Hausdorff dimension of Lebesgue sets for W α p classes on metric spaces

Let (X, μ, d) be a space of homogeneous type, where d and p are a metric and a measure, respectively, related to each other by the doubling condition with γ > 0. Let W _α ^p (X) be generalized Sobolev classes, let Cap_{α p} (where p > 1 and 0 < α ≤ 1) be the corresponding capacity, and let dim_H be the Hausdorff dimension. We show that the capacity Cap_{α p} is related to the Hausdorff dimension; we also prove that, for each function u ∈ W _α/p (X), p > 1, 0 < a < γ/p, there exists a set E ? X such that dim _H (E) ≤ γ - αp, the limit $$\mathop {\lim }\limits_{r \to + 0} \frac{1}{{\mu (B(x,r))}}\int_{B(x,r)} {u d\mu = u * (x)} $$ exists for each x ∈ X\E, and moreover $$\mathop {\lim }\limits_{r \to + 0} \frac{1}{{\mu (B(x,r))}}\int_{B(x,r)} {\left| {u - u * (x)} \right|^q d\mu = 0, \frac{1}{q} = \frac{1}{p} - \frac{\alpha }{\gamma }.} $$ .     

Divided differences for functions of two variables for irregularly spaced arguments

Divided differences forf (x, y) for completely irregular spacing of points (x _i ,y _i ) are developed here by a natural generalization of Newton's scheme. Existing bivariate schemes either iterate the one-dimensional scheme, thus constraining (x _i ,y _i ) to be at corners of rectangles, or give polynomials Σa _jk x ^j y ^k having more coefficients than interpolation conditions. Here the generalizedn ^th divided difference is defined by (1)\(\left[ {01... n} \right] = \sum\limits_{i = 0}^n {A_i f\left( {x_i , y_i } \right)} \) where (2)\(\sum\limits_{i = 0}^n {A_i x_i^j , y_i^k = 0} \), and 1 for the last or (n+1)^th equation, for every (j, k) wherej+k=0, 1, 2,... in the usual ascending order. The gen. div. diff. [01...n] is symmetric in (x _i ,y _i ), unchanged under translation, 0 forf (x, y) an, ascending binary polynomial as far asn terms, degree-lowering with respect to (X, Y) whenf(x, y) is any polynomialP(X+x, Y+y), and satisfies the 3-term recurrence relation (3) [01...n]=λ{[1...n]?[0...n?1]}, where (4) λ= |1...n|·|01...n?1|/|01...n|·|1...n?1|, the |...i...| denoting determinants inx _i ^j y _i ^k . The generalization of Newton's div. diff. formula is (5)

$$\begin{gathered} f\left( {x, y} \right) = f\left( {x_0 , y_0 } \right) - \frac{{\left| {\alpha 0} \right|}}{{\left| 0 \right|}}\left[ {01} \right] + \frac{{\left| {\alpha 01} \right|}}{{\left| {01} \right|}}\left[ {012} \right] - \frac{{\left| {\alpha 012} \right|}}{{\left| {012} \right|}}\left[ {0123} \right] + \cdots + \hfill \\ + \left( { - 1} \right)^n \frac{{\left| {\alpha 01 \ldots n - 1} \right|}}{{\left| {01 \ldots n - 1} \right|}}\left[ {01 \ldots n} \right] + \left( { - 1} \right)^{n + 1} \frac{{\left| {\alpha 01 \ldots n} \right|}}{{\left| {01 \ldots n} \right|}}\left[ {01 \ldots n} \right], \hfill \\ \end{gathered} $$     

On the best constants in the Khintchine inequality for Steinhaus variables

Let S _j : (Ω, P) → S ¹ ? ? be an i.i.d. sequence of Steinhaus random variables, i.e. variables which are uniformly distributed on the circle S ¹. We determine the best constants a _p in the Khintchine-type inequality $${a_p}{\left\| x \right\|_2} \leqslant {\left( {{\text{E}}{{\left| {\sum\limits_{j = 1}^n {{x_j}{S_j}} } \right|}^p}} \right)^{1/p}} \leqslant {\left\| x \right\|_2};{\text{ }}x = ({x_j})_{j = 1}^n \in {{\Bbb C}^n}$$ for 0 < p < 1, verifying a conjecture of U. Haagerup that $${a_p} = \min \left( {\Gamma {{\left( {\frac{p}{2} + 1} \right)}^{1/p}},\sqrt 2 {{\left( {{{\Gamma \left( {\frac{{p + 1}}{2}} \right)} \mathord{\left/ {\vphantom {{\Gamma \left( {\frac{{p + 1}}{2}} \right)} {\left[ {\Gamma \left( {\frac{p}{2} + 1} \right)\sqrt \pi } \right]}}} \right. \kern-\nulldelimiterspace} {\left[ {\Gamma \left( {\frac{p}{2} + 1} \right)\sqrt \pi } \right]}}} \right)}^{1/p}}} \right)$$ . Both expressions are equal for p = p ₀ }~ 0.4756. For p ≥ 1 the best constants a _p have been known for some time. The result implies for a norm 1 sequence x ∈ ? ⁿ , ‖x‖₂ = 1, that $${\text{E}}\ln \left| {\frac{{{S_1} + {S_2}}}{{\sqrt 2 }}} \right| \leqslant {\text{E}}\ln \left| {\sum\limits_{j = 1}^n {{x_j}{S_j}} } \right|$$ , answering a question of A. Baernstein and R. Culverhouse.     

Singular oscillatory integrals on {\mathbb{R}^n}

Let ${\mathcal{P}_{d,n}}Let P_d,n{\mathcal{P}_{d,n}} denote the space of all real polynomials of degree at most d on \mathbbRⁿ{\mathbb{R}^n} . We prove a new estimate for the logarithmic measure of the sublevel set of a polynomial P ? P_d,1{P\in \mathcal{P}_{d,1}} . Using this estimate, we prove that

supP ? Pd,n| p.v.ò\mathbbRneiP(x)\fracW(x/|x|)|x|ndx| ￡ c log d (||W||L logL(Sn-1)+1),\mathop{\rm sup}\limits_ {P \in \mathcal{P}_{d,n}}\left| p.v.\int_{\mathbb{R}^{n}}{e^{iP(x)}}{\frac{\Omega(x/|x|)}{|x|^n}dx}\right | \leq c\,{\rm log}\,d\,(||\Omega||_L \log L(S^{n-1})+1),      5. Vertical versus horizontal Poincaré inequalities on the Heisenberg group      Vincent Lafforgue  Assaf Naor 《Israel Journal of Mathematics》2014,203(1):309-339 Let ? = 〈a, b|a[a, b] = [a, b]a ∧ b[a, b] = [a, b]b〉 be the discrete Heisenberg group, equipped with the left-invariant word metric d W (·, ·) associated to the generating set {a, b, a ?1, b ?1}. Letting B n = {x ∈ ?: d W (x, e ?) ? n} denote the corresponding closed ball of radius n ∈ ?, and writing c = [a, b] = aba ?1 b ?1, we prove that if (X, ‖ · ‖X) is a Banach space whose modulus of uniform convexity has power type q ∈ [2,∞), then there exists K ∈ (0, ∞) such that every f: ? → X satisfies $$\sum\limits_{k = 1}^{{n^2}} {\sum\limits_{x \in {B_n}} {\frac{{\left\| {f(x{c^k}) - f(x)} \right\|_X^q}}{{{k^{1 + q/2}}}}} } \leqslant K\sum\limits_{x \in {B_{21n}}} {(\left\| {f(xa) - f(x)} \right\|_X^q + \left\| {f(xb) - f(x)} \right\|_X^q)} $$ . It follows that for every n ∈ ? the bi-Lipschitz distortion of every f: B n → X is at least a constant multiple of (log n)1/q , an asymptotically optimal estimate as n → ∞.      6. Pointwise estimates for Lagrange interpolation polynomials      T. F. Xie 《Acta Mathematica Hungarica》2007,117(1-2):77-89 Let f ∈ C[?1, 1]. Let the approximation rate of Lagrange interpolation polynomial of f based on the nodes $ \left\{ {\cos \frac{{2k - 1}} {{2n}}\pi } \right\} \cup \{ - 1,1\} $ be Δ n + 2(f, x). In this paper we study the estimate of Δ n + 2(f,x), that keeps the interpolation property. As a result we prove that $$ \Delta _{n + 2} (f,x) = \mathcal{O}(1)\left\{ {\omega \left( {f,\frac{{\sqrt {1 - x^2 } }} {n}} \right)\left| {T_n (x)} \right|\ln (n + 1) + \omega \left( {f,\frac{{\sqrt {1 - x^2 } }} {n}\left| {T_n (x)} \right|} \right)} \right\}, $$ where T n (x) = cos (n arccos x) is the Chebeyshev polynomial of first kind. Also, if f ∈ C r [?1, 1] with r ≧ 1, then $$ \Delta _{n + 2} (f,x) = \mathcal{O}(1)\left\{ {\frac{{\sqrt {1 - x^2 } }} {{n^r }}\left| {T_n (x)} \right|\omega \left( {f^{(r)} ,\frac{{\sqrt {1 - x^2 } }} {n}} \right)\left( {\left( {\sqrt {1 - x^2 } + \frac{1} {n}} \right)^{r - 1} \ln (n + 1) + 1} \right)} \right\}. $$       7. Simultaneouse approximation to a differentiable function and its derivatives by Lagrange interpolating polynomials      T.F.Xie  S.P.Zhou 《分析论及其应用》1994,10(4):100-109 This paper establishes the following pointwise result for simultancous Lagrange imterpolating approxima-tion:then|f~(k)(x)-P_n~(k)(f,x)|=O(1)△_n~(q-k)(x)ωwhere P_n(f,x)is the Lagrange interpolating potynomial of deereeon the nodesX_nUY_n(see the definition of the next).      8. On functional equations associated with characters of unitary representations of groups      Zbigniew Gajda 《Aequationes Mathematicae》1992,44(1):109-121 This is mainly concerned with the following functional equation:       9. Approximation operators and tauberian constants      A. Jakimovski  A. Livne 《Israel Journal of Mathematics》1969,7(3):263-292 The explicit expression of the smallest constantC satisfying $$\mathop {lim}\limits_{\lambda \to \infty } \left| {t_{n(\lambda )}^{(1)} - t_{m(\lambda )}^{(2)} } \right| \leqq C. \mathop {lim sup}\limits_{n \to \infty } \left| {d_n } \right|$$ for all sequences {s n} satisfying lim sup n→∞ |d n| <∞, where {t n (1) }, {t n (2) } are two generalised Hausdorff transforms of {s n }, {d n} is the generalised (C, α)-transform (0≦α≦1) of {λ n a n} andn(λ, m(λ) are suitably related, is obtained. These results are obtained by using new properties of positive approximation operators and generalised Bernstein approximation operators.      10. Интегро-дифференциа ляные формулы сферических средних      А. П. Терехин  A. P. Terekhin 《Analysis Mathematica》1999,25(1):155-166 Реэуме. Установлена формула для средних $$S(t)f(\mu ): = \frac{1}{{\left| {\sigma ^\prime } \right|}}\smallint _{\sigma (\mu )} f(\mu cos t + \nu sin t)d\nu $$ функцииf, определеннои на единичнои сфере а пространства R n ,n>2; σ(μ) — Экватор с полусом в μ, |σ′| — мера Экватора. Для проиэводнои ? t (r) дано представление.      11. On the existence of diffusions with singular drift coefficient      严加安 《应用数学学报(英文版)》1988,4(1):23-29 Let L:be an operator in R~d,where the matrix(a) is bounded,Holder continuous and uniformly positive definite,and (b~t(x)) is Borel measurable.In this paper we prove the existence of L-diffusion under the hypothesis that where g_1(z)=1,g_2(z)=-ln|z|and g_c(z)=|z|~(2-d) for d≥3.      12. A sharp weightedL 2-estimate for the solution to the time-dependent Schrödinger equation      Björn G. Walther 《Arkiv f?r Matematik》1999,37(2):381-393 For Ξ∈R n ,t∈R andf∈S(R n ) define $\left( {S^2 f} \right)\left( t \right)\left( \xi \right) = \exp \left( {it\left| \xi \right|^2 } \right)\hat f\left( \xi \right)$ . We determine the optimal regularitys 0 such that $\int_{R^n } {\left\| {(S^2 f)[x]} \right\|_{L^2 (R)}^2 \frac{{dx}}{{(1 + |x|)^b }} \leqslant C\left\| f \right\|_{H^s (R^n )}^2 ,s > s_0 } ,$ holds whereC is independent off∈S(R n ) or we show that such optimal regularity does not exist. This problem has been treated earlier, e.g. by Ben-Artzi and Klainerman [2], Kato and Yajima [4], Simon [6], Vega [9] and Wang [11]. Our theorems can be generalized to the case where the exp(it|ξ|2) is replaced by exp(it|ξ|a),a≠2. The proof uses Parseval's formula onR, orthogonality arguments arising from decomposingL 2(R n ) using spherical harmonics and a uniform estimate for Bessel functions. Homogeneity arguments are used to show that results are sharp with respect to regularity.      13. Some properties of elliptic Riesz means at critical index on Hp(TH)      Jiang Yinsheng  Liu Heping  Lu Shanzhen 《分析论及其应用》1990,6(2):28-37 Suppose f∈Hp(Tn), 0 r δ , δ=n/p?(n+1)/2. In this paper we eastablish the following inequality $$\mathop {\sup }\limits_{R > 1} \left\{ {\frac{1}{{\log R}}\int_1^R {\left\| {\sigma _r^\delta } \right\|_{H^p (T^R )}^p \frac{{dr}}{r}} } \right\}^{1/p} \leqslant C_{R,p} \left\| f \right\|_{H^p (T^R )} $$ It implies that $$\mathop {\lim }\limits_{R \to \infty } \frac{1}{{\log R}}\int_1^R {\left\| {\sigma _r^\delta - f} \right\|_{H^p (T^R )}^p \frac{{dr}}{r}} = 0$$ Moreover we obtain the same conclusion when p=1 and n=1.      14. Concave iteration semigroups of linear continuous set-valued functions      Andrzej Smajdor  Wilhelmina Smajdor 《Central European Journal of Mathematics》2012,10(6):2272-2282 Let {F t : t ≥ 0} be a concave iteration semigroup of linear continuous set-valued functions defined on a convex cone K with nonempty interior in a Banach space X with values in cc(K). If we assume that the Hukuhara differences F 0(x) ? F t (x) exist for x ∈ K and t > 0, then D t F t (x) = (?1)F t ((?1)G(x)) for x ∈ K and t ≥ 0, where D t F t (x) denotes the derivative of F t (x) with respect to t and $G(x) = \mathop {\lim }\limits_{s \to 0} {{\left( {F^0 \left( x \right) - F^s \left( x \right)} \right)} \mathord{\left/ {\vphantom {{\left( {F^0 \left( x \right) - F^s \left( x \right)} \right)} {\left( { - s} \right)}}} \right. \kern-0em} {\left( { - s} \right)}}$ for x ∈ K.      15. Asymptotic properties of least squares estimators for discrete compound distributions      Keewhan Choi 《Annals of the Institute of Statistical Mathematics》1978,30(1):45-50 Let (X, Λ) be a pair of random variables, where Λ is an Ω (a compact subset of the real line) valued random variable with the density functiong(Θ: α) andX is a real-valued random variable whose conditional probability function given Λ=Θ is P {X=x|Θ} withx=x 0, x1, …. Based onn independent observations ofX, x (n), we are to estimate the true (unknown) parameter vectorα=(α 1, α2, ...,αm) of the probability function ofX, Pα(X=∫ΩP{X=x|Θ}g(Θ:α)dΘ. A least squares estimator of α is any vector \(\hat \alpha \left( {X^{\left( n \right)} } \right)\) which minimizes $$n^{ - 1} \sum\limits_{i = 1}^n {\left( {P_\alpha \left( {x_i } \right) - fn\left( {x_i } \right)} \right)^2 } $$ wherex (n)=(x1, x2,…,x n) is a random sample ofX andf n(xi)=[number ofx i inx (n)]/n. It is shown that the least squares estimators exist as a unique solution of the normal equations for all sufficiently large sample size (n) and the Gauss-Newton iteration method of obtaining the estimator is numerically stable. The least squares estimators converge to the true values at the rate of \(O\left( {\sqrt {2\log \left( {{{\log n} \mathord{\left/ {\vphantom {{\log n} n}} \right. \kern-0em} n}} \right)} } \right)\) with probability one, and has the asymptotically normal distribution.      16. The maximal variation of a bounded martingale      Jean-Francois Mertens  Shmuel Zamir 《Israel Journal of Mathematics》1977,27(3-4):252-276 Let \(\chi _0^n = \left\{ {X_t } \right\}_0^n \) be a martingale such that 0≦Xi≦1;i=0, …,n. For 0≦p≦1 denote by ? p n the set of all such martingales satisfying alsoE(X0)=p. Thevariation of a martingale χ 0 n is denoted byV 0 n and defined by \(V(\chi _0^n ) = E\left( {\sum {_{l = 0}^{n - 1} } \left| {X_{l + 1} - X_l } \right|} \right)\) . It is proved that $$\mathop {\lim }\limits_{n \to \infty } \left\{ {\mathop {Sup}\limits_{x_0^n \in \mathcal{M}_p^n } \left[ {\frac{1}{{\sqrt n }}V(\chi _0^n )} \right]} \right\} = \phi (p)$$ , where ?(p) is the well known normal density evaluated at itsp-quantile, i.e. $$\phi (p) = \frac{1}{{\sqrt {2\pi } }}\exp ( - \frac{1}{2}\chi _p^2 ) where \int_{ - \alpha }^{x_p } {\frac{1}{{\sqrt {2\pi } }}\exp ( - \frac{1}{2}\chi ^2 )} dx = p$$ . A sequence of martingales χ 0 n ,n=1,2, … is constructed so as to satisfy \(\lim _{n \to \infty } (1/\sqrt n )V(\chi _0^n ) = \phi (p)\) .      17. Bi-Lipschitz bijection between the Boolean cube and the Hamming ball      Itai Benjamini  Gil Cohen  Igor Shinkar 《Israel Journal of Mathematics》2016,212(2):677-703 We construct a bi-Lipschitz bijection from the Boolean cube to the Hamming ball of equal volume. More precisely, we show that for all even n ∈ N there exists an explicit bijection ψ: {0, 1}n → {x ∈ {0, 1}n+1 : |x| > n/2} such that for every x ≠ y ∈ {0, 1}n, \(\frac{1}{5} \leqslant \frac{{dis\tan ce\left( {\psi \left( x \right),\psi \left( y \right)} \right)}}{{dis\tan ce\left( {x,y} \right)}} \leqslant 4,\) where distance(·, ·) denotes the Hamming distance. In particular, this implies that the Hamming ball is bi-Lipschitz transitive.      18. Refinement of an inequality of P. L. Chebyshev      D. Dryanov  R. Fournier 《Acta Mathematica Hungarica》2009,122(1-2):59-69 Let P n denote the linear space of polynomials p(z:=Σ k=0 n a k (p)z k of degree ≦ n with complex coefficients and let |p|[?1,1]: = max x∈[?1,1]|p(x)| be the uniform norm of a polynomial p over the unit interval [?1, 1]. Let t n ∈ P n be the n th Chebyshev polynomial. The inequality $$ \frac{{\left| p \right|_{\left[ { - 1,1} \right]} }} {{\left| {a_n (p)} \right|}} \geqq \frac{{\left| {t_n } \right|_{\left[ { - 1,1} \right]} }} {{\left| {a_n (t_n )} \right|}},p \in P_n $$ due to P. L. Chebyshev can be considered as an extremal property of the Chebyshev polynomial t n in P n . The present note contains various extensions and improvements of the above inequality obtained by using complex analysis methods.      19. On the global wellposedness for the nonlinear Schrödinger equations with L p -large initial data      Ryosuke Hyakuna  Masayoshi Tsutsumi 《NoDEA : Nonlinear Differential Equations and Applications》2011,18(3):309-327 We consider the Cauchy problem for the nonlinear Schrödinger equations $ \begin{array}{l} iu_t + \triangle u \pm |u|^{p-1}u =0, \qquad x \in \mathbb{R}^d, \quad t \in \mathbb{R} \\ u(x,0)= u_0(x), \qquad x \in \mathbb{R}^d \end{array} $ for 1 < p < 1 + 4/d and prove that there is a ${\rho (p ,d) \in (1,2)}We consider the Cauchy problem for the nonlinear Schr?dinger equations l iut + \triangle u ±|u|p-1u = 0,        x ? \mathbbRd,     t ? \mathbbR u(x,0) = u0(x),        x ? \mathbbRd \begin{array}{l} iu_t + \triangle u \pm |u|^{p-1}u =0, \qquad x \in \mathbb{R}^d, \quad t \in \mathbb{R} \\ u(x,0)= u_0(x), \qquad x \in \mathbb{R}^d \end{array}      20. On the existence and smoothness of a generalized solution of a nonlinear differential equation with degeneration      S. A. Iskhokov 《Differential Equations》2008,44(2):241-255 Let Ω be an arbitrary open set in R n , and let σ(x) and g i (x), i = 1, 2, ..., n, be positive functions in Ω. We prove a embedding theorem of different metrics for the spaces W p r (Ω, σ, $ \vec g $ ), where r ∈ N, p ≥ 1, and $ \vec g $ (x) = (g 1(x), g 2(x), ..., g n (x)), with the norm $$ \left\| {u;W_p^r (\Omega ;\sigma ,\vec g)} \right\| = \left\{ {\left\| {u;L_{p,r}^r (\Omega ;\sigma ,\vec g)} \right\|^p + \left\| {u;L_{p,r}^0 (\Omega ;\sigma ,\vec g)} \right\|^p } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} , $$ where $$ \left\| {u;L_{p,r}^m (\Omega ;\sigma ,\vec g)} \right\| = \left\{ {\sum\limits_{\left| k \right| = m} {\int\limits_\Omega {(\sigma (x)g_1^{k_1 - r} (x)g_2^{k_2 - r} (x) \cdots g_n^{k_n - r} (x)\left| {u^{(k)} (x)} \right|)^p dx} } } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} , $$ We use this theorem to prove the existence and uniqueness of a minimizing element U(x) ∈ W p r (Ω, σ, $ \vec g $ ) for the functional $$ \Phi (u) = \sum\limits_{\left| k \right| \leqslant r} {\frac{1} {{p_k }}\int\limits_\Omega {a_k (x)} \left| {u^{(k)} (x)} \right|^{p_k } } dx - \left\langle {F,u} \right\rangle , $$ where F is a given functional. We show that the function U(x) is a generalized solution of the corresponding nonlinear differential equation. For the case in which Ω is bounded, we study the differential properties of the generalized solution depending on the smoothness of the coefficients and the right-hand side of the equation.      设为首页 | 免责声明 | 关于勤云 | 加入收藏 Copyright©北京勤云科技发展有限公司  京ICP备09084417号

Vertical versus horizontal Poincaré inequalities on the Heisenberg group

Let ? = 〈a, b|a[a, b] = [a, b]a ∧ b[a, b] = [a, b]b〉 be the discrete Heisenberg group, equipped with the left-invariant word metric d _W (·, ·) associated to the generating set {a, b, a ^?1, b ^?1}. Letting B _n = {x ∈ ?: d _W (x, e _?) ? n} denote the corresponding closed ball of radius n ∈ ?, and writing c = [a, b] = aba ^?1 b ^?1, we prove that if (X, ‖ · ‖X) is a Banach space whose modulus of uniform convexity has power type q ∈ [2,∞), then there exists K ∈ (0, ∞) such that every f: ? → X satisfies $$\sum\limits_{k = 1}^{{n^2}} {\sum\limits_{x \in {B_n}} {\frac{{\left\| {f(x{c^k}) - f(x)} \right\|_X^q}}{{{k^{1 + q/2}}}}} } \leqslant K\sum\limits_{x \in {B_{21n}}} {(\left\| {f(xa) - f(x)} \right\|_X^q + \left\| {f(xb) - f(x)} \right\|_X^q)} $$ . It follows that for every n ∈ ? the bi-Lipschitz distortion of every f: B _n → X is at least a constant multiple of (log n)^1/q , an asymptotically optimal estimate as n → ∞.     

Pointwise estimates for Lagrange interpolation polynomials

Let f ∈ C[?1, 1]. Let the approximation rate of Lagrange interpolation polynomial of f based on the nodes $ \left\{ {\cos \frac{{2k - 1}} {{2n}}\pi } \right\} \cup \{ - 1,1\} $ be Δ _{n + 2}(f, x). In this paper we study the estimate of Δ _{n + 2}(f,x), that keeps the interpolation property. As a result we prove that $$ \Delta _{n + 2} (f,x) = \mathcal{O}(1)\left\{ {\omega \left( {f,\frac{{\sqrt {1 - x^2 } }} {n}} \right)\left| {T_n (x)} \right|\ln (n + 1) + \omega \left( {f,\frac{{\sqrt {1 - x^2 } }} {n}\left| {T_n (x)} \right|} \right)} \right\}, $$ where T _n (x) = cos (n arccos x) is the Chebeyshev polynomial of first kind. Also, if f ∈ C ^r [?1, 1] with r ≧ 1, then $$ \Delta _{n + 2} (f,x) = \mathcal{O}(1)\left\{ {\frac{{\sqrt {1 - x^2 } }} {{n^r }}\left| {T_n (x)} \right|\omega \left( {f^{(r)} ,\frac{{\sqrt {1 - x^2 } }} {n}} \right)\left( {\left( {\sqrt {1 - x^2 } + \frac{1} {n}} \right)^{r - 1} \ln (n + 1) + 1} \right)} \right\}. $$      

Simultaneouse approximation to a differentiable function and its derivatives by Lagrange interpolating polynomials

This paper establishes the following pointwise result for simultancous Lagrange imterpolating approxima-tion:then|f~(k)(x)-P_n~(k)(f,x)|=O(1)△_n~(q-k)(x)ωwhere P_n(f,x)is the Lagrange interpolating potynomial of deereeon the nodesX_nUY_n(see the definition of the next).     

On functional equations associated with characters of unitary representations of groups

This is mainly concerned with the following functional equation:

Approximation operators and tauberian constants

The explicit expression of the smallest constantC satisfying $$\mathop {lim}\limits_{\lambda \to \infty } \left| {t_{n(\lambda )}^{(1)} - t_{m(\lambda )}^{(2)} } \right| \leqq C. \mathop {lim sup}\limits_{n \to \infty } \left| {d_n } \right|$$ for all sequences {s _n} satisfying lim sup _n→∞ |d _n| <∞, where {t _n ⁽¹⁾ }, {t _n ⁽²⁾ } are two generalised Hausdorff transforms of {s _n }, {d _n} is the generalised (C, α)-transform (0≦α≦1) of {λ _n a _n} andn(λ, m(λ) are suitably related, is obtained. These results are obtained by using new properties of positive approximation operators and generalised Bernstein approximation operators.     

Интегро-дифференциа ляные формулы сферических средних

Реэуме. Установлена формула для средних $$S(t)f(\mu ): = \frac{1}{{\left| {\sigma ^\prime } \right|}}\smallint _{\sigma (\mu )} f(\mu cos t + \nu sin t)d\nu $$ функцииf, определеннои на единичнои сфере а пространства R ⁿ ,n>2; σ(μ) — Экватор с полусом в μ, |σ′| — мера Экватора. Для проиэводнои ? _t ^(r) дано представление.     

On the existence of diffusions with singular drift coefficient

Let L:be an operator in R~d,where the matrix(a) is bounded,Holder continuous and uniformly positive definite,and (b~t(x)) is Borel measurable.In this paper we prove the existence of L-diffusion under the hypothesis that where g_1(z)=1,g_2(z)=-ln|z|and g_c(z)=|z|~(2-d) for d≥3.     

A sharp weightedL 2-estimate for the solution to the time-dependent Schrödinger equation

For Ξ∈R ⁿ ,t∈R andf∈S(R ⁿ ) define $\left( {S^2 f} \right)\left( t \right)\left( \xi \right) = \exp \left( {it\left| \xi \right|^2 } \right)\hat f\left( \xi \right)$ . We determine the optimal regularitys ₀ such that $\int_{R^n } {\left\| {(S^2 f)[x]} \right\|_{L^2 (R)}^2 \frac{{dx}}{{(1 + |x|)^b }} \leqslant C\left\| f \right\|_{H^s (R^n )}^2 ,s > s_0 } ,$ holds whereC is independent off∈S(R ⁿ ) or we show that such optimal regularity does not exist. This problem has been treated earlier, e.g. by Ben-Artzi and Klainerman [2], Kato and Yajima [4], Simon [6], Vega [9] and Wang [11]. Our theorems can be generalized to the case where the exp(it|ξ|²) is replaced by exp(it|ξ|^a),a≠2. The proof uses Parseval's formula onR, orthogonality arguments arising from decomposingL ²(R ⁿ ) using spherical harmonics and a uniform estimate for Bessel functions. Homogeneity arguments are used to show that results are sharp with respect to regularity.     

Some properties of elliptic Riesz means at critical index on Hp(TH)

Suppose f∈H^p(Tⁿ), 0 _r ^δ , δ=n/p?(n+1)/2. In this paper we eastablish the following inequality $$\mathop {\sup }\limits_{R > 1} \left\{ {\frac{1}{{\log R}}\int_1^R {\left\| {\sigma _r^\delta } \right\|_{H^p (T^R )}^p \frac{{dr}}{r}} } \right\}^{1/p} \leqslant C_{R,p} \left\| f \right\|_{H^p (T^R )} $$ It implies that $$\mathop {\lim }\limits_{R \to \infty } \frac{1}{{\log R}}\int_1^R {\left\| {\sigma _r^\delta - f} \right\|_{H^p (T^R )}^p \frac{{dr}}{r}} = 0$$ Moreover we obtain the same conclusion when p=1 and n=1.     

Concave iteration semigroups of linear continuous set-valued functions

Let {F ^t : t ≥ 0} be a concave iteration semigroup of linear continuous set-valued functions defined on a convex cone K with nonempty interior in a Banach space X with values in cc(K). If we assume that the Hukuhara differences F ⁰(x) ? F ^t (x) exist for x ∈ K and t > 0, then D _t F ^t (x) = (?1)F ^t ((?1)G(x)) for x ∈ K and t ≥ 0, where D _t F ^t (x) denotes the derivative of F ^t (x) with respect to t and $G(x) = \mathop {\lim }\limits_{s \to 0} {{\left( {F^0 \left( x \right) - F^s \left( x \right)} \right)} \mathord{\left/ {\vphantom {{\left( {F^0 \left( x \right) - F^s \left( x \right)} \right)} {\left( { - s} \right)}}} \right. \kern-0em} {\left( { - s} \right)}}$ for x ∈ K.     

Asymptotic properties of least squares estimators for discrete compound distributions

Let (X, Λ) be a pair of random variables, where Λ is an Ω (a compact subset of the real line) valued random variable with the density functiong(Θ: α) andX is a real-valued random variable whose conditional probability function given Λ=Θ is P {X=x|Θ} withx=x ₀, x₁, …. Based onn independent observations ofX, x ⁽ⁿ⁾, we are to estimate the true (unknown) parameter vectorα=(α ₁, α₂, ...,α_m) of the probability function ofX, P_α(X=∫_ΩP{X=x|Θ}g(Θ:α)dΘ. A least squares estimator of α is any vector \(\hat \alpha \left( {X^{\left( n \right)} } \right)\) which minimizes $$n^{ - 1} \sum\limits_{i = 1}^n {\left( {P_\alpha \left( {x_i } \right) - fn\left( {x_i } \right)} \right)^2 } $$ wherex ⁽ⁿ⁾=(x₁, x₂,…,x _n) is a random sample ofX andf _n(x_i)=[number ofx _i inx ⁽ⁿ⁾]/n. It is shown that the least squares estimators exist as a unique solution of the normal equations for all sufficiently large sample size (n) and the Gauss-Newton iteration method of obtaining the estimator is numerically stable. The least squares estimators converge to the true values at the rate of \(O\left( {\sqrt {2\log \left( {{{\log n} \mathord{\left/ {\vphantom {{\log n} n}} \right. \kern-0em} n}} \right)} } \right)\) with probability one, and has the asymptotically normal distribution.     

The maximal variation of a bounded martingale

Let \(\chi _0^n = \left\{ {X_t } \right\}_0^n \) be a martingale such that 0≦X_i≦1;i=0, …,n. For 0≦p≦1 denote by ? _p ⁿ the set of all such martingales satisfying alsoE(X₀)=p. Thevariation of a martingale χ ₀ ⁿ is denoted byV ₀ ⁿ and defined by \(V(\chi _0^n ) = E\left( {\sum {_{l = 0}^{n - 1} } \left| {X_{l + 1} - X_l } \right|} \right)\) . It is proved that $$\mathop {\lim }\limits_{n \to \infty } \left\{ {\mathop {Sup}\limits_{x_0^n \in \mathcal{M}_p^n } \left[ {\frac{1}{{\sqrt n }}V(\chi _0^n )} \right]} \right\} = \phi (p)$$ , where ?(p) is the well known normal density evaluated at itsp-quantile, i.e. $$\phi (p) = \frac{1}{{\sqrt {2\pi } }}\exp ( - \frac{1}{2}\chi _p^2 ) where \int_{ - \alpha }^{x_p } {\frac{1}{{\sqrt {2\pi } }}\exp ( - \frac{1}{2}\chi ^2 )} dx = p$$ . A sequence of martingales χ ₀ ⁿ ,n=1,2, … is constructed so as to satisfy \(\lim _{n \to \infty } (1/\sqrt n )V(\chi _0^n ) = \phi (p)\) .     

Bi-Lipschitz bijection between the Boolean cube and the Hamming ball

We construct a bi-Lipschitz bijection from the Boolean cube to the Hamming ball of equal volume. More precisely, we show that for all even n ∈ N there exists an explicit bijection ψ: {0, 1}ⁿ → {x ∈ {0, 1}ⁿ⁺¹ : |x| > n/2} such that for every x ≠ y ∈ {0, 1}ⁿ, \(\frac{1}{5} \leqslant \frac{{dis\tan ce\left( {\psi \left( x \right),\psi \left( y \right)} \right)}}{{dis\tan ce\left( {x,y} \right)}} \leqslant 4,\) where distance(·, ·) denotes the Hamming distance. In particular, this implies that the Hamming ball is bi-Lipschitz transitive.     

Refinement of an inequality of P. L. Chebyshev

Let P _n denote the linear space of polynomials p(z:=Σ _k=0 ⁿ a _k (p)z ^k of degree ≦ n with complex coefficients and let |p|_[?1,1]: = max _x∈[?1,1]|p(x)| be the uniform norm of a polynomial p over the unit interval [?1, 1]. Let t _n ∈ P _n be the n ^th Chebyshev polynomial. The inequality $$ \frac{{\left| p \right|_{\left[ { - 1,1} \right]} }} {{\left| {a_n (p)} \right|}} \geqq \frac{{\left| {t_n } \right|_{\left[ { - 1,1} \right]} }} {{\left| {a_n (t_n )} \right|}},p \in P_n $$ due to P. L. Chebyshev can be considered as an extremal property of the Chebyshev polynomial t _n in P _n . The present note contains various extensions and improvements of the above inequality obtained by using complex analysis methods.     

On the global wellposedness for the nonlinear Schrödinger equations with L p -large initial data

We consider the Cauchy problem for the nonlinear Schrödinger equations $ \begin{array}{l} iu_t + \triangle u \pm |u|^{p-1}u =0, \qquad x \in \mathbb{R}^d, \quad t \in \mathbb{R} \\ u(x,0)= u_0(x), \qquad x \in \mathbb{R}^d \end{array} $ for 1 < p < 1 + 4/d and prove that there is a ${\rho (p ,d) \in (1,2)}We consider the Cauchy problem for the nonlinear Schr?dinger equations

l iut + \triangle u ±|u|p-1u = 0,        x ? \mathbbRd,     t ? \mathbbR u(x,0) = u0(x),        x ? \mathbbRd \begin{array}{l} iu_t + \triangle u \pm |u|^{p-1}u =0, \qquad x \in \mathbb{R}^d, \quad t \in \mathbb{R} \\ u(x,0)= u_0(x), \qquad x \in \mathbb{R}^d \end{array}      20. On the existence and smoothness of a generalized solution of a nonlinear differential equation with degeneration      S. A. Iskhokov 《Differential Equations》2008,44(2):241-255 Let Ω be an arbitrary open set in R n , and let σ(x) and g i (x), i = 1, 2, ..., n, be positive functions in Ω. We prove a embedding theorem of different metrics for the spaces W p r (Ω, σ, $ \vec g $ ), where r ∈ N, p ≥ 1, and $ \vec g $ (x) = (g 1(x), g 2(x), ..., g n (x)), with the norm $$ \left\| {u;W_p^r (\Omega ;\sigma ,\vec g)} \right\| = \left\{ {\left\| {u;L_{p,r}^r (\Omega ;\sigma ,\vec g)} \right\|^p + \left\| {u;L_{p,r}^0 (\Omega ;\sigma ,\vec g)} \right\|^p } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} , $$ where $$ \left\| {u;L_{p,r}^m (\Omega ;\sigma ,\vec g)} \right\| = \left\{ {\sum\limits_{\left| k \right| = m} {\int\limits_\Omega {(\sigma (x)g_1^{k_1 - r} (x)g_2^{k_2 - r} (x) \cdots g_n^{k_n - r} (x)\left| {u^{(k)} (x)} \right|)^p dx} } } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} , $$ We use this theorem to prove the existence and uniqueness of a minimizing element U(x) ∈ W p r (Ω, σ, $ \vec g $ ) for the functional $$ \Phi (u) = \sum\limits_{\left| k \right| \leqslant r} {\frac{1} {{p_k }}\int\limits_\Omega {a_k (x)} \left| {u^{(k)} (x)} \right|^{p_k } } dx - \left\langle {F,u} \right\rangle , $$ where F is a given functional. We show that the function U(x) is a generalized solution of the corresponding nonlinear differential equation. For the case in which Ω is bounded, we study the differential properties of the generalized solution depending on the smoothness of the coefficients and the right-hand side of the equation.

On the existence and smoothness of a generalized solution of a nonlinear differential equation with degeneration

Let Ω be an arbitrary open set in R _n , and let σ(x) and g _i (x), i = 1, 2, ..., n, be positive functions in Ω. We prove a embedding theorem of different metrics for the spaces W _p ^r (Ω, σ, $ \vec g $ ), where r ∈ N, p ≥ 1, and $ \vec g $ (x) = (g ₁(x), g ₂(x), ..., g _n (x)), with the norm $$ \left\| {u;W_p^r (\Omega ;\sigma ,\vec g)} \right\| = \left\{ {\left\| {u;L_{p,r}^r (\Omega ;\sigma ,\vec g)} \right\|^p + \left\| {u;L_{p,r}^0 (\Omega ;\sigma ,\vec g)} \right\|^p } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} , $$ where $$ \left\| {u;L_{p,r}^m (\Omega ;\sigma ,\vec g)} \right\| = \left\{ {\sum\limits_{\left| k \right| = m} {\int\limits_\Omega {(\sigma (x)g_1^{k_1 - r} (x)g_2^{k_2 - r} (x) \cdots g_n^{k_n - r} (x)\left| {u^{(k)} (x)} \right|)^p dx} } } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} , $$ We use this theorem to prove the existence and uniqueness of a minimizing element U(x) ∈ W _p ^r (Ω, σ, $ \vec g $ ) for the functional $$ \Phi (u) = \sum\limits_{\left| k \right| \leqslant r} {\frac{1} {{p_k }}\int\limits_\Omega {a_k (x)} \left| {u^{(k)} (x)} \right|^{p_k } } dx - \left\langle {F,u} \right\rangle , $$ where F is a given functional. We show that the function U(x) is a generalized solution of the corresponding nonlinear differential equation. For the case in which Ω is bounded, we study the differential properties of the generalized solution depending on the smoothness of the coefficients and the right-hand side of the equation.