A Tauberian theorem for ℓ-adic sheaves on \mathbb{A} 1

Let K ∈ L ¹(?) and let f ∈ L ^∞(?) be two functions on ?. The convolution $$ \left( {K*F} \right)\left( x \right) = \int_\mathbb{R} {K\left( {x - y} \right)f\left( y \right)dy} $$ can be considered as an average of f with weight defined by K. Wiener’s Tauberian theorem says that under suitable conditions, if $$ \mathop {\lim }\limits_{x \to \infty } \left( {K*F} \right)\left( x \right) = \mathop {\lim }\limits_{x \to \infty } \int_\mathbb{R} {\left( {K*A} \right)\left( x \right)} $$ for some constant A, then $$ \mathop {\lim }\limits_{x \to \infty } f\left( x \right) = A $$ We prove the following ?-adic analogue of this theorem: Suppose K, F, G are perverse ?-adic sheaves on the affine line $ \mathbb{A} $ over an algebraically closed field of characteristic p (p ≠ ?). Under suitable conditions, if $ \left( {K*F} \right)|_{\eta _\infty } \cong \left( {K*G} \right)|_{\eta _\infty } $ , then $ F|_{\eta _\infty } \cong G|_{\eta _\infty } $ , where η _∞ is the spectrum of the local field of $ \mathbb{A} $ at ∞.     

Symmetry breaking bifurcation from solutions concentrating on the equator of \mathbb{S}^N

We are concerned with the elliptic problem $${\varepsilon ^2}{\Delta _{{S^n}}}u - u + {u^p} = 0{\text{ in }}{S^n},u > 0{\text{ in }}{S^n}$$ , where ${\Delta _{{S^n}}}$ is the Laplace-Beltrami operator on $\mathbb{S}^n : = \left\{ {x \in \mathbb{R}^{n + 1} ;\left\| x \right\| = 1} \right\}\left( {n \geqslant 3} \right)$ , and p ? 2. We construct a smooth branch C of solutions concentrating on the equator S ⁿ ∩ {x _{n+1 = 0}}. Using the Crandall-Rabinowitz bifurcation theorem, we show that C has infinitely many bifurcation points from which continua of nonradial solutions emanate. In applying the bifurcation theorem, we verify the transversality condition directly.     

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 \(\lim _{n \to \infty } (1/\sqrt n )V(\chi _0^n ) = \phi (p)\) .     

Представление целых функций рядами по фун кциям Миттаг-Леффлера

The class \(B_{\varrho _1 } \) is introduced and thoroughly studied in the paper. By definition,H∈ \(B_{\varrho _1 } \) if there exist sequences {А _n } and {μ _n }, ¦μ _n ¦ ↑ ∞ (depending onH(?)) such that $$\mathop {\lim \sup }\limits_{t \to \infty } \frac{{\ln \Phi \left( {re^{i\varphi } } \right)}}{{r^{\varrho _1 } }} = H\left( \varphi \right), \Phi \left( z \right) = \mathop \Sigma \limits_{k = 1}^\infty \left| {A_k E_\varrho \left( {\lambda _k z} \right)} \right|,$$ whereE _? (z) is a Mittag—Leffler function and? ₁>?>1/2. The significance of the class \(B_{\varrho _1 } \) is confirmed by the following theorem. For each functionH∈ \(B_{\varrho _1 } \) there exists a sequence {λ _n } with the following property: every entire functionF(z) of order? ₁ with the growth indicatorh _F (?)< <H(?) can be expanded into the series $$F\left( z \right) = \mathop \Sigma \limits_{n = 1}^\infty a_n E_\varrho \left( {\lambda _n z} \right),$$ furthermore, $$\mathop {\lim sup}\limits_{r \to \infty } \frac{{\ln \Phi \left( {re^{i\varphi } } \right)}}{{r^{\varrho 1} }}< H\left( \varphi \right), \Phi \left( z \right) = \mathop \Sigma \limits_{n = 1}^\infty \left| {a_n E_\varrho \left( {\lambda _n z} \right)} \right|.$$ The coefficientsa _n are explicitly defined. The results were previously announced by the author inDokl. AN SSSR,264 (1982), 1313–1315.     

A generalization of the katona theorem for crosst-intersecting families

LetX be ann-element set and letA and? be families of subsets ofX. We say thatA and? are crosst-intersecting if |A ∩ B| ≥ t holds for all A ∈A and for allB ∈ ?. Suppose thatA and ? are crosst-intersecting. This paper first proves a crosst-intersecting version of Harper's Theorem:

1. There are two crosst-intersecting Hamming spheresA ₀,? ₀ with centerX such that |A| ≤ |A ₀| and|?| ≤ |? ₀| hold.
2. Suppose thatt ≥ 2 and that the pair of integers (|A) is maximal with respect to direct product ordering among pairs of crosst-intersecting families. Then,A and? are Hamming spheres with centerX.

Using these claims, the following conjecture of Frankl is proven:

1. Ifn + t = 2k ? 1 then |A| |?| ≤ max \(\left\{ {\left( {K_k^n + \left( {_{k - 1}^{n - 1} } \right)} \right)^2 ,K_k^n K_{k - 1}^n } \right\}\) holds, whereK _l ⁿ is defined as \(\left( {_n^n } \right)\left( {_{n - 1}^n } \right) + \cdots + \left( {_l^n } \right).\)
2. Ifn + t = 2k then |A| |? ≤ (K _k ⁿ )² holds.

The extremal configurations are also determined.     

On a transformation of power series

Изучаются ряды Риман а, рассматривавшиеся ранее в работах [1] и [2]. Пустьa _n (n=1, 2,…) — последов ательность комплекс ных чисел иr _n =a _n +a _2n +…. Предполо жим, чтоΣ¦a _n ¦<∞. Тогда выпо лняются неравенства $$\begin{array}{*{20}c} {\sum\limits_n {\left| {r_n } \right| \leqq } \sum\limits_n {\left| {a_n } \right|} d(n),} & {\sum\limits_n {\left| {a_n } \right|} } \\ \end{array} \leqq \sum\limits_n {\left| {r_n } \right|2^{\omega (n)} ,} $$ гдеd(n) иω(n) — соответств енно число делителей и число простых делителейn. Е сли $$\begin{array}{*{20}c} {F(z) = \sum\limits_n {a_n z^n ,} } & {p_n (z) = \sum\limits_{s|n} {\mu \left( {\frac{n}{s}} \right)z^s ,} } \\ \end{array} $$ то \(F(z) = \sum\limits_n {r_n p_n (z)} \) для ¦z¦<1. В статье с одержатся некоторые результаты о сходимо сти рядов РиманаΣt _n p _n (z) на окружно сти ¦z¦=1. Например, если числаt _n неотрицатель ны, монотонно убывают и \(\sum\limits_n {t_n< \infty } \) , то ряд равн омерно сходится для ¦z¦=1. Сформулированы неко торые новые задачи.     

Index formulae for subspaces of Kreîn spaces

For a subspaceS of a Kreîn spaceK and an arbitrary fundamental decompositionK=K ^?[+]K ⁺ ofK, we prove the index formula $$\kappa ^ - \left( \mathcal{S} \right) + \dim \left( {\mathcal{S}^ \bot \cap \mathcal{K}^ + } \right) = \kappa ^ + \left( {\mathcal{S}^ \bot } \right) + \dim \left( {\mathcal{S} \cap \mathcal{K}^ - } \right)$$ where κ^±(S) stands for the positive/negative signature ofS. The difference dim(S∩K ^?)?dim(S ^⊥∩K ⁺), provided it is well defined, is called the index ofS. The formula turns out to unify other known index formulac for operators or subspaces in a Kreîn space.     

ОБ ИНтЕРпОлИРОВАНИИ В кОМплЕксНОИ ОБлАст И

LetD be a simply connected domain, the boundary of which is a closed Jordan curveγ; \(\mathfrak{M} = \left\{ {z_{k, n} } \right\}\) , 0≦k≦n; n=1, 2, 3, ..., a matrix of interpolation knots, \(\mathfrak{M} \subset \Gamma ; A_c \left( {\bar D} \right)\) the space of the functions that are analytic inD and continuous on \(\bar D; \left\{ {L_n \left( {\mathfrak{M}; f, z} \right)} \right\}\) the sequence of the Lagrange interpolation polynomials. We say that a matrix \(\mathfrak{M}\) satisfies condition (B _m ), \(\mathfrak{M}\) ∈(B _m ), if for some positive integerm there exist a setB _m containingm points and a sequencen _{p p=1} ^∞ of integers such that the series \(\mathop \Sigma \limits_{p = 1}^\infty \frac{1}{{n_p }}\) diverges and for all pairsn _i ,n _j ∈{n _p } _p=1 ^∞ the set \(\left( {\bigcap\limits_{k = 0}^{n_i } {z_{k, n_i } } } \right)\bigcap {\left( {\bigcup\limits_{k = 0}^{n_j } {z_{k, n_j } } } \right)} \) is contained inB _m . The main result reads as follows. {Let D=z: ¦z¦ \(\Gamma = \partial \bar D\) and let the matrix \(\mathfrak{M} \subset \Gamma \) satisfy condition (B_m). Then there exists a function \(f \in A_c \left( {\bar D} \right)\) such that the relation $$\mathop {\lim \sup }\limits_{n \to \infty } \left| {L_n \left( {\mathfrak{M}, f, z} \right)} \right| = \infty $$ holds almost everywhere on γ.     

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.     

Classification of 7-dimensional einstein nilradicals

The problem of classifying Einstein solvmanifolds, or equivalently, Ricci soliton nilmanifolds, is known to be equivalent to a question on the variety $ {\mathfrak{N}_n}\left( \mathbb{C} \right) $ of n-dimensional complex nilpotent Lie algebra laws. Namely, one has to determine which GL _n (?)-orbits in $ {\mathfrak{N}_n}\left( \mathbb{C} \right) $ have a critical point of the squared norm of the moment map. In this paper, we give a classification result of such distinguished orbits for n?=?7. The set $ {{{{\mathfrak{N}_n}\left( \mathbb{C} \right)}} \left/ {{{\text{G}}{{\text{L}}_7}\left( \mathbb{C} \right)}} \right.} $ is formed by 148 nilpotent Lie algebras and 6 one-parameter families of pairwise non-isomorphic nilpotent Lie algebras. We have applied to each Lie algebra one of three main techniques to decide whether it has a distinguished orbit or not.     

Lower bounds for the modulus of the logarithmic derivative of a polynomial

Estimates are given for the measure of a section of an arbitrary straight line of the set $$E_\delta = \left\{ {z:\left| {P' {{\left( z \right)} \mathord{\left/ {\vphantom {{\left( z \right)} {\left( {nP \left( z \right)} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {nP \left( z \right)} \right)}} \leqslant \delta } \right|} \right\} \left( {\delta > 0} \right)$$ where P (z) is a polynomial of degree n. THEOREM. Suppose P (x) = (x ? x₁) ... (x ? x_n) is a polynomial with real zeros. Then, for any δ > 0, on any intervala ?x ?b, containing all of the x_k (k=1, 2, ..., n), outside an exceptional set E_δ?[a,b] such that $$mes E_\delta \leqslant \left( {\sqrt {1 + \delta ^2 \left( {b - a} \right)^2 } - 1} \right)/\delta $$ , we have the inequality $$\left| {P' {{\left( x \right)} \mathord{\left/ {\vphantom {{\left( x \right)} {\left( {nP \left( x \right)} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {nP \left( x \right)} \right)}}} \right| > \delta $$ . A similar estimate is given for polynomials whose roots lie either in Imz ? 0 or in Imz ? 0.     

A decomposition of the universal embedding space for the near polygon {{\mathbb H}_n}

Let ${{\mathbb H}_n, n \geq 1}$ , be the near 2n-gon defined on the 1-factors of the complete graph on 2n?+?2 vertices, and let e denote the absolutely universal embedding of ${{\mathbb H}_n}$ into PG(W), where W is a ${\frac{1}{n+2} \left(\begin{array}{c}2n+2 \\ n+1\end{array}\right)}$ -dimensional vector space over the field ${{\mathbb F}_2}$ with two elements. For every point z of ${{\mathbb H}_n}$ and every ${i \in {\mathbb N}}$ , let Δ _i (z) denote the set of points of ${{\mathbb H}_n}$ at distance i from z. We show that for every pair {x, y} of mutually opposite points of ${{\mathbb H}_n, W}$ can be written as a direct sum ${W_0 \oplus W_1 \oplus \cdots \oplus W_n}$ such that the following four properties hold for every ${i \in \{0,\ldots,n \}}$ : (1) ${\langle e(\Delta_i(x) \cap \Delta_{n-i}(y)) \rangle = {\rm PG}(W_i)}$ ; (2) ${\left\langle e \left( \bigcup_{j \leq i} \Delta_j(x) \right) \right\rangle = {\rm PG}(W_0 \oplus W_1 \oplus \cdots \oplus W_i)}$ ; (3) ${\left\langle e \left( \bigcup_{j \leq i} \Delta_j(y) \right) \right\rangle = {\rm PG}(W_{n-i}\oplus W_{n-i+1} \oplus \cdots \oplus W_n)}$ ; (4) ${\dim(W_i) = |\Delta_i(x) \cap \Delta_{n-i}(y)| = \left(\begin{array}{c}n \\ i\end{array}\right)^2 - \left(\begin{array}{c}n \\ i-1\end{array}\right) \cdot \left(\begin{array}{c}n \\ i+1\end{array}\right)}$ .     

More quantitative results on walsh equiconvergence: I. Lagrange Case

Letf∈A _ρ (ρ>1), whereA _ρ denotes the class of functions analytic in ¦z¦ <ρ but not in ¦z¦≤ρ. For any positive integerl, the quantity Δ _l,n?1(f; z) (see (2.3)) has been studied extensively. Recently, V. Totik has obtained some quantitative estimates for \(\overline {\lim _{n \to \infty } } \max _{\left| z \right| = R} \left| {\Delta _{l,n - 1}^ - \left( {f;z} \right)} \right|^{1/n} \) . Here we investigate the order of pointwise convergence (or divergence) of Δ _l,n?1(f; z), i.e., we study \(B_1 \left( {f;z} \right) = \overline {\lim _{n \to \infty } } \left| {\Delta _{l,n - 1} \left( {f;z} \right)} \right|^{1/n} \) . We also study some problems arising from the results of Totik.     

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.     

Convergence of biorthogonal series in the system of contractions and translations of functions in the spaces L p [0, 1]

We obtain conditions for the convergence in the spaces L ^p [0, 1], 1 ≤ p < ∞, of biorthogonal series of the form $$ f = \sum\limits_{n = 0}^\infty {(f,\psi _n )\phi _n } $$ in the system {? _n } _n≥0 of contractions and translations of a function ?. The proposed conditions are stated with regard to the fact that the functions belong to the space $ \mathfrak{L}^p $ of absolutely bundleconvergent Fourier-Haar series with norm $$ \left\| f \right\|_p^ * = \left| {f,\chi _0 } \right| + \sum\limits_{k = 0}^\infty {2^{k({1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2} - {1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p})} } \left( {\sum\limits_{n = 2^k }^{2^{k + 1} - 1} {\left| {f,\chi _n } \right|^p } } \right)^{{1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-\nulldelimiterspace} p}} , $$ where (f,χ _n ), n = 0, 1, ..., are the Fourier coefficients of a function f ? L ^p [0, 1] in the Haar system {χ _n } _n≥0. In particular, we present conditions for the system {? _n } _n≥0 of contractions and translations of a function ? to be a basis for the spaces L ^p [0, 1] and $ \mathfrak{L}^p $ .     

Оценки отклонений оп тимальных сеток вL p -норме и теория квад ратурных формул

Let E^s=[0, 1]^s be then-dimensional unit cube, 1<p<∞, anda=(a ₁, ...,a _s ) some set of natural numbers. Denote byL _p ^(a) , (E ^s ) the class of functionsf: E ^s → C for which $$\left\| {\frac{{\partial ^{b_1 + \cdots + b_s } f}}{{\partial x_1^{b_1 } \cdots \partial x_s^{b_s } }}} \right\|_p \leqslant 1,$$ where $$0< b_1< a_1 , ..., 0< b_s< a_s .$$ Set $$R_p^{\left( a \right)} \left( N \right) = \mathop {\inf }\limits_{card \mathfrak{S} = N} R_\mathfrak{S} \left( {L_p^{\left( a \right)} \left( {E^s } \right)} \right),$$ where $R_\mathfrak{S} \left( {L_p^{\left( a \right)} \left( {E^s } \right)} \right)$ is the error of the quadrature formulas on the mesh $\mathfrak{S}$ (for the classL _p ^(a) (E ^s )), consisting of N nodes and weights, and the infimum is taken with respect to all possibleN nodes and weights. In this paper, the two-sided estimate $$\frac{{\left( {\log N} \right)^{{{\left( {l - 1} \right)} \mathord{\left/ {\vphantom {{\left( {l - 1} \right)} 2}} \right. \kern-\nulldelimiterspace} 2}} }}{{N^d }} \ll _{p, a} R^{\left( a \right)} \left( N \right) \ll _{p, a} \frac{{\left( {\log N} \right)^{{{\left( {l - 1} \right)} \mathord{\left/ {\vphantom {{\left( {l - 1} \right)} 2}} \right. \kern-\nulldelimiterspace} 2}} }}{{N^d }}$$ is proved for every natural numberN > 1, whered=min{a ₁, ...,a _s }, whilel is the number of those components of a which coincide withd. An analogous result is proved for theL _p -norm of the deviation of meshes.     

Schwache Lösungen des Frankl-Morawetz-Problems im ℝ3

In a bounded simple connected region G ? ?³ we consider the equation $$L\left[ u \right]: = k\left( z \right)\left( {u_{xx} + u_{yy} } \right) + u_{zz} + d\left( {x,y,z} \right)u = f\left( {x,y,z} \right)$$ where k(z)? 0 whenever z ? 0.G is surrounded forz≥0 by a smooth surface Γ₀ with S:=Γ₀ ? {(x,y,z)|=0} and forz<0 by the characteristic \(\Gamma _2 :---(x^2 + y^2 )^{{\textstyle{1 \over 2}}} + \int\limits_z^0 {(---k(t))^{{\textstyle{1 \over 2}}} dt = 0} \) and a smooth surface Γ₁ which intersect the planez=0 inS and where the outer normal n=(n_x, n_y, n_z) fulfills \(k(z)(n_x^2 + n_y^2 ) + n_z^2 |_{\Gamma _1 } > 0\) . Under conditions on Γ₁ and the coefficientsk(z), d(x,y,z) we prove the existence of weak solutions for the boundary value problemL[u]=f inG with \(u|_{\Gamma _0 \cup \Gamma _1 } = 0\) . The uniqueness of the classical solution for this problem was proved in [1].