Strong capacity-estimates for “fractional” norms

It is proved that for all fractionall the integral \(\int\limits_0^\infty {(p,\ell ) - cap(M_t )} dt^p\) is majorized by the P-th power norm of the functionu in the space ? _p ^l (Rⁿ) (here M_t={x∶¦u(x)¦?t} and (p,l)-cap(e) is the (p,l)-capacity of the compactum e?Rⁿ). Similar results are obtained for the spaces W _p ^l (Rⁿ) and the spaces of M. Riesz and Bessel potentials. One considers consequences regarding imbedding theorems of “fractional” spaces in ?_q(dμ), whereμ is a nonnegative measure in Rⁿ. One considers specially the case p=1.     

On the class of limits of lacunary trigonometric series

Let (n _k ) _k≧1 be a lacunary sequence of positive integers, i.e. a sequence satisfying n _k+1/n _k > q > 1, k ≧ 1, and let f be a “nice” 1-periodic function with ∝ ₀ ¹ f(x) dx = 0. Then the probabilistic behavior of the system (f(n _k x)) _k≧1 is very similar to the behavior of sequences of i.i.d. random variables. For example, Erd?s and Gál proved in 1955 the following law of the iterated logarithm (LIL) for f(x) = cos 2πx and lacunary $ (n_k )_{k \geqq 1} $ : (1) $$ \mathop {\lim \sup }\limits_{N \to \infty } (2N\log \log N)^{1/2} \sum\limits_{k = 1}^N {f(n_k x)} = \left\| f \right\|_2 $$ for almost all x ∈ (0, 1), where ‖f‖₂ = (∝ ₀ ¹ f(x)² dx)^1/2 is the standard deviation of the random variables f(n _k x). If (n _k ) _k≧1 has certain number-theoretic properties (e.g. n _k+1/n _k → ∞), a similar LIL holds for a large class of functions f, and the constant on the right-hand side is always ‖f‖₂. For general lacunary (n _k ) _k≧1 this is not necessarily true: Erd?s and Fortet constructed an example of a trigonometric polynomial f and a lacunary sequence (n _k ) _k≧1, such that the lim sup in the LIL (1) is not equal to ‖f‖₂ and not even a constant a.e. In this paper we show that the class of possible functions on the right-hand side of (1) can be very large: we give an example of a trigonometric polynomial f such that for any function g(x) with sufficiently small Fourier coefficients there exists a lacunary sequence (n _k ) _k≧1 such that (1) holds with √‖f‖ ₂ ² + g(x) instead of ‖f‖₂ on the right-hand side.     

A short note on the Frobenius norm of the commutator

This note mainly aims to improve the inequality, proposed by Böttcher and Wenzel, giving the upper bound of the Frobenius norm of the commutator of two particular matrices in ? ^n×n . We first propose a new upper bound on basis of the Böttcher and Wenzel’s inequality. Motivated by the method used, the inequality ‖XY ? XY‖ _F ² ≤ 2‖X‖ _F ² ‖Y‖ _F ² is finally improved into $$ \left\| {XY - YX} \right\|_F^2 \leqslant 2\left\| X \right\|_F^2 \left\| Y \right\|_F^2 - 2[tr(X^T Y)]^2 . $$ . In addition, a further improvement is made.     

Asymptotic expansions of solutions of equations with a deviating argument in Banach spaces

For the equation $$Lu = \frac{1}{i}\frac{{du}}{{dt}}\sum\nolimits_{j = 0}^m {A_j u} (l - h_j^0 - h_j^1 (t)) = f(t),$$ whereh ₀ ^o =0,h ₀ ¹ =0 (t) ≡ 0,h _j ^o = const > 0,h ₁ ^j (t),j= 1, ...,m are nonnegative continuously differentiable functions in [0, ∞), Aj are bounded linear operators, under conditions on the resolvent and on the right hand sidef(t), we have obtained an asymptotic formula for any solution u(t) from L₂ in terms of the exponential solutions u_k(t), k=1, ..., n, of the equation $$\frac{1}{i}\frac{{du}}{{dt}} - A_0 u - \sum\nolimits_{j = 0}^m {A_j u} (t - h_j^0 ) = 0,$$ connected with the poles λ_k, k=1, ..., n, of the resolvent R_λ in a certain strip.     

Convergence of the Vallée-Poussin means for Fourier-Jacobi sums

Letf εC[?1, 1], ?1<α,β≤0, let $f \in C[ - 1, 1], - 1< \alpha , \beta \leqslant 0$ , letS _n ^{α, β} (f, x) be a partial Fourier-Jacobi sum of ordern, and let $$\nu _{m, n}^{\alpha , \beta } = \nu _{m, n}^{\alpha , \beta } (f) = \nu _{m, n}^{\alpha , \beta } (f,x) = \frac{1}{{n + 1}}[S_m^{\alpha ,\beta } (f,x) + ... + S_{m + n}^{\alpha ,\beta } (f,x)]$$ be the Vallée-Poussin means for Fourier-Jacobi sums. It was proved that if 0<a≤m/n≤b, then there exists a constantc=c(α, β, a, b) such that ‖ν _{m, n} ^{α, β} ‖ ≤c, where ‖ν _{m, n} ^{α, β} ‖ is the norm of the operator ν _{m, n} ^{α, β} inC[?1,1].     

The semigroup approach to first order quasilinear equations in several space variables

The Cauchy problem for u _t + Σ _{i = 1} ⁿ (φ _i (u)) _xi = 0 is treated via the theory of semigroups of nonlinear transformations. This treatment requires the development of results concerning the time-independent equation u + Σ _{i = 1} ⁿ (φ _i (u)) _xi = h for h∈L ¹(Rⁿ), which in turn is studied via the regularized equation $$ u + \sum\nolimits_{i = 1}^n {\left( {\phi _i \left( u \right)} \right)} _{xi} - \varepsilon \Delta u = h $$ .     

Wavelet characterization of growth spaces of harmonic functions

We consider the space h _ν ^∞ of harmonic functions in R ₊ ⁿ⁺¹ with finite norm ‖u‖ _ν = sup |u(x, t)|/v(t), where the weight ν satisfies the doubling condition. Boundary values of functions in h _ν ^∞ are characterized in terms of their smooth multiresolution approximations. The characterization yields the isomorphism of Banach spaces h _ν ^∞ ～ l ^∞. The results are also applied to obtain the law of the iterated logarithm for the oscillation of functions in h _ν ^∞ along vertical lines.     

Comparison Inequalities for One Sided Normal Probabilities

Several sharp upper and lower bounds for the ratio of two normal probabilities $\mathbb{P}\Biggl(\,\bigcap_{i=1}^{n}\bigl\{\xi^{(1)}_i\leq \mu_i\bigr\}\Biggr)\Big/\mathbb{P}\Biggl(\,\bigcap_{i=1}^{n}\bigl\{\xi^{(0)}_i\leq \mu_i\bigr\}\Biggr)$ are given in this paper for various cases, where (ξ ₁ ⁽⁰⁾ ,ξ ₂ ⁽⁰⁾ ,…,ξ _n ⁽⁰⁾ ) and (ξ ₁ ⁽¹⁾ ,ξ ₂ ⁽¹⁾ , …,ξ _n ⁽¹⁾ ) are standard normal random variables with covariance matrices R ⁰=(r _ij ⁰ ) and R ¹=(r _ij ¹ ), respectively.     

Freud's conjecture and approximation of reciprocals of weights by polynomials

LetW(x) be a function that is nonnegative inR, positive on a set of positive measure, and such that all power moments ofW ² (x) are finite. Let {p _n (W ²;x)} ₀ ^∞ denote the sequence of orthonormal polynomials with respect to the weightW ², and let {α _n } ₁ ^∞ and {β _n } ₁ ^∞ denote the coefficients in the recurrence relation $$xp_n (W^2 ,x) = \alpha _{n + 1} p_{n + 1} (W^2 ,x) + \beta _n p_n (W^2 ,x) + \alpha _n p_{n - 1} (W^2 ,x).$$ We obtain a sufficient condition, involving mean approximation ofW ^?1 by reciprocals of polynomials, for $$\mathop {\lim }\limits_{n \to \infty } {{\alpha _n } \mathord{\left/ {\vphantom {{\alpha _n } {c_n }}} \right. \kern-\nulldelimiterspace} {c_n }} = \tfrac{1}{2}and\mathop {\lim }\limits_{n \to \infty } {{\beta _n } \mathord{\left/ {\vphantom {{\beta _n } {c_{n + 1} }}} \right. \kern-\nulldelimiterspace} {c_{n + 1} }} = 0,$$ wherec _{n 1} ^∞ is a certain increasing sequence of positive numbers. In particular, we obtain a sufficient condition for Freud's conjecture associated with weights onR.     

On the regularity of solutions of model nonlinear elliptic systems with the oblique derivative type boundary condition

Properties of generalized solutions of model nonlinear elliptic systems of second order are studied in the semiball $B_1^ + = B_1 (0) \cap \{ x_n > 0\} \subset $ ? ⁿ , with the oblique derivative type boundary condition on $\Gamma _1 = B_1 (0) \cap \{ x_n = 0\} $ . For solutionsu∈H ¹(B ₁ ⁺ ) of systems of the form $\frac{d}{{dx_\alpha }}a_\alpha ^k (u_x ) = 0, k \leqslant {\rm N}$ , it is proved that the derivatives u_x are Hölder in $B_1^ + \cup \Gamma _1 )\backslash \Sigma $ , where H_n?p(σ)=0,p>2. It is shown for continuous solutions u from H¹(B₁/⁺) of systems $\frac{d}{{dx_\alpha }}a_\alpha ^k (u,u_x ) = 0$ that the derivatives u_x are Hölder on the set $(B_1^ + \cup \Gamma _1 )\backslash \Sigma , dim_\kappa \Sigma \leqslant n - 2$ . Bibliography: 13 titles.     

Pointwise estimates of potentials for higher-order capacities

In a domain D=Ω\E∈ R ⁿ , we consider a nonlinear higher-order elliptic equation such that the corresponding energy space is W _p ^m (D)?W _q ¹ (D), q>mp, and estimate a solution u(x) of this equation satisfying the condition u(x)?kf(x)∈W _p ^m (D)?W _q ¹ (D), where k∈R ¹, f(x)∈ C ₀ ^∞ (Ω), and f(x)=1 for x∈F. We establish a pointwise estimate for u(x) in terms of the higher-order capacity of the set F and the distance from the point x to the set F.     

Asymptotics for the minimization of a Ginzburg-Landau functional

LetΩ ? ?² be a smooth bounded simply connected domain. Consider the functional $$E_\varepsilon (u) = \frac{1}{2}\int\limits_\Omega {\left| {\nabla u} \right|^2 + \frac{1}{{4\varepsilon ^2 }}} \int\limits_\Omega {(|u|^2 - 1)^2 } $$ on the classH _g ¹ ={u εH ¹(Ω; ?);u=g on ?Ω} whereg:?Ω? → ? is a prescribed smooth map with ¦g¦=1 on ?Ω? and deg(g, ?Ω)=0. Let uu _ε be a minimizer for E_ε onH _g ¹ . We prove that u_ε → u₀ in \(C^{1,\alpha } (\bar \Omega )\) as ε → 0, where u₀ is identified. Moreover \(\left\| {u_\varepsilon - u_0 } \right\|_{L^\infty } \leqslant C\varepsilon ^2 \) .     

On coefficients of null-series and on sets of uniqueness of trigonometric and Walsh systems

В работе доказываютс я следующие утвержде ния. Теорема I.Пусть ? _n ↓0u \(\sum\limits_{n = 0}^\infty {\varepsilon _n^2 = + \infty } \) .Тогд а существует множест во Е?[0, 1]с μЕ=0 такое что:1. Существует ряд \(\sum\limits_{n = 0}^\infty {a_n W_n } (t)\) с к оеффициентами ¦а _n ¦≦{in¦n¦, который сх одится к нулю всюду вне E и ε∥a_n∥>0.2. Если b _n ¦=о(ε _n )и ряд \(\sum\limits_{n = 0}^\infty {b_n W_n (t)} \) сх одится к нулю всюду вн е E за исключением быть может некоторого сче тного множества точе к, то b _n =0для всех п. Теорема 3.Пусть ? _n ↓0u \(\mathop {\lim \sup }\limits_{n \to \infty } \frac{{\varepsilon _n }}{{\varepsilon _{2n} }}< \sqrt 2 \) Тогд а существует множест во E?[0, 1] с υ E=0 такое, что:

1. Существует ряд \(\sum\limits_{n = - \infty }^{ + \infty } {a_n e^{inx} ,} \sum\limits_{n = - \infty }^{ + \infty } {\left| {a_n } \right|} > 0,\) кот орый сходится к нулю в сюду вне E и ¦a_n≦¦n¦ для n=±1, ±2, ...
2. Если ряд \(\sum\limits_{n = - \infty }^{ + \infty } {b_n e^{inx} } \) сходится к нулю всюду вне E и ¦b_v¦=о(ε ¦n¦), то b_n=0 для всех я. Теорема 5. Пусть послед овательности S⁽¹⁾={ε ₀ ⁽¹⁾ , ε ₁ ⁽¹⁾ , ε ₂ ⁽¹⁾ , ...} u S²=ε ₀ ⁽²⁾ , ε ₁ ⁽²⁾ . ε ₂ ⁽²⁾ монотонно стремятся к нулю, \(\mathop {\lim \sup }\limits_{n \to \infty } \varepsilon ^{(i)} /\varepsilon _{2n}^{(i)}< 2,i = 1,2\) , причем \(\mathop {\lim }\limits_{n \to \infty } \varepsilon _n^{(2)} /\varepsilon _n^{(i)} = + \infty \) . Тогда для каждого ε>O н айдется множество Е? [-π,π], μE >2π — ε, которое является U(S¹), но не U(S¹) — множеством для тригонометричес кой системы. Аналог теоремы 5 для си стемы Уолша был устан овлен в [7].

     

Power Values of Generalized Derivations with Annihilator Conditions in Prime Rings

Let R be a prime ring, H a nonzero generalized derivation of R and L a noncommutative Lie ideal of R. Suppose that there exists ${0 \neq a \in R}$ such that a(u ^s H(u)u ^t ) ⁿ = 0 for all ${u \in L}$ , where s ≥ 0, t ≥ 0, n ≥ 1 are fixed integers. Then s = 0, H(x) = bx for all ${x \in R}$ with ab = 0, unless R satisfies s ₄, the standard identity in four variables. We also describe completely this last case.     

An initial value problem for a singular parabolic equation of even order

At first Cauchy-problem for the equation: \(L[u(X,t)] \equiv \sum\limits_{i = 1}^n {\frac{{\partial ^2 u}}{{\partial x_1^2 }} + \frac{{2v}}{{\left| X \right|^2 }}} \sum\limits_{i = 1}^n {x_i \frac{{\partial u}}{{\partial x_i }} - \frac{{\partial u}}{{\partial t}} = 0} \) wheren≥1,v—an arbitrary constant,t>0,X=(x ₁, …, x_n)∈E ⁿ/{0}, |X|= =(x ₁ ² +…+x _n ² )^1/2, with 0 being a centre of coordinate system, is studied. Basing on the above, the solution of Cauchy-Nicolescu problem is given which consist in finding a solution of the equationL ^p [u (X, t)]=0, withp∈N subject the initial conditions \(\mathop {\lim }\limits_{t \to \infty } L^k [u(X,t)] = \varphi _k (X)\) ,k=0, 1,…,p?1 and ?_k(X) are given functions.     

Small into-isomorphism from L∞ (A, μ) into L∞(B, υ)

In this paper we shall assert that if T is an isomorphism of L_∞(Ω₁, A, μ) into L_∞(Ω₂, B, υ) satisfying the condition ‖T‖·‖T ^?1‖?1+? for ?∈ $\left( {0,\frac{1}{5}} \right)$ , then $\frac{T}{{\parallel T\parallel }}$ is close to an isometry with an error less than 6ε in some conditions.     

Weak solutions for elliptic systems with variable growth in Clifford analysis

In this paper we consider the following Dirichlet problem for elliptic systems: $$\begin{array}{*{20}c} {\overline {DA\left( {x,u\left( x \right),Du\left( x \right)} \right)} = B\left( {x,u\left( x \right),Du\left( x \right)} \right), x \in \Omega ,} \\ {u\left( x \right) = 0, x\partial \Omega } \\ \end{array}$$ where D is a Dirac operator in Euclidean space, u(x) is defined in a bounded Lipschitz domain Ω in ? ⁿ and takes value in Clifford algebras. We first introduce variable exponent Sobolev spaces of Clifford-valued functions, then discuss the properties of these spaces and the related operator theory in these spaces. Using the Galerkin method, we obtain the existence of weak solutions to the scalar part of the above-mentioned systems in the space W ₀ ^1,p(x) (Ω,C? _n ) under appropriate assumptions.     

Uniform stability of the inverse Sturm-Liouville problem with respect to the spectral function in the scale of Sobolev spaces

We consider the inverse problem of recovering the potential for the Sturm-Liouville operator Ly = ?y″ + q(x)y on the interval [0, π] from the spectrum of the Dirichlet problem and norming constants (from the spectral function). For a fixed θ ≥ 0, with this problem we associate a map F: W ₂ ^θ → l _D ^θ , F(σ) = {s _k } ₁ ^∞ , where W ₂ ^θ = W ₂ ^θ [0, π] is the Sobolev space, σ = ∫ q is a primitive of the potential q ∈ W ₂ ^{θ ? 1} , and l _D ^θ is a specially constructed finite-dimensional extension of the weighted space l ₂ ^θ ; this extension contains the regularized spectral data s = {s _k } ₁ ^∞ for the problem of recovering the potential from the spectral function. The main result consists in proving both lower and upper uniform estimates for the norm of the difference ‖σ ? σ ₁‖ _θ in terms of the l _D ^θ norm of the difference of the regularized spectral data ‖s ? s₁‖ _θ . The result is new even for the classical case q ∈ L ₂, which corresponds to the case θ = 1.     

An infinite-dimensional law of large numbers in Cesaro's sense

Let (X _k) _k≥0 be a sequence of independent copies of a random variableX taking its values in a real separable Banach space (B, ¦ ¦). For every real number β>?1 one defines the following coefficients: $$A_0^\beta = 1, A_1^\beta = \beta + 1,..., A_k^\beta = (\beta + 1) \cdots (\beta + k)/k!,...$$ It is shown that for all α∈]0, 1[ the sequenceV _n =(1/A _n ^α )∑_0?k?n A _n?k ^α?1 X _k converges almost surely toE(X) if and only if ‖X‖^1/α is integrable. This extends results obtained earlier by several authors for scalar-valued random variables: Lorentz (case 1/2<α<1), Chow and Lai (case 0<α<1/2), Déniel and Derriennic (case α=1/2).     

Exact constants in generalized inequalities for intermediate derivatives

Consider the Sobolev space W ₂ ⁿ (?₊) on the semiaxis with norm of general form defined by a quadratic polynomial in derivatives with nonnegative coefficients. We study the problem of exact constants A _n,k in inequalities of Kolmogorov type for the values of intermediate derivatives |f ^(k)(0)| ≤ A _n,k ‖f‖. In the general case, the expression for the constants A _n,k is obtained as the ratio of two determinants. Using a general formula, we obtain an explicit expression for the constants A _n,k in the case of the following norms: $$ \left\| f \right\|_1^2 = \left\| f \right\|_{L_2 }^2 + \left\| {f^{(n)} } \right\|_{L_2 }^2 and\left\| f \right\|_2^2 = \sum\limits_{l = 0}^n {\left\| {f^{(l)} } \right\|_{L_2 }^2 } . $$ In the case of the norm ‖ · ‖₁, formulas for the constants A _n,k were obtained earlier by another method due to Kalyabin. The asymptotic behavior of the constants A _n,k is also studied in the case of the norm ‖ · ‖₂. In addition, we prove a symmetry property of the constants A _n,k in the general case.