Two inequalities for pseudo-differential operators

Изучается ограничен ность псевдодиффере нциальных операторов на \(L^2 (R^n )\) и на пр остранствах Харди в \(R^n \) . Пусть \(D_k = \{ \xi \in R^n :2^{k - 1} \leqq \left| \xi \right|< 2^k \} , k = 1,2,3, \ldots ,\) и \(D_0 = \{ \xi \in R^n :\left| \xi \right|< 1\} \) . Псевдодиффер енциальный операторP с символом p определяется соотно шением $$Pf(x) = \int\limits_{R^n } {e^{ix \cdot \xi } p(x,\xi )\hat f(\xi )d\xi ,x \in R^n .} $$ Будем говорить, что p пр инадлежит классу \(\bar S_{\varrho ,} {}_\delta (M,N), 0 \leqq \delta ,\varrho \leqq 1\) , ес ли $$\left| {D_x^a p(x,\xi )} \right| \leqq C_a (1 + \left| \xi \right|)^{\delta \left| a \right|} , x,\xi \in R^n ,\left| a \right| \leqq M,$$ и $$\int\limits_{D_k } {\left| {D_x^a D_\xi ^\beta p(x,\xi )} \right|d\xi \leqq C_{a\beta } 2^{kn} 2^{k(\delta |a| - \varrho |\beta |)} , x} \in R^n , k = 0,1,2, \ldots ;|a| \leqq M, |\beta | \leqq N.$$ Изучаются условия, ко торым должны удовлет ворять ?. δ,M иN, чтобы для каждого символа \(p \in \bar S_\varrho , {}_\delta (M,N)\) соответствующий оп ераторP был ограниче н на \(L^2 (R^n )\) . Далее, пусть \(p \in S_\varrho , {}_\delta \) , если дл я всех мультииндексо в а и β выполнено условие $$|D_x^a D_\xi ^\beta p(x,\xi )| \leqq C_{a\beta } (1 + |\xi |)^{\delta |\alpha | - \varrho |\beta |} , x,\xi \in R^n .$$ Доказывается, что при 0≦δ<1 операторP отображ ает пространство Харди \(H^p (R^n )\) в локальное пространство Харди ? ^p , если символp принадл ежит классуS ₁, δ.     

A note on the mean value of L-series

Using Hilbert’s inequality, we give a new asymptotic formula (uniform inq andT) for $$\mathop \Sigma \limits_{\begin{array}{*{20}c} {\chi (mod q)} \hfill \\ {\chi primitive} \hfill \\ \end{array} } \smallint _T^{2T} |L(\tfrac{1}{2} + it,\chi )^4 |dt$$      

Chen’ s theorem in arithmetical progressions

LetN be a sufficiently large even integer and $$\begin{gathered} q \geqslant 1, (l_i ,q) = 1 (i = 1, 2), \hfill \\ l_1 + l_2 \equiv N(\bmod q). \hfill \\ \end{gathered} $$ . It is proved that the equation $$N = p + P_2 ,p \equiv l_1 (\bmod q), P_2 \equiv l_2 (\bmod q)$$ has infinitely many solutions for almost all $q \leqslant N^{\frac{1}{{37}}} $ , wherep is a prime andP ₂ is an almost prime with at most two prime factors.     

О РАцИОНАльНых сОстА ВльУЩИх МЕРОМОРФНых ФУНкцИИ И Их пРОИжВОД Ных

LetG be an arbitrary domain in \(\bar C\) ,f a function meromorphic inG, $$M_f \mathop = \limits^{def} \mathop {\lim \sup }\limits_{G \mathrel\backepsilon z \to \partial G} \left| {f(z)} \right|< \infty ,$$ andR the sum of the principal parts in the Laurent expansions off with respect to all its poles inG. We set $$f_G (z) = R(z) - \alpha ,{\mathbf{ }}where{\mathbf{ }}\alpha = \mathop {\lim }\limits_{z \to \infty } (f(z) - R(z))$$ in case ∞?G, andα=0 in case ∞?G. It is proved that $$\left\| {f_G } \right\|_{C(\partial G)} \leqq 50(\deg f_G )M_f ,{\mathbf{ }}\left\| {f'_G } \right\|_{L_1 (\partial G)} \leqq 50(\deg f_G )V(\partial G)M_f ,$$ where $$V(\partial G) = \sup \left\{ {\left\| {r'} \right\|_{L_1 (\partial G)} :r(z) = a/(z - b),{\mathbf{ }}\left\| r \right\|_{G(\partial G)} \leqq 1} \right\}.$$      

On orthogonal polynomials

Пустьw(х)∈L[-1, +1] — неотрица тельная функция така я, что $$\frac{{\log ^ + \frac{1}{{w(x)}}}}{{\sqrt {1 - x^2 } }} \in L[ - 1, + 1]$$ и пусть {(р _n (х)} — много члены, ортогональные и нормированные с весо мw(x). Мы доказываем следующие две теорем ы, являющиеся обобщен ием одного известного результа та Н. Винера. I. Для каждого δ, 0<δ<1, суще ствует числоB=B(δ, w) тако е, что если $$f_N (x) = \sum\limits_{j = 1}^N {a_j p_{v_j } (x)} $$ причем выполнено сле дующее условие лакун арности $$\begin{gathered} v_{j + 1} - v_j \geqq B(\delta ,w) (j = 1,2,...,N - 1), \hfill \\ v_1 \geqq B(\delta ,w) \hfill \\ \end{gathered} $$ , то для некоторого С(δ, w) и всехh и δ, для которых $$ - 1 \leqq h - \delta< h + \delta \leqq + 1$$ , имеет место неравенс тво $$\int\limits_{ - 1}^1 {|f_N (x)|^2 w(x)dx \leqq C(\delta ,w)} \int\limits_{h - \delta }^{h + \delta } {|f_N (x)|^2 w(x)dx} $$ каковы бы ни былиa _j ,N и h. II. Если формальный ряд $$\sum\limits_{j = 1}^\infty {b_j p_{\mu _j } (x)} $$ удовлетворяет услов ию лакунарности μ_j+1-μ_j→∞ и суммируем, например, м етодом Абеля на произвольно малом отрезке [а, Ь] ?[0,1] к ф ункцииf(x) такой, что \(f(x)\sqrt {w(x)} \in L_2 [a,b]\) , то $$\sum\limits_j {|b_j |^2< \infty } $$ Теорема I — это первый ш аг в направлении проб лемы типа Мюнтца-Саса о замкнут ости подпоследовательно сти pv_j(x)} последовател ьности {р_n(х)} на отрезке [а, Ь] в метрике С[а, Ь] (см. теорему II стать и).     

Order function for almost all numbers

For almost all pointsξ? R^m (m≥2) the inequality $$\sup {\text{ }}ln \frac{1}{{|P (\xi )|}} \ll (ln u)^{m + 2} ,$$ is valid, where the upper bound is taken over all nonzero polynomials P for which $$\exp {\text{ }}(\deg {\text{ }}P){\text{ }}L{\text{ }}(P) \leqslant u,$$ where L(P) is the sum of the moduli of the coefficients of P. When m=1 the exponent of the right side is equal to 2.     

On the divergence of Lagrange interpolation processes on a set of second category

В статье даны полные д оказательства следу ющих утверждений. Пустьω — непрерывная неубывающая полуадд итивная функций на [0, ∞),ω(0)=0 и пусть M?[0, 1] — матрица узл ов интерполирования. Если $$\mathop {\lim sup}\limits_{n \to \infty } \omega \left( {\frac{1}{n}} \right)\log n > 0$$ то существует точкаx ₀∈[0,1] и функцияf ∈ С[0,1] таки е, чтоω(f, δ)=О(ω(δ)), для которой $$\mathop {\lim sup}\limits_{n \to \infty } |L_n (\mathfrak{M},f,x_0 ) - f(x_0 )| > 0$$ Если же $$\mathop {\lim sup}\limits_{n \to \infty } \omega \left( {\frac{1}{n}} \right)\log n = \infty$$ , то существуют множес твоE второй категори и и функцияf ∈ С[0,1],ω(f, δ)=o(ω(δ)) та кие, что для всехx∈E $$\mathop {\lim sup}\limits_{n \to \infty } |L_n (\mathfrak{M},f,x)| = \infty$$ . Исправлена погрешно сть, допущенная автор ом в [5], и отмеченная в работе П. Вертеши [9].     

Fourier coefficients of Siegel cusp forms of genus n

Let F(Z) be a cusp form of integral weight k relative to the Siegel modular group Sp_n(Z) and let f(N) be its Fourier coefficient with index N. Making use of Rankin's convolution, one proves the estimate (1) $$f(\mathcal{N}) = O(\left| \mathcal{N} \right|^{\tfrac{k}{2} - \tfrac{1}{2}\delta (n)} ),$$ where $$\delta (n) = \frac{{n + 1}}{{\left( {n + 1} \right)\left( {2n + \tfrac{{1 + ( - 1)^n }}{2}} \right) + 1}}.$$ Previously, for n ≥ 2 one has known Raghavan's estimate $$f(\mathcal{N}) = O(\left| \mathcal{N} \right|^{\tfrac{k}{2}} )$$ In the case n=2, Kitaoka has obtained a result, sharper than (1), namely: (2) $$f(\mathcal{N}) = O(\left| \mathcal{N} \right|^{\tfrac{k}{2} - \tfrac{1}{4} + \varepsilon } ).$$ At the end of the paper one investigates specially the case n=2. It is shown that in some cases the result (2) can be improved to, apparently, unimprovable estimates if one assumes some analogues of the Petersson conjecture. These results lead to a conjecture regarding the optimal estimates of f(N), n=2.     

Some approximation theorems for modified Bernstein-Durrmeyer operators

The modified Bernstein-Durrmeyer operators discussed in this paper are given byM_nf≡M_n(f,x)=(n+2)P_(n,k)∫_0~1p_n+1.k(t)f(t)dt,whereWe will show,for 0<α<1 and 1≤p≤∞     

Eventual uniform asymptotic stability for stochastic differential equation based on semimartingale with spatial parameter

Given a stochastic differential equation based on semimartingale with spatial parameter (1) $$\varphi _t = x_0 + \int_{t_0 }^t {F(\varphi _s ,ds) } on t \geqslant t_0 $$ and it perturbed system (2) $$\psi _t = x_0 + \int_{t_0 }^t {F\left( {\psi \alpha _s , ds} \right)} + \int_{t_0 }^t {G\left( {\psi _s , ds} \right)} on t \geqslant t_0 $$ In this paper we give some sufficient conditions under which the eventual uniform asymptotic stability of Eq. (1) is shared by Eq. (2).     

On spectra ofp-hyponormal operators

The purpose of this paper is to show the following: Let 0<p<1/2. IfT=U|T| is a p-hyponormal operator with a unitaryU on a Hilbert space, then $$\sigma (T) = \mathop \cup \limits_{0 \leqslant k \leqslant 1} \sigma (T_{\left[ k \right]} ),$$ where $$T_{\left[ k \right]} = U[(1 - k)S_U^ - (\left| T \right|^{2p} ) + kS_U^ + (\left| T \right|^{2p} ]^{\tfrac{1}{{2p}}} $$ andS _U ^± (T) denote the polar symbols ofT.     

Mean value of weyl sums

We give a simple proof of a mean value theorem of I. M. Vinogradov in the following form. Suppose P, n, k, τ are integers, P≥1, n≥2, k≥n (τ+1), τ≥0. Put $$J_{k,n} (P) = \int_0^1 \cdots \int_0^1 {\left| {\sum\nolimits_{x = 1}^P {e^{2\pi i(a_1 x + \cdots + a_n x^n )} } } \right|^{2k} da_1 \ldots da_n .} $$ Then $$J_{k,n} \leqslant n!k^{2n\tau } n^{\sigma n^2 u} \cdot 2^{2n^2 \tau } P^{2k - \Delta } ,$$ where $$\begin{gathered} u = u_\tau = min(n + 1,\tau ), \hfill \\ \Delta = \Delta _\tau = n(n + 1)/2 - (1 - 1/n)^{\tau + 1} n^2 /2. \hfill \\ \end{gathered} $$      

Some consequences of the Lindelöf conjecture

Suppose that the Lindelöf conjecture is valid in the following quantitative form: $$|\zeta (\frac{1}{2} + it)| \leqslant c_0 |t|^{\varepsilon (|t|)} $$ , where ε(t) is a monotone decreasing function, $\varepsilon (2t) \geqslant \tfrac{1}{2}\varepsilon (t),\varepsilon (t) \geqslant \tfrac{1}{{\sqrt {log t} }}$ . Then it is proved that for |t|≥T₀ the disk $\{ s:|s - \tfrac{1}{2} - it| \leqslant v\} $ contains at most 20v log |t| zeros of ζ(s) if $\tfrac{1}{2} \geqslant v \geqslant \sqrt {\varepsilon (t)} $ . There exists an absolute constant A such that for |t|≥T₁ the disk $\{ s:|s - \tfrac{1}{2} - it| \leqslant A\varepsilon ^{\tfrac{1}{3}} (t)\} $ contains at least one zero of ζ(s). Bibliography: 2 titles.     

Асимптотические рав енства для интеграло в от модуля суммы кратн ого ряда из синусов

Assume that the coefficients of the series $$\mathop \sum \limits_{k \in N^m } a_k \mathop \Pi \limits_{i = 1}^m \sin k_i x_i $$ satisfy the following conditions: a) a_k → 0 for k₁ + k₂ + ...+k_m →∞, b) \(\delta _{B,G}^M (a) = \mathop {\mathop \sum \limits_{k_i = 1}^\infty }\limits_{i \in B} \mathop {\mathop \sum \limits_{k_j = 2}^\infty }\limits_{j \in G} \mathop {\mathop \sum \limits_{k_v = 0}^\infty }\limits_{v \in M\backslash (B \cup G)} \mathop \Pi \limits_{i \in B} \frac{1}{{k_i }}|\mathop \sum \limits_{I_j = 1}^{[k_j /2]} (\nabla _{l_G }^G (\Delta _1^{M\backslash B} a_k ))\mathop \Pi \limits_{j \in G} l_j^{ - 1} |< \infty ,\) for ∨B?M, ∨G?M,B∩G, where M={1,2, ...,m}, $$\begin{gathered} \,\,\,\,\,\,\,\,\,\,\,\,\Delta _1^j a_k = a_k - a_{k_{M\backslash \{ j\} } ,k_{j + 1} } ,\Delta _1^B a_k = \Delta _1^{B\backslash \{ j\} } (\Delta _1^j a_k ), \hfill \\ \Delta _{l_j }^j a_k = a_{k_{M\backslash \{ j\} } ,k_j - l_j } - a_{k_{M\backslash \{ j\} } ,k_j + l_j } ,\nabla _{l_G }^G a_k = \nabla _{l_{G\backslash \{ j\} } }^{G\backslash \{ j\} } (\nabla _{l_j }^j a_k ). \hfill \\ \end{gathered} $$ Then for all n∈N^m the following asymptotic equation is valid: $$\mathop \smallint \limits_{{\rm T}_{\pi /(2n + 1)}^m } |\mathop \sum \limits_{k \in N^m } a_k \mathop \Pi \limits_{i \in M} \sin k_i x_i |dx = \mathop \sum \limits_{k = 1}^n \left| {a_k } \right|\mathop \Pi \limits_{i \in M} k^{ - 1} + O(\mathop {\mathop \sum \limits_{B,{\mathbf{ }}G \subset M} }\limits_{B \ne M} \delta _{B,G}^M (a)).$$ Here \(T_{\pi /(2n + 1)}^m = \left\{ {x = (x1,x2,...,xm):\pi /(2n + 1) \leqq xi \leqq \pi ;i = \overline {1,m} } \right\}\) . In the one-dimensional case such an equation was proved by S. A. Teljakovskii.     

{\mathcal{C}^{\alpha}}-regularity for a class of non-diagonal elliptic systems with p-growth

We consider weak solutions to nonlinear elliptic systems in a W ^1,p -setting which arise as Euler equations to certain variational problems. The solutions are assumed to be stationary in the sense that the differential of the variational integral vanishes with respect to variations of the dependent and independent variables. We impose new structure conditions on the coefficients which yield everywhere ${\mathcal{C}^{\alpha}}$ -regularity and global ${\mathcal{C}^{\alpha}}$ -estimates for the solutions. These structure conditions cover variational integrals like ${\int F(\nabla u)\; dx}$ with potential ${F(\nabla u):=\tilde F (Q_1(\nabla u),\ldots, Q_N(\nabla u))}$ and positively definite quadratic forms in ${\nabla u}$ defined as ${Q_i(\nabla u)=\sum_{\alpha \beta} a_i^{\alpha \beta} \nabla u^\alpha \cdot \nabla u^\beta}$ . A simple example consists in ${\tilde F(\xi_1,\xi_2):= |\xi_1|^{\frac{p}{2}} + |\xi_2|^{\frac{p}{2}}}$ or ${\tilde F(\xi_1,\xi_2):= |\xi_1|^{\frac{p}{4}}|\xi_2|^{\frac{p}{4}}}$ . Since the Q _i need not to be linearly dependent our result covers a class of nondiagonal, possibly nonmonotone elliptic systems. The proof uses a new weighted norm technique with singular weights in an L ^p -setting.     

Approximation of the function sign x in the uniform and integral metrics by means of rational functions

Estimates are obtained for the nonsymmetric deviations R_n [sign x] and R_n [sign x]_L of the function sign x from rational functions of degree ≤n, respectively, in the metric $$c([ - 1, - \delta ] \cup [\delta ,1]), 0< \delta< exp( - \alpha \surd \overline n ), \alpha > 0,$$ and in the metric L[?1, 1]: $$\begin{gathered} R_n [sign x] _{\frown }^\smile exp \{ - \pi ^2 n/(2 ln 1/\delta )\} , n \to \infty , \hfill \\ 10^{ - 3} n^{ - 2} \exp ( - 2\pi \surd \overline n )< R_n [sign x_{|L}< \exp ( - \pi \surd \overline {n/2} + 150). \hfill \\ \end{gathered} $$ Let 0 < δ < 1, Δ (δ)=[?1, ? δ] ∪ [δ, 1]; $$\begin{gathered} R_n [f;\Delta (\delta )] = R_n [f] = inf max |f(x) - R(x)|, \hfill \\ R_n [f;[ - 1,1] ]_L = R_n [f]_L = \mathop {inf}\limits_{R(x)} \smallint _{ - 1}^1 |f(x) - R(x)|dx, \hfill \\ \end{gathered} $$ where R(x) is a rational function of order at most n. Bulanov [1] proved that for δ ε [e^?n, e^?1] the inequality $$\exp \left( {\frac{{\pi ^2 n}}{{2\ln (1/\delta }}} \right) \leqslant R_n [sign x] \leqslant 30 exp\left( {\frac{{\pi ^2 n}}{{2\ln (1/\delta + 4 ln ln (e/\delta ) + 4}}} \right)$$ is valid. The lower estimate in this inequality was previously obtained by Gonchar ([2], cf. also [1]).     

Well-posedness of a semilinear heat equation with weak initial data

This article mainly consists of two parts. In the first part the initial value problem (IVP) of the semilinear heat equation $$\begin{gathered} \partial _t u - \Delta u = \left| u \right|^{k - 1} u, on \mathbb{R}^n x(0,\infty ), k \geqslant 2 \hfill \\ u(x,0) = u_0 (x), x \in \mathbb{R}^n \hfill \\ \end{gathered} $$ with initial data in $\dot L_{r,p} $ is studied. We prove the well-posedness when $$1< p< \infty , \frac{2}{{k(k - 1)}}< \frac{n}{p} \leqslant \frac{2}{{k - 1}}, and r =< \frac{n}{p} - \frac{2}{{k - 1}}( \leqslant 0)$$ and construct non-unique solutions for $$1< p< \frac{{n(k - 1)}}{2}< k + 1, and r< \frac{n}{p} - \frac{2}{{k - 1}}.$$ In the second part the well-posedness of the avove IVP for k=2 with μ₀?H ^s (? ⁿ ) is proved if $$ - 1< s, for n = 1, \frac{n}{2} - 2< s, for n \geqslant 2.$$ and this result is then extended for more general nonlinear terms and initial data. By taking special values of r, p, s, and u₀, these well-posedness results reduce to some of those previously obtained by other authors [4, 14].     

On a problem of L. Leindler concerning strong approximation by Fourier series and Lipschitz classes

Пусть? — возрастающа я непрерывная фцнкци я на [0,π],?(0)=0 и $$\mathop \smallint \limits_0^h \frac{{\varphi \left( t \right)}}{t}dt = O\left( {\varphi \left( h \right)} \right){\text{ }}\left( {h \to 0} \right).$$ Положим $$\psi \left( h \right) = h\mathop \smallint \limits_h^\pi \frac{{\varphi \left( t \right)}}{{t^2 }}dt \left( {h \in (0, \pi ]} \right).$$ Доказывается следую щая теорема.Пусть f∈ С[?π, π], ω(f, δ)=О(?(δ))) и $$\mathop {\lim }\limits_{h \to 0} \frac{1}{{\varphi \left( {\left| h \right|} \right)}}\left| {f\left( {x + h} \right) - f\left( x \right)} \right| = 0$$ для x∈E?[?π, π], ¦E¦>0. Тогда д ля сопряженной функц ии f почти всюду на E выполн яется соотношение $$\mathop {\lim }\limits_{h \to 0} \frac{1}{{\psi \left( {\left| h \right|} \right)}}\left| {\tilde f\left( {x + h} \right) - \tilde f\left( x \right)} \right| = 0.$$ Из этой теоремы вытек ает положительное ре шение одной задачи Л. Лейндлера.