisotropic spaces , σ ∈ ℝ₊, p, q ∈ (1, ∞), of the heat equation u_t − Δu = f in Ω^T ≡ (0, T) × Ω is studied under the boundary and initial conditions u = g on S^T, u|_t=0 = u₀ in Ω, where S is the boundary of a bounded domain Ω ⊂ ℝⁿ. The existence of a unique solution of the above problem is proved under the assumptions that and under some additional conditions on the data. The existence is proved by the technique of regularizers. For this purpose the local-in-space solvability near the boundary and near an interior point of Ω is needed. To show the local-in-space existence, the definition of Besov spaces by the dyadic decomposition of a partition of unity is used. This enables us to get an appropriate estimate in a new and promising way without applying either the potential technique or the resolvent estimates or the interpolation. Bibliography: 26 titles. Published in Zapiski Nauchnykh Seminarov POMI, Vol. 348, 2007, pp. 40–97.     

The Balian-Low theorem and regularity of Gabor systems

For any positive real numbers A, B, and d satisfying the conditions , d>2, we construct a Gabor orthonormal basis for L²(ℝ), such that the generating function g∈L²(ℝ) satisfies the condition:∫_ℝ|g(x)|²(1+|x| ^A )/log ^d (2+|x|)dx < ∞ and .     

Oscillatory integrals on unit square along surfaces

Let Q ² = [0, 1]² be the unit square in two-dimensional Euclidean space ℝ². We study the L ^p boundedness of the oscillatory integral operator T _α,β defined on the set ℒ(ℝ²⁺ⁿ ) of Schwartz test functions by

Essential and discrete spectra of partially integral operators

Let Ω₁, Ω₂ ⊂ ℝ^ν be compact sets. In the Hilbert space L ₂(Ω₁ × Ω₂), we study the spectral properties of selfadjoint partially integral operators T ₁, T ₂, and T ₁ + T ₂, with

On One Extremal Problem for a Seminorm on the Space l<Subscript>1</Subscript> with Weight

Let be a nondecreasing sequence of positive numbers and let l _1,α be the space of real sequences for which . We associate every sequence ξ from l _1,α with a sequence , where ϕ(·) is a permutation of the natural series such that , j ∈ ℕ. If p is a bounded seminorm on l _1,α and , then

Uniqueness and normality for M. riesz potentials and solutions of the darboux equation

M. Riesz potentials are considered, where Ω is a domain in ℝⁿ⁺¹ with a nice boundary ∂Ω, and μ is a Borel charge on ∂Ω. These potentials satisfy the Darboux equation

Multiple recurrence and almost sure convergence for weakly mixing dynamical systems

We prove the following: Let (X, β, μ,T) be a weakly mixing dynamical system such that the restriction ofT to its Pinsker algebra has singular spectrum, then for all positive integersH, for allf _i ∈L ^∞, 1≤i≤H, the averages

Lower semicontinuity of some functionals under PDE constraints: An \mathcal{A}-quasiconvex pair

The problem of establishing necessary and sufficient conditions for l.s.c. under PDE constraints is studied for a special class of functionals:

On best localized bandlimited functions

Let, for σ > 0, be the set of complex functions f ∈ L ¹ (ℝ) with the Fourier transforms vanishing outside the interval [−σ; σ]. In this paper, we study the problem of the best approximation of the Dirac function δ (which has the Fourier transform with widest support supp ) by functions . More precisely, we consider the quantity inf and its extremal functions . __________ Translated from Lietuvos Matematikos Rinkinys, Vol. 46, No. 4, pp. 548–564, October–December, 2006.