On the strong approximation by the (C, α)-means of Fourier series. II

стАтьь ьВльЕтсь пРОД ОлжЕНИЕМ пРЕДыДУЩЕИ ОДНОИМЕННОИ РАБОты АВтОРА, гДЕ ИжУ ЧАлсь пОРьДОк ВЕлИЧИН пРИ УслОВИьх, ЧтО α>-1/2, Рα >- 1 И ЧтО МАтРИцАt _nk УДОВлЕтВОРьЕт НЕкОт ОРОМУ УслОВИУ РЕгУльРНОстИ. жДЕсь ДОкАжыВАЕтсь, Ч тО ЕслИf∈H ^Ω, тО ВыпОлНь Етсь ОцЕНкА $$\left\{ {\frac{1}{{\lambda _n }}\mathop \Sigma \limits_{k = n - \lambda _n + 1}^n \left| {\sigma _k^\alpha \left( x \right) - f\left( x \right)} \right|^p } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} = O\left( {\left\{ {\frac{1}{{\lambda _n }}\mathop \Sigma \limits_{k = n - \lambda _n + 1}^n \left( {\frac{1}{k}\mathop \smallint \limits_{{1 \mathord{\left/ {\vphantom {1 k}} \right. \kern-\nulldelimiterspace} k}}^{2\pi } \frac{{\omega \left( t \right)}}{{t^2 }}dt} \right)^p } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} + \left( {\frac{{\lambda _n }}{n}} \right)^\alpha \omega \left( {\frac{1}{n}} \right)} \right)$$ (λ₁=1, λ_n+1-λ_n≦1), А тАкжЕ ЧтО Ёт А ОцЕНкА ОкОНЧАтЕльН А В сВОИх тЕРМИНАх; пОДОБ НыИ РЕжУль-тАт спРАВЕДлИВ тАкжЕ И Дль сОпРьжЕННОИ ФУНкцИИ . ДОкАжыВАЕтсь, ЧтО Усл ОВИьα>?1/2 Иpα>?1, кОтОРыЕ Б ылИ НАлОжЕНы В УпОМьНУтО И ВышЕ ЧАстИ I, сУЩЕстВЕН Ны.     

Generalized Knopp identities for homogeneous Hardy sums and Cochrane-Hardy sums

Let q, h, a, b be integers with q > 0. The classical and the homogeneous Dedekind sums are defined by $$s(h,q) = \sum\limits_{j = 1}^q {\left( {\left( {{j \over q}} \right)} \right)\left( {\left( {{{hj} \over q}} \right)} \right),{\rm{ }}s(a,b,q) = \sum\limits_{j = 1}^q {\left( {\left( {{{aj} \over q}} \right)} \right)\left( {\left( {{{bj} \over q}} \right)} \right),} } $$ respectively, where $((x)) = \left\{ \begin{gathered} x - [x] - \tfrac{1} {2},if x is not an integer; \hfill \\ 0,if x is an integer. \hfill \\ \end{gathered} \right. $ The Knopp identities for the classical and the homogeneous Dedekind sum were the following: $$\sum\limits_{d|n} {\sum\limits_{r = 1}^d {s\left( {{n \over d}a + rq,dq} \right) = \sigma (n)s(a,q),} } $$ $$\sum\limits_{d|n} {\sum\limits_{{r_1} = 1}^d {\sum\limits_{{r_2} = 1}^d s \left( {{n \over d}a + {r_1}q,{n \over d}b + {r_2}q,dq} \right) = n\sigma (n)s(a,b,q),} } $$ where σ(n) =Σ _d|n d. In this paper generalized homogeneous Hardy sums and Cochrane-Hardy sums are defined, and their arithmetic properties are studied. Generalized Knopp identities for homogeneous Hardy sums and Cochrane-Hardy sums are given.     

Estimation of Kloosterman sums with primes and its application

Suppose that p is a large prime. In this paper, we prove that, for any natural number N < p the following estimate holds: $$ \left. {\mathop {\max }\limits_{\left( {a,p} \right) = 1} } \right|\left. {\sum\limits_{q \leqslant N} {e^{{{2\pi iaq*} \mathord{\left/ {\vphantom {{2\pi iaq*} p}} \right. \kern-\nulldelimiterspace} p}} } } \right| \leqslant \left( {N^{{{15} \mathord{\left/ {\vphantom {{15} {16}}} \right. \kern-\nulldelimiterspace} {16}}} + N^{{2 \mathord{\left/ {\vphantom {2 3}} \right. \kern-\nulldelimiterspace} 3}} p^{{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-\nulldelimiterspace} 4}} } \right)p^{0\left( 1 \right)} , $$ where q is a prime and q* is the least natural number satisfying the congruence qq* ≡ 1 (modp). This estimate implies the following statement: if p > N > p ^16/17+? , where ? > 0, and if we have λ ? 0 (modp), then the number J of solutions of the congruence $$ q_1 \left( {q_2 + q_3 } \right) \equiv \lambda \left( {\bmod p} \right) $$ for the primes q ₁, q ₂, q ₃ ≤ N can be expressed as $$ J = \frac{{\pi \left( N \right)^3 }} {p}\left( {1 + O\left( {p^{ - \delta } } \right)} \right), \delta = \delta \left( \varepsilon \right) > 0. $$ This statement improves a recent result of Friedlander, Kurlberg, and Shparlinski in which the condition p > N > p ^38/39+? was required.     

Regularity results for a class of obstacle problems in Heisenberg groups

We study regularity results for solutions u ∈ HW ^1,p (Ω) to the obstacle problem $$\int_\Omega \mathcal{A} \left( {x,\nabla _{\mathbb{H}^u } } \right)\nabla _\mathbb{H} \left( {v - u} \right)dx \geqslant 0 \forall v \in \mathcal{K}_{\psi ,u} \left( \Omega \right)$$ such that u ? ψ a.e. in Ω, where $xxx$ , in Heisenberg groups ? ⁿ . In particular, we obtain weak differentiability in the T-direction and horizontal estimates of Calderon-Zygmund type, i.e. $$\begin{gathered} T\psi \in HW_{loc}^{1,p} \left( \Omega \right) \Rightarrow Tu \in L_{loc}^p \left( \Omega \right), \hfill \\ \left| {\nabla _{\mathbb{H}\psi } } \right|^p \in L_{loc}^q \left( \Omega \right) \Rightarrow \left| {\nabla _{\mathbb{H}^u } } \right|^p \in L_{loc}^q \left( \Omega \right), \hfill \\ \end{gathered}$$ where 2 < p < 4, q > 1.     

Strong and best approximation of Fourier series and the Lipschitz classes

Статья содержит весьма общее утверждение о сильном приближении суммами Фурье (теорема 1), из котороговытекают многие ранее известные, а также некоторые новые резуль-таты В частности доказано, что неравенство $$E_n (f) \leqq K\left\| {\left. {\left\{ {\frac{1}{{n^\beta }}\sum\limits_{v = 0}^n {(v + 1)^{\beta - 1} |S_v - f|^p } } \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} } \right\|} \right.$$ выполнено для любых положительныхp иβ. Теорема 4 содержит достаточное условие для того, чтобы производнаяf(r) почти всюду удовлетворяла условию типа Lip α (0 α1). Теорема 5 показывает, что для любогор0 и неотрицательного целогоr из условия (20) следует, чтоf(r)еLiр 1.     

Small complete arcs in Galois planes

In this paper we construct a class of k-arcs in PG(2, q), q=p ^h, h>1, p≠3 and prove its completeness for h large enough. The main result states that this class contains complete k-arcs with $$k \leqslant 2 \cdot q^{{9 \mathord{\left/ {\vphantom {9 {10}}} \right. \kern-\nulldelimiterspace} {10}}} {\text{ }}\left( {10{\text{ divides }}h{\text{ and }}q{\text{ }} \geqslant {\text{ }}q_{\text{0}} } \right).$$ Such complete k-arcs are the unique known complete k-arcs with $$k \leqslant {q \mathord{\left/ {\vphantom {q 4}} \right. \kern-\nulldelimiterspace} 4}.$$      

L 1 decay properties for a semilinear parabolic system

This article is concerned with the decay property in theL ¹ norm ast»∞ of the nonnegative solutions of the initial value problem in ? ⁿ $\left\{ {\begin{array}{*{20}c} {u_t = \Delta u + \mu |\nabla \upsilon |^q } \\ {\upsilon _t = \Delta \upsilon + \upsilon |\nabla \upsilon |^p } \\ \end{array} } \right.$ for different values of the parametersp, q≥1 and when μ, ν<0. If $pq > \frac{{\inf \left( {p,q} \right)}}{{n + 1}} + \left( {n + 2} \right)/\left( {n + 1} \right)$ then lim _t→∞∥u(t)+v(t)∥₁>0 and when $pq< \frac{{\inf \left( {p,q} \right)}}{{n + 1}} + \left( {n + 2} \right)/\left( {n + 1} \right)$ then lim _t→∞∥u(t)+v(t)∥₁>0.     

Some embedding theorems for generalized Nikol'skii classes from Lorentz spaces

Quasi-normed Lorentz spaces Λ_{ψ, q} of 2π-periodic functions with quasinorms $$\left\| f \right\|_{\psi ,q} = \left\{ {\int\limits_0^{2\pi } {\psi ^q (t)\left[ {\frac{1}{t}\int\limits_0^t {f * (x)} dx} \right]} ^q \frac{{dt}}{t}} \right\}^{{1 \mathord{\left/ {\vphantom {1 q}} \right. \kern-\nulldelimiterspace} q}} $$ (0<q<∞,ω(t): [0,2π]→R is a continuous concave function with finite derivative everywhere on (0, 2gp)) and classes of functions $$H_{\psi ,q}^\omega \equiv \{ f(x):f(x) \in \Lambda _{\psi ,q} ;\mathop {\sup }\limits_{0 \leqq h \leqq \delta } \left\| {f(x + h) - f(x)} \right\|_{\psi ,q} = O\{ \omega (\delta )\} , \delta \to + 0\} $$ (ω(δ) — modulus of continuity) are studied. Precise embedding conditions of classes H _{ψ, q} ^ω into Lorentz spaces and into each other are obtained: $$\begin{array}{*{20}c} {H_{\psi ,q_1 }^\omega \subset \Lambda _{\psi ,q_2 } ;} & {H_{\psi ,q_1 }^\omega \subset {\rm H}_{\psi ,q_2 }^{\omega * } ,} & {0< q_2< q_1< \infty ,} \\ \end{array} $$ under conditions \(\mathop {\lim }\limits_{t \to \infty } \frac{{\psi (2t)}}{{\psi (t)}} > 1,\mathop {\overline {\lim } }\limits_{x \to \infty } \frac{{\psi (2t)}}{{\psi (t)}}< 2\) andω(δ)=O{ω(δ ²)},δ→+0, andω ^* (δ) is an arbitrary modulus of continuity.     

Sharp constants in the doubly weighted Hardy-Littlewood-Sobolev inequality

It is well known that the doubly weighted Hardy-Littlewood-Sobolev inequality is as follows,Z Rn Z Rn f(x)g(y)|x||x.y||y|dxdy6 B(p,q,,,,n)kfkLp(Rn)kgkLq(Rn).The main purpose of this paper is to give the sharp constants B(p,q,,,,n)for the above inequality for three cases:(i)p=1 and q=1;(ii)p=1 and 1q 6∞,or 1p 6∞and q=1;(iii)1p,q∞and 1p+1q=1.In addition,the explicit bounds can be obtained for the case 1p,q∞and 1p+1q1.     

Sharp Estimates for Integral Means for Three Classes of Domains

In this paper, the following sharp estimate is proved: $$\int_{0}^{2{\pi }} {\left| {F\prime \left( {e^{i\theta } } \right)} \right|^p d\theta \leqslant \sqrt {\pi } 2^{1 + p} \frac{{\gamma \left( {{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} + {p \mathord{\left/ {\vphantom {p 2}} \right. \kern-\nulldelimiterspace} 2}} \right)}} {{\gamma \left( {1 + {p \mathord{\left/ {\vphantom {p 2}} \right. \kern-\nulldelimiterspace} 2}} \right)}}} ,\quad p > - 1,$$ where F is the conformal mapping of the domain $D^ - = \left\{ {\zeta :\left| \zeta \right| > 1} \right\}$ onto the exterior of a convex curve, with $F\prime \left( \infty \right) = 1$ . For p=1, this result is due to Pólya and Shiffer. We also obtain several generalizations of this estimate under other geometric assumptions about the structure of the domain F(D ^-).     

Sharp weak-type inequalities for Hilbert transform and Riesz projection

The paper is devoted to the study of the weak norms of the classical operators in the vector-valued setting.

1. Let S, H denote the singular integral involution operator and the Hilbert transform on $L^p \left( {\mathbb{T}, \ell _\mathbb{C}^2 } \right)$ , respectively. Then for 1 ≤ p ≤ 2 and any f, $$\left\| {\mathcal{S}f} \right\|_{p,\infty } \leqslant \left( {\frac{1} {\pi }\int_{ - \infty }^\infty {\frac{{\left| {\tfrac{2} {\pi }\log \left| t \right|} \right|^p }} {{t^2 + 1}}dt} } \right)^{ - 1/p} \left\| f \right\|p,$$ $$\left\| {\mathcal{H}f} \right\|_{p,\infty } \leqslant \left( {\frac{1} {\pi }\int_{ - \infty }^\infty {\frac{{\left| {\tfrac{2} {\pi }\log \left| t \right|} \right|^p }} {{t^2 + 1}}dt} } \right)^{ - 1/p} \left\| f \right\|p.$$ Both inequalities are sharp.
2. Let P ₊ and P _? stand for the Riesz projection and the co-analytic projection on $L^p \left( {\mathbb{T}, \ell _\mathbb{C}^2 } \right)$ , respectively. Then for 1 ≤ p ≤ 2 and any f, $$\left\| {P + f} \right\|_{p,\infty } \leqslant \left\| f \right\|_p ,$$ $$\left\| {P - f} \right\|_{p,\infty } \leqslant \left\| f \right\|_p .$$ Both inequalities are sharp.
3. We establish the sharp versions of the estimates above in the nonperiodic case.

The results are new even if the operators act on complex-valued functions. The proof rests on the construction of an appropriate plurisubharmonic function and probabilistic techniques.     

The convergence of moments in the central limit theorem for stationary ϕ-mixing processes

Пусть {X_j} - строго стац ионарная последоват ельностьс ?перемешиванием, EXj-Q,E¦-X _j¦^r<∞ для некоторогоr>2. Положим \(S_n = \mathop \sum \limits_{j = 1}^n X_j \) . Ибрагимов (1962) доказал, что если приn →∞, то 1 $$\mathop {\lim }\limits_{n \to \infty } P\{ S_n /\sigma _n< x\} = (2\pi )^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \mathop \smallint \limits_{ - \infty }^x e^{{{ - u^2 } \mathord{\left/ {\vphantom {{ - u^2 } 2}} \right. \kern-\nulldelimiterspace} 2}} du.$$ В работе установлено, что при указанных выш е условиях в этой центральной пр едельной теореме имеет место т акже и сходимостьr-ых абсолютных моментов, т.е. если σ _n ² →∞ приn→ ∞, то $$\mathop {\lim }\limits_{n \to \infty } E|S_n /\sigma _n |^r = (2\pi )^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} \mathop \smallint \limits_{ - \infty }^{ + \infty } |u|^r e^{ - u^2 /2} du.$$ Этот результат обобщ ает один более ранний результат автора (1980 г.).     

Some estimates of differentiable functions

Suppose thatx(t) ∈ C _[a,b ^(n)] and has n zeros at the pointsa and b. It is shown that if x⁽ⁿ⁾(t) preserves sign on [a, b], then $$\left| {x\left( t \right)} \right| \geqslant \frac{{p_0 }}{{n - 1}}\mathop {\left[ {\mathop {\sup }\limits_{\tau \in \left( {a, b} \right)} \frac{{\left| {x\left( \tau \right)} \right|}}{{\left( {\tau - a} \right)^{p - 1} \left( {b - \tau } \right)^{q - 1} }}} \right]}\limits_{\left( {a< t< b} \right),} \left( {t - a} \right)^p \left( {b - t} \right)^q $$ where p and q are the multiplicities of the zeros of x(t) ata and b, respectively, and p_o=min{p,q}. Two-sided estimates of the Green's function for a two-point interpolation problem for the operator Lx ≡ x⁽ⁿ⁾ are established in the proof. As an application, new conditions for the solvability of de la Vallée Poussin's two-point boundary problems are obtained.     

On the best polynomial approximation of entire transcendental functions in banach spaces. I

We study the behavior of the best approximationsE _n (?) _p of entire transcendental functions ?(z) of the order ρ=∞ by polynomials of at mostn th degree in the metric of the Banach space E′_p(Ω) of functions /tf(z) analytic in a bounded simply connected domain Ω with rectifiable Jordan boundary and such that $$\left\| f \right\|_{E'_p } = \left\{ {\iint_\Omega {\left| {f\left( z \right)} \right|^p }dxdy} \right\}^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}}< \infty $$ . In particular, we describe the relationship between the best approximationsE _n (?)_p and theq-order andq-type of the function ?(z).     

A Landesman-Lazer Type Theorem for Periodic Solutions of the Resonant Asymmetric ρ-Laplacian Equation

In this paper, we give a Landesman-Lazer type theorem for periodic solutions of the asymmetric 1-dimensional p-Laplacian equation -（｜x＇｜^p-2x＇）＇=λ｜x｜^p-2x＋＋μ｜x｜^p-2x-＋f（t,x）with periodic boundary value.     

Multiple rational trigonometric sums and multiple integrals

We obtain an estimate of the modulus of a complete multiple rational trigonometric sum: $$\left| {\sum {_{x_{1, \ldots ,} x_r = 1^{\exp \left( {{{2\pi if\left( {x_{1, \ldots ,} x_r } \right)} \mathord{\left/ {\vphantom {{2\pi if\left( {x_{1, \ldots ,} x_r } \right)} q}} \right. \kern-\nulldelimiterspace} q}} \right)} }^q } } \right| \ll q^{{{r - 1} \mathord{\left/ {\vphantom {{r - 1} {n + \varepsilon }}} \right. \kern-\nulldelimiterspace} {n + \varepsilon }}} ,$$ where $$\begin{gathered} f\left( {x_{1, \ldots ,} x_r } \right) = \sum {_{0 \leqslant t_1 , \ldots ,t_r \leqslant n^a t_1 , \ldots ,t_r x_1^{t_1 } \ldots x_r^{t_r } ,} } \hfill \\ a_{0, \ldots ,0} = 0,\left( {a_{0, \ldots ,0,1} , \ldots ,a_{n, \ldots ,n,} q} \right) = 1 \hfill \\ \end{gathered} $$ , and an estimate of the modulus of a multiple trigonometric integral.     

Complete rational arithmetic sums

Let {itq} > 1 be an integer number, \(f\left( x \right) = {a_n}{x^n} + \ldots + {a_1}x + {a_0}\) be a polynomial with integer coefficients, and ({ita}{in{itn}}, . . . ,{ita}{in1},{itq}) = 1. The following estimate is valid: \(\left| {S\left( {\frac{{f\left( x \right)}}{q}} \right)} \right| = \left| {\sum\limits_{x = 1}^q \rho \left( {\frac{{f\left( x \right)}}{q}} \right)} \right| \ll {q^{1 - 1/n}}\), where \(\rho \left( t \right) = 0,5 - \left\{ t \right\}\).     

On the asymptotics of the Riesz-Bochner kernel

В работе рассматрива ется асимптотика в ме трике пространстваL _p (T ^N ),T ^N ={x∈R ^N , 0<x _i <2π} ядра Р исса-Бохнера $$\Theta ^s \left( {x, \lambda } \right) = \left( {2\pi } \right)^{ - N} \mathop \Sigma \limits_{\left| n \right|^2< \lambda } \left( {1 - \frac{{\left| n \right|^2 }}{\lambda }} \right)^s e^{inx} \left( {x \in T^N , s \geqq 0, \lambda \geqq 0} \right)$$ при λ→∞. Доказывается, что есл иN≧4,p≧2N/(N?1) иs>N((N?1)/2N?1/p), то для произвольной точкиx∈T ^N существует п остояннаяC=C _p (x, s) такая, что выполняется неравен ство $$\parallel \Theta ^s \left( {x - y, \lambda } \right) - \left( {2\pi } \right)^{ - {N \mathord{\left/ {\vphantom {N 2}} \right. \kern-\nulldelimiterspace} 2}} 2^s \Gamma \left( {s + 1} \right)\lambda ^{{N \mathord{\left/ {\vphantom {N 2}} \right. \kern-\nulldelimiterspace} 2}} J_{{N \mathord{\left/ {\vphantom {N {2 + s}}} \right. \kern-\nulldelimiterspace} {2 + s}}} {{\left( {\left| {x - y} \right|\sqrt \lambda } \right)} \mathord{\left/ {\vphantom {{\left( {\left| {x - y} \right|\sqrt \lambda } \right)} {\left( {\left| {x - y} \right|\sqrt \lambda } \right)^{{N \mathord{\left/ {\vphantom {N {2 + s}}} \right. \kern-\nulldelimiterspace} {2 + s}}} \parallel _{L_p \left( {T^N } \right)} \leqq }}} \right. \kern-\nulldelimiterspace} {\left( {\left| {x - y} \right|\sqrt \lambda } \right)^{{N \mathord{\left/ {\vphantom {N {2 + s}}} \right. \kern-\nulldelimiterspace} {2 + s}}} \parallel _{L_p \left( {T^N } \right)} \leqq }}$$ где нормаL _p (T ^N ) берется по пе ременнойy, а черезJ _v обозначена функция Б есселя первого рода порядкаv. СлучаиN=2 иN=3 рассматриваются отдельно.     

Real Clifford Algebras as Tensor Products over Centers

Denote by ${\mathcal{C}\ell_{p,q}}$ the Clifford algebra on the real vector space ${\mathbb{R}^{p,q}}$ . This paper gives a unified tensor product expression of ${\mathcal{C}\ell_{p,q}}$ by using the center of ${\mathcal{C}\ell_{p,q}}$ . The main result states that for nonnegative integers p, q, ${\mathcal{C}\ell_{p,q} \simeq \otimes^{\kappa-\delta}\mathcal{C}_{1,1} \otimes Cen(\mathcal{C}\ell_{p,q}) \otimes^{\delta} \mathcal{C}\ell_{0,2},}$ where ${p + q \equiv \varepsilon}$ mod 2, ${\kappa = ((p + q) - \varepsilon)/2, p - |q - \varepsilon| \equiv i}$ mod 8 and ${\delta = \lfloor i / 4 \rfloor}$ .