A Schwarz Lemma at the Boundary of Hilbert Balls

In this paper, the authors prove a general Schwarz lemma at the boundary for the holomorphic mapping f between unit balls B and B′in separable complex Hilbert spaces H and H′, respectively. It is found that if the mapping f ∈ C~(1+α)at z_0∈ ?B with f(z_0) = w_0∈ ?B′, then the Fr′echet derivative operator Df(z_0) maps the tangent space Tz_0(?B~n) to Tw_0(?B′), the holomorphic tangent space T_(z_0)~(1,0)(?B~n) to T_(w_0)~(1,0)(?B′),respectively.     

On a Composition Preserving Inequalities between Polynomials

The Schur-Szegö composition of two polynomials \(f\left( z \right) = \sum\nolimits_{j = 0}^n {{A_j}{z^j}} \) and \(g\left( z \right) = \sum\nolimits_{j = 0}^n {{B_j}{z^j}} \), both of degree n, is defined by \(f * g\left( z \right) = \sum\nolimits_{j = 0}^n {{A_j}{B_j}{{\left( {\begin{array}{*{20}{c}}n \\ j \end{array}} \right)}^{ - 1}}{z^j}} \). In this paper, we estimate the minimum and the maximum of the modulus of f * g(z) on z = 1 and thereby obtain results analogues to Bernstein type inequalities for polynomials.     

A Supplement to Hölder’s Inequality. The Resonance Case. I

Suppose that m ≥ 2, numbers p₁, …, p _m ∈ (1, +∞] satisfy the inequality \(\frac{1}{{{p_1}}} + ... + \frac{1}{{{p_m}}} < 1\), and functions γ₁ ∈ \({L^{{p_1}}}\)(?¹), …, γ _m ∈ \({L^{{p_m}}}\)(?¹) are given. It is proved that if the set of “resonance points” of each of these functions is nonempty and the so-called “resonance condition” holds, then there are arbitrarily small (in norm) perturbations Δγ_k ∈ \({L^{{p_k}}}\)(?¹) under which the resonance set of each function γ_k + Δγ_k coincides with that of γ_k for 1 ≤ k ≤ m, but \({\left\| {\int\limits_0^t {\prod\limits_{k = 0}^m {\left[ {{\gamma _k}\left( \tau \right) + \Delta {\gamma _k}\left( \tau \right)} \right]d\tau } } } \right\|_{{L^\infty }\left( {{\mathbb{R}^1}} \right)}} = \infty \). The notion of a resonance point and the resonance condition for functions in the spaces L ^p (?¹), p ∈ (1, +∞], were introduced by the author in his previous papers.     

Determination of cusp forms by central values of Rankin–Selberg <Emphasis Type="BoldItalic">L</Emphasis>-functions<Superscript>*</Superscript>

Let f be a fixed holomorphic Hecke eigen cusp form of weight k for \( SL\left( {2,{\mathbb Z}} \right) \), and let \( {\mathcal U} = \left\{ {{u_j}:j \geqslant 1} \right\} \) be an orthonormal basis of Hecke–Maass cusp forms for \( SL\left( {2,{\mathbb Z}} \right) \). We prove an asymptotic formula for the twisted first moment of the Rankin–Selberg L-functions \( L\left( {s,f \otimes {u_j}} \right) \) at \( s = \frac{1}{2} \) as u _j runs over \( {\mathcal U} \). It follows that f is uniquely determined by the central values of the family of Rankin–Selberg L-functions \( \left\{ {L\left( {s,f \otimes {u_j}} \right):{u_j} \in {\mathcal U}} \right\} \).     

Hrmander Type Theorem for Fourier Multipliers with Optimal Smoothness on Hardy Spaces of Arbitrary Number of Parameters

The main purpose of this paper is to establish the Hormander-Mihlin type theorem for Fourier multipliers with optimal smoothness on k-parameter Hardy spaces for k≥ 3 using the multiparameter Littlewood-Paley theory. For the sake of convenience and simplicity, we only consider the case k = 3, and the method works for all the cases k≥ 3:■where x =(x_1,x_2,x_3)∈R~(n_1)×R~(n_2)×R~(n_3) and ξ =(ξ_1,ξ_2,ξ_3)∈R~(n_1)×R~(n_2)×R~(n_3). One of our main results is the following:Assume that m(ξ) is a function on R~(n_1+n_2+n_3) satisfying ■ with s_i n_i(1/p-1/2) for 1≤i≤3. Then T_m is bounded from H~p(R~(n_1)×R~(n_2)×R~(n_3) to H~p(R~(n_1)×R~(n_2)×R~(n_3)for all 0 p≤1 and ■ Moreover, the smoothness assumption on s_i for 1≤i≤3 is optimal. Here we have used the notations m_(j,k,l)(ξ)=m(2~jξ_1,2~kξ_2,2~lξ_3)Ψ(ξ_1)Ψ(ξ_2)Ψ(ξ_3) and Ψ(ξ_i) is a suitable cut-off function on R~(n_i) for1≤i≤3, and W~(s_1,s_2,s_3) is a three-parameter Sobolev space on R~(n_1)×R~(n_2)× R~(n_3).Because the Fefferman criterion breaks down in three parameters or more, we consider the L~p boundedness of the Littlewood-Paley square function of T_mf to establish its boundedness on the multi-parameter Hardy spaces.     

Interlacing of zeros of Gegenbauer polynomials of non-consecutive degree from different sequences

A theorem due to Stieltjes’ states that if \({\{p_n\}_{n=0}^\infty}\) is any orthogonal sequence then, between any two consecutive zeros of p _k , there is at least one zero of p _n whenever k < n, a property called Stieltjes interlacing. We show that Stieltjes interlacing extends to the zeros of Gegenbauer polynomials \({C_{n+1}^{\lambda}}\) and \({C_{n-1}^{\lambda+t}}\), \({\lambda > -\frac 12}\), if 0 < t ≤ k + 1, and also to the zeros of \({C_{n+1}^{\lambda}}\) and \({C_{n-2}^{\lambda +k}}\) if \({k\in\{1,2,3\}}\). More generally, we prove that Stieltjes interlacing holds between the zeros of the kth derivative of \({C_{n}^{\lambda}}\) and the zeros of \({C_{n+1}^{\lambda}}\), \({k\in\{1,2,\dots,n-1\}}\) and we derive associated polynomials that play an analogous role to the de Boor–Saff polynomials in completing the interlacing process of the zeros.     

Estimates for sums of moduli of blocks in trigonometric Fourier series

We consider the following two problems. Problem 1: what conditions on a sequence of finite subsets A _k ? ? and a sequence of functions λ _k : A _k → ? provide the existence of a number C such that any function f ∈ L ₁ satisfies the inequality ‖U _A,Λ(f)‖ _p ≤ C‖f‖₁ and what is the exact constant in this inequality? Here, \(U_{\mathcal{A},\Lambda } \left( f \right)\left( x \right) = \sum\nolimits_{k = 1}^\infty {\left| {\sum\nolimits_{m \in A_k } {\lambda _k \left( m \right)c_m \left( f \right)e^{imx} } } \right|}\) and c _m (f) are Fourier coefficients of the function f ∈ L ₁. Problem 2: what conditions on a sequence of finite subsets A _k ? ? guarantee that the function \(\sum\nolimits_{k = 1}^\infty {\left| {\sum\nolimits_{m \in A_k } {c_m \left( h \right)e^{imx} } } \right|}\) belongs to L _p for every function h of bounded variation?     

On one complement to the Hölder inequality: I

Let m ≥ 2, the numbers p ₁,…, p _m ∈ (1, +∞] satisfy the inequality \(\frac{1}{{{p_1}}} + ...\frac{1}{{{p_m}}} < 1\), and γ₁ ∈ L ^p1(?¹), …, γ _m ∈ \({L^{{p_m}}}\)(?¹). We prove that, if the set of “resonance” points of each of these functions is nonempty and the “nonresonance” condition holds (both concepts have been introduced by the author for functions of spaces L ^p (?¹), p ∈ (1, +∞]), we have the inequality \(\mathop {\sup }\limits_{a,b \in {R^1}} \left| {\int\limits_a^b {\prod\limits_{k = 1}^m {\left[ {{\gamma _k}\left( \tau \right) + \Delta {\gamma _k}\left( \tau \right)} \right]} d\tau } } \right| \leqslant C{\prod\limits_{k = 1}^m {\left\| {{\gamma _k} + \Delta {\gamma _k}} \right\|} _{L_{{a_k}}^{{p_k}}}}\left( {{\mathbb{R}^1}} \right)\), where the constant C > 0 is independent of functions \(\Delta {\gamma _k} \in L_{{a_k}}^{{p_k}}\left( {{\mathbb{R}^1}} \right)\) and \(L_{{a_k}}^{{p_k}}\left( {{\mathbb{R}^1}} \right) \subset {L^{{p_k}}}\left( {{\mathbb{R}^1}} \right)\), 1 ≤ k ≤ m are some specially constructed normed spaces. In addition, we give a boundedness condition for the integral of the product of functions over a subset of ?¹.     

Cotorsion and wild homology

We solve the problem of describing the solutions of E-operators of order μ ≥ 1 admitting at z = 0 a basis over C of local solutions which are all holomorphic at z = 0. We prove that the components of such a basis can be taken of the form \(\sum {_{j = 1}^\ell } {P_j}\left( z \right){e^{{\beta _{{j^z}}}}}\), where ? ≤ μ, β ₁,...,β _? ∈ \(\overline {\mathbb{Q}} \) ^x, and P ₁(z),..., P _?(z) ∈ \(\overline {\mathbb{Q}} \)[z].     

Sharp Estimates for Mean Square Approximations of Classes of Differentiable Periodic Functions by Shift Spaces

Let L₂ be the space of 2π-periodic square-summable functions and E(f, X)₂ be the best approximation of f by the space X in L₂. For n ∈ ? and B ∈ L₂, let \({{\Bbb S}_{B,n}}\) be the space of functions s of the form \(s\left( x \right) = \sum\limits_{j = 0}^{2n - 1} {{\beta _j}B\left( {x - \frac{{j\pi }}{n}} \right)} \). This paper describes all spaces \({{\Bbb S}_{B,n}}\) that satisfy the exact inequality \(E{\left( {f,{S_{B,n}}} \right)_2} \leqslant \frac{1}{{^{{n^r}}}}\parallel {f^{\left( r \right)}}{\parallel _2}\). (2n–1)-dimensional subspaces fulfilling the same estimate are specified. Well-known inequalities are for approximation by trigonometric polynomials and splines obtained as special cases.     

Riemann Sums and Möbius

Let S be the set of square-free natural numbers. A Hilbert-Schmidt operator, A, associated to the Möbius function has the property that it maps from \({ \cup _{0 < r < \infty }}{l^r}(s)\) to \({ \cap _{0 < r < \infty }}{l^r}(s)\), injectively. If 0 < r< 2 and ξ ∈ l^r (S), the series \({f_\zeta } = \sum\nolimits_{n \in s} {A\zeta (x)cos2\pi nx} \) converges uniformly to an element of f_ξR₀, i.e., a periodic, even, continuous function with equally spaced Riemann sums, \(\sum\nolimits_{j = 0}^{N - 1} {{f_\zeta }} (j/N) = 0,N = 1,2....\) If \({A_{\zeta \lambda }} = \lambda {\zeta _\lambda },{\zeta _\lambda }(1) = 1\), then ξ_λ is multiplicative. If \({f_{{\zeta _\lambda }}} \in {\Lambda _a}\), the space of α-Lipschitz continous functions, for some α > 0, and if χ is any Dirichlet character, then L(s, χ) ≠ 0, Res > 1 ? α. Conjecturally, the Generalized Riemann Hypothesis (GRH) is equivalent to f_ξ ∈ Λ_α, α < 1/2, ξ ∈ l^r (S), 0 < r < 2. Using a 1991 estimate by R. C. Baker and G. Harman, one finds GRH implies f_ξ ∈ Λ_α, α < 1/4, ξ ∈ l^r (S), 0 < r < 2. The question of whether R₀ ∩ Λ_α ≠ {0} for some positive α > 0 is open.     

Existence of Multi-bump Solutions for a Class of Quasilinear Schrödinger Equations in $${\mathbb{R}^{N}}$$ Involving Critical Growth

In this paper we prove the existence of multi-bump solutions for a class of quasilinear Schrödinger equations of the form \({-\Delta{u} + (\lambda{V} (x) + Z(x))u - \Delta(u^{2})u = \beta{h}(u) + u^{22*-1}}\) in the whole space, where h is a continuous function, \({V, Z : \mathbb{R}^{N} \rightarrow \mathbb{R}}\) are continuous functions. We assume that V(x) is nonnegative and has a potential well \({\Omega : = {\rm int} V^{-1}(0)}\) consisting of k components \({\Omega_{1}, \ldots , \Omega{k}}\) such that the interior of Ω _i is not empty and \({\partial\Omega_{i}}\) is smooth. By using a change of variables, the quasilinear equations are reduced to a semilinear one, whose associated functionals are well defined in the usual Sobolev space and satisfy the geometric conditions of the mountain pass theorem for suitable assumptions. We show that for any given non-empty subset. \({\Gamma \subset \{1, \ldots ,k\}}\), a bump solution is trapped in a neighborhood of \({\cup_{{j}\in\Gamma}\Omega_{j}}\) for\({\lambda > 0}\) large enough.     

On the towers of torsion Bertrandias and Payan modules

For an odd prime p, let K/k be a Galois p-extension and S be a set of primes of k containing the primes lying over p. For the p ^r th roots \({\mu _{{p^r}}}\left( K \right)\) of unity in K, we describe the so-called Sha group Sha _S (G(K/k), \({\mu _{{p^r}}}\left( K \right)\)) in terms of the Galois groups of certain subfields of K corresponding to S. As an application, we investigate a tower of extension fields \({\left\{ {{k_{{T^i}}}} \right\}_i} \geqslant 0\) where \({k_{{T^{i + 1}}}}\) is defined as the fixed field of a free part of the Galois group of the Bertrandias and Payan extension of \({k_{{T^i}}}\) over \({k_{{T^i}}}\). This is called a tower of torsion parts of the Bertrandias and Payan extensions over k. We find a relation between the degrees \({\left\{ {\left[ {{k_{{T^{i + 1}}}}:{k_{{T^i}}}} \right]} \right\}_{i \geqslant 0}}\) over the towers. Using this formula we investigate whether the towers are stationary or not.     

Bounded Sequences,Orbits and Invariant Subspaces

Let {x _n } be a sequence of complex numbers and let \({\Delta^nx_j = \sum\nolimits_{k=0}^{n} (-1)^k\break\left(\begin{array}{l}n\\ k\\\end{array} \right)x_{n-k+j}}\) . In this paper, we will show that if \({ |x_n| = O(n^k)}\) , as n → ∞ for some positive integer k, and \({n|\Delta^n x_j|^{\frac{1}{n}} \to 0}\) as n→ ∞, then \({\Delta^{k+1} x_j = 0}\) . More importantly, applications to the orbits of operators and invariant subspace problem are also given; this helps to improve former results obtained by Gelfand–Hille, Mbekhta–Zemánek and others.     

Supplement to Hölder’s inequality. II

Suppose that m ≥ 2, numbers p ₁, …, p _m ∈ (1, +∞] satisfy the inequality \(\frac{1}{{{p_1}}} + \cdots + \frac{1}{{{p_m}}} < 1\), and functions \({\gamma _1} \in {L^{{p_1}}}\left( {{?^1}} \right), \cdots ,{\gamma _m} \in {L^{{p_m}}}\left( {{?^1}} \right)\) are given. It is proved that if the set of “resonance” points of each of these functions is nonempty and the “nonresonance” condition holds (both notions were defined by the author for functions in L ^p (?¹), p ∈ (1, +∞]), then \(\mathop {\sup }\limits_{a,b \in {R^1}} \left| {\mathop \smallint \limits_a^b \prod\limits_{k = 1}^m {[{\gamma _k}\left( \tau \right) + \Delta {\gamma _k}\left( \tau \right)]} d\tau } \right| \leqslant C\prod\limits_{k = 1}^m {{{\left\| {{\gamma _k} + \Delta {\gamma _k}} \right\|}_{L_{ak}^{pk}\left( {{R^1}} \right)}}} \) where the constant C > 0 is independent of the functions \(\Delta {\gamma _k} \in L_{ak}^{pk}\left( {{?^1}} \right)\) and \(L_{ak}^{pk}\left( {{?^1}} \right) \subset {L^{pk}}\left( {{?^1}} \right)\), 1 ≤ k ≤ m, are special normed spaces. A condition for the integral over ?¹ of a product of functions to be bounded is also given.     

On the derivative of the Minkowski question mark function ?(<Emphasis Type="Italic">x</Emphasis>)

Let x?=?[0; a ₁ , a ₂ , …] be the regular continued fraction expansion of an irrational number x ∈ [0, 1]. For the derivative of the Minkowski function ?(x) we prove that ?′(x)?=?+∞, provided that \( \mathop {{\lim \sup }}\limits_{t \to \infty } \frac{{{a_1} + \cdots + {a_t}}}{t} < {\kappa_1} = \frac{{2\log {\lambda_1}}}{{\log 2}} = {1.388^{+} } \), and ?′(x)?=?0, provided that \( \mathop {{\lim \inf }}\limits_{t \to \infty } \frac{{{a_1} + \cdots + {a_t}}}{t} > {\kappa_2} = \frac{{4{L_5} - 5{L_4}}}{{{L_5} - {L_4}}} = {4.401^{+} } \), where \( {L_j} = \log \left( {\frac{{j + \sqrt {{{j^2} + 4}} }}{2}} \right) - j \cdot \frac{{\log 2}}{2} \). Constants κ₁, κ₂ are the best possible. It is also shown that ?′(x)?=?+∞ for all x with partial quotients bounded by 4.     

One-Sided Integral Approximations of the Generalized Poisson Kernel by Trigonometric Polynomials

We consider the generalized Poisson kernel Π _q,α = cos(απ/2)P + sin(απ/2)Q with q ∈ (?1, 1) and α ∈ ?, which is a linear combination of the Poisson kernel \(P(t) = 1/2 + \sum\nolimits_{k = 1}^\infty {{q^k}} \cos kt\)and the conjugate Poisson kernel \(Q(t) = \sum\nolimits_{k = 1}^\infty {{q^k}} \sin kt\). The values of the best integral approximation to the kernel Π _q,α from below and from above by trigonometric polynomials of degree not exceeding a given number are found. The corresponding polynomials of the best one-sided approximation are obtained.     

On behavior of Fourier coefficients by Walsh system

The paper proves that for any ε > 0 there exists ameasurable set E ? [0, 1] with measure |E| > 1 ? ε such that for each f ∈ L¹[0, 1] there is a function \(\tilde f \in {L^1}\left[ {0,1} \right]\) coinciding with f on E whose Fourier-Walsh series converges to \(\tilde f\) in L¹[0, 1]-norm, and the sequence \(\left\{ {\left| {{c_k}\left( {\tilde f} \right)} \right|} \right\}_{n = 0}^\infty \) is monotonically decreasing, where \(\left\{ {{c_k}\left( {\tilde f} \right)} \right\}\) is the sequence of Fourier-Walsh coefficients of \(\left\{ {\left| {{c_k}\left( {\tilde f} \right)} \right|} \right\}_{n = 0}^\infty \).     

Generalized Kloosterman sum with primes

The work is devoted to generalized Kloosterman sums modulo a prime, i.e., trigonometric sums of the form \(\sum\nolimits_{p \leqslant x} {\exp \left\{ {2\pi i\left( {a\bar p + {F_k}\left( p \right)} \right)/q} \right\}} \) and \(\sum\nolimits_{n \leqslant x} {\mu \left( n \right)\exp \left\{ {2\pi i\left( {a\bar n + {F_k}\left( n \right)} \right)/q} \right\}} \), where q is a prime number, \(\left( {a,q} \right) = 1,m\bar m \equiv 1\left( {\bmod {\kern 1pt} q} \right)\), F _k (u) is a polynomial of degree k ≥ 2 with integer coefficients, and p runs over prime numbers. An upper estimate with a power saving is obtained for the absolute values of such sums for x ≥ q^1/2+ε.     

A Supplement to Hölder’s Inequality. The Resonance Case. II

Suppose that m ≥ 2, and numbers p₁, …, p _m ∈ (1, +∞] satisfy the inequality 1/p₁+…+1/p _m < 1, and functions \({\gamma _1} \in {L^{{p_1}}}\left( {{\mathbb{R}^1}} \right),...,{\gamma _m} \in {L^{{p_m}}}\left( {{\mathbb{R}^1}} \right)\) are given. It is proven that, if the set of resonance points of each of these functions is nonempty and the so-called resonance condition holds, there will always exist arbitrarily small (in norm) perturbations \(\Delta {\gamma _k} \in {L^{{p_k}}}\left( {{\mathbb{R}^1}} \right)\) under which the set of resonance points of the function _γk + Δ_γk coincides with that of the function γ _k for 1 ≤ k ≤ m, but in this case, \({\left\| {\int\limits_0^t {\prod\limits_{k = 1}^m {[{\gamma _k}\left( \tau \right) + \Delta {\gamma _k}\left( \tau \right)]d\tau } } } \right\|_{{L^\infty }\left( {{\mathbb{R}^1}} \right)}} = \infty\) The notion of a resonance point and the resonance condition for the functions of the spaces L ^p (R¹), p ∈ (1, +∞], were introduced by the author in his previous papers.