L p Boundedness for the Multilinear Singular Integral Operator

L^p (\mathbbRⁿ )L^{p} (\mathbb{R}^{n} ) boundedness is considered for the maximal multilinear singular integral operator which is defined by

On the homogenization of noncoercive variational problems

The paper deals with variational problems of the form $$\mathop {\inf }\limits_{u \in W^{1,p} (\Omega )} \int\limits_\Omega {a(\varepsilon ^{ - 1} x)(\left| {\nabla u} \right|^p + \left| {u - g} \right|^p )} dx,$$ where Ω is a bounded Lipschitzian domain in ? ^N , g∈L^p(Ω). The function a(x) is assumed to satisfy the following conditions:

1. a(x) is periodic and lower semicontinuous;
2. 0≤a(x)≤1 and the set {∈? ^N , a(x)>0} is connected in ? ^N Under these conditions, basic properties of homogenization (convergence of energies and generalized solutions) and properties of Г-convergence type are proved. Bibliography: 3 titles.

     

Exponents for three-dimensional simultaneous Diophantine approximations

Let Θ = (θ ₁,θ ₂,θ ₃) ∈ ℝ³. Suppose that 1, θ ₁, θ ₂, θ ₃ are linearly independent over ℤ. For Diophantine exponents

Error bounds for the finite element approximation of a degenerate quasilinear parabolic variational inequality

In this paper, we establish some error bounds for the continuous piecewise linear finite element approximation of the following problem: Let Ω be an open set in ? ^d , withd=1 or 2. GivenT>0,p ∈ (1, ∞),f andu ₀; findu ∈K, whereK is a closed convex subset of the Sobolev spaceW ₀ ^1,p (Ω), such that for anyv∈K $$\begin{gathered} \int\limits_\Omega {u_1 (\upsilon - u) dx + } \int\limits_\Omega {\left| {\nabla u} \right|^{p - 2} } \nabla u \cdot \nabla (\upsilon - u) dx \geqslant \int\limits_\Omega {f(\upsilon - u) dx for} a.e. t \in (0,T], \hfill \\ u = 0 on \partial \Omega \times (0,T] and u(0,x) = u_0 (x) for x \in \Omega . \hfill \\ \end{gathered} $$ We prove error bounds in energy type norms for the fully discrete approximation using the backward Euler time discretisation. In some notable cases, these error bounds converge at the optimal rate with respect to the space discretisation, provided the solutionu is sufficiently regular.     

ON SMOOTHNESS AND DIFFERENTIABILITY OF FUNCTIONS

Let \(f(x)\) be a bounded real function on [-1,1],we define the modulus of continuity of f as \[\omega (f,\delta ) = \mathop {\sup }\limits_{x,y \in [ - 1,1],\left| {x - y} \right| \le \delta } \left| {f(x) - f(y)} \right|\] and the modulus of smoothness of f as \[{\omega _2}(f,\delta ) = \mathop {\sup }\limits_{x \pm h \in [ - 1,1],\left| h \right| \le \delta } \left| {f(x + h) + f(x - h) - 2f(x)} \right|\] Functions \(f(x)\), continuous on [-1,1] and \({\omega _2}(f,\delta ) = o(\delta )\) ,are called uniformly smooth functions. It is well known that there is a uniformly smooth functions whose derivative exisits on a null-set only. It would is of interest to discuss what condition should be added on the nonnegative function \(\varphi (\delta )\), \(\left( {0 \le \delta \le \frac{1}{2}} \right)\),in order that every bounded function f satisfying\[{\omega _2}(f,\delta ) = O(\varphi (\delta ))\] possess continous (or finite) derivative. the main result of this paper are the following two theorems. Theorem 1 let \(\varphi (\delta )\),\(\left( {0 \le \delta \le \frac{1}{2}} \right)\) ,be a nonnegative function, then, in order that every bounded function \(f(x)\) satisfying condition \[{\omega _2}(f,\delta ) = O(\varphi (\delta ))\] possess continous (or finite) derivative \(f'(x)\) on [-1,1],it is necessary and sufficient that the following condition hold \[\int_0^{\frac{1}{2}} {\frac{{\tilde \varphi (t)}}{t}} dt < \infty \] where \[\tilde \varphi (\delta ) = {\delta ^2}\mathop {\inf }\limits_{0 \le \eta \le \delta } \left\{ {{\eta ^{ - 2}}\mathop {\inf }\limits_{\eta \le \xi \le 1/2} \varphi (\xi )} \right\}\] Theorm 2 Let \(f(x)\) be a bounded function with \[\int_0^{\frac{1}{2}} {\frac{{{\omega _2}(f,t)}}{{{t^2}}}} dt < \infty \] then \(f'(x)\) is a continous function and \[{\omega _2}(f',\delta ) = O\left\{ {\int_0^\delta {\frac{{{\omega _2}(f,t)}}{{{t^2}}}} dt} \right\}\].     

GLOBAL EXISTENCE OF THE SOLUTION OF NONLINEAR SCHRDINGER EQUATIONS IN EXTERIOR DOMAINS

In this paper,we discuss the problem for the nonlinear Schr(?)dinger equation(?)where Ω is the exterior domain of a compact set in B~n,a_j(u)=O(|u|),b_j(u)=O(|u|)(1≤j≤n),c(u)=O(|u|~2)near u=0.If n≥5,some Sobolev norm of u_0(x)is sufficiently small,under suitableassumptions on smoothnessand and compatibility and the shape of Ω,we get that the problem has aunique global solution u(t,x),with the decay estimate‖u(t,·)‖_(L(?)(Ω))=O(t~(-n/4)),‖u(t,·)‖_(L~4(Ω))=O(t~(-n/4)),t→+∞.     

Strong Convergence for Weighted Sums of Negatively Associated Arrays

Let {Xni} be an array of rowwise negatively associated random variables and Tnk=k∑i=1 i^a Xni for a ≥ -1, Snk =∑｜i｜≤k Ф（i/nη）1/nη Xni for η∈（0,1],where Ф is some function. The author studies necessary and sufficient conditions of ∞∑n=1 AnP（max 1≤k≤n｜Tnk｜〉εBn）〈∞ and ∞∑n=1 CnP（max 0≤k≤mn｜Snk｜〉εDn）〈∞ for all ε 〉 0, where An, Bn, Cn and Dn are some positive constants, mn ∈ N with mn /nη →∞. The results of Lanzinger and Stadtmfiller in 2003 are extended from the i.i.d, case to the case of the negatively associated, not necessarily identically distributed random variables. Also, the result of Pruss in 2003 on independent variables reduces to a special case of the present paper; furthermore, the necessity part of his result is complemented.     

Bombieri's theorem in short intervals

The well-known Bombieri-A. I. Vinogradov theorem states that (1) $$\sum\limits_{q \leqslant x^{\tfrac{1}{2}} (\log x)^{ - s} } {\mathop {\max }\limits_{(a,q) = 1} \mathop {\max }\limits_{y \leqslant x} } \left| {\psi (y,q;a) - \frac{y}{{\varphi (q)}}} \right| \ll \frac{x}{{(\log x)^A }},$$ whereA is an arbitrary positive constant,B=B(A)>0, and as usual, $$\psi (x,q;a) = \sum\limits_{\mathop {n \leqslant x}\limits_{n = a(q)} } {\Lambda (n),}$$ Λ being the Von Mangoldt's function. The problem of finding a result analogous to (1) for short intervals was investigated by many authors. Using Heath-Brown's identity and the approximate functional equation for DirichletL-functions, A. Perelli, J. Pintz and S. Salerno in 1985 established the following extension of Bombieri's theorem: Theorem 1. (2) $$\sum\limits_{q \leqslant Q} {\mathop {\max }\limits_{(a,q) = 1} \mathop {\max }\limits_{h \leqslant y} \mathop {\max }\limits_{\frac{x}{2}< \approx \leqslant x} } \left| {\psi (z + h,q;a) - \psi (z,q;a) - \frac{h}{{\varphi (q)}}} \right| \ll \frac{y}{{(\log x)^A }}$$ where A>0 is an arbitrary constant,y=x ^θ $$\frac{7}{{12}}< \theta \leqslant 1, Q = x^{\frac{1}{{40}}} .$$ ,Q=x ^1/40. By improving the basic lemma which A. Perelli, J. Pintz and S. Salerno used as the main tool to prove Theorem 1, we obtain Theorem 2.Under the same condition as in Theorem 1,for Q=x ^1/38.5, (2)still holds.     

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.     

Comparison results for some types of relaxation of variational integral functionals

A comparison between some relaxation methods of an integral functional is carried out. The following relaxed functionals of the variational integral I(, u)= :

A sharp maximal inequality for continuous martingales and their differential subordinates

Assume that X, Y are continuous-path martingales taking values in ? ^ν , ν ? 1, such that Y is differentially subordinate to X. The paper contains the proof of the maximal inequality $$\left\| {\mathop {\sup }\limits_{t \geqslant 0} \left| {Y_t } \right|} \right\|_1 \leqslant 2\left\| {\mathop {\sup }\limits_{t \geqslant 0} \left| {X_t } \right|} \right\|_1 .$$ The constant 2 is shown to be the best possible, even in the one-dimensional setting of stochastic integrals with respect to a standard Brownian motion. The proof uses Burkholder’s method and rests on the construction of an appropriate special function.     

OnL 1-convergence of Walsh-Fourier series I

We prove the convergence in theL ¹(0, 1)-metric of Walsh-Fourier series \(\sum\limits_{k = 0}^\infty {a_k w_k \left( x \right)} \) of an integrable function with coefficients such that lim_n→∞ and the following Tauberian condition of Hardy-Karamata kind is satisfied: $$\mathop {lim}\limits_{\lambda \to 1 + 0} {\text{ }}\mathop {lim}\limits_{n \to \infty } \sum\limits_{k = n}^{\left[ {\lambda n} \right]} {k^{p - 1} \left| {\Delta a_k } \right|^p } = 0,$$ , wherep>1, [·] denotes the integral part, and Δa _k=a_k?a_k+1.     

Dichotomy laws for the behaviour near the unit element of group representations

A well known “zero-two law" shows that if is a strongly continuous one-parameter group of bounded operators on a Banach space X, and if then Here we discuss analogous problems for general unital representations θ of a topological group G on a unital Banach algebra A. Let 1 be the unit of G, and I the unit element of A. We show that either or if, moreover, θ admits “continuous division by any positive integer”, then, either or Our argument also gives automatic continuity results for representations of abelian Baire groups on a separable Banach algebra and representations of compact non abelian groups on a Banach algebra which are locally bounded and satisfy Received: 8 June 2005; revised: 13 October 2005     

THE ASYMPTOTIC BEHAVIOUR OF ANALYTIC FUNCTIONS

AIn this paper, the author obtains the following results:(1) If Taylor coeffiients of a function satisfy the conditions:(i),(ii),(iii)A_k=O(1/k) the for any h>0 the function φ(z)=exp{w(z)} satisfies the asymptotic equality the case h>1/2 was proved by Milin.(2) If f(z)=z α_2z~2 …∈S~* and,then for λ>1/2     

Extinction of solutions of semilinear higher order parabolic equations with degenerate absorption potential

We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation ${u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1}We study the first vanishing time for solutions of the Cauchy–Dirichlet problem for the 2m-order (m ≥ 1) semilinear parabolic equation u_t + Lu + a(x) |u|^q-1u=0, 0 < q < 1{u_t + Lu + a(x) |u|^{q-1}u=0,\,0 < q < 1} with a(x) ≥ 0 bounded in the bounded domain W ì \mathbb R^N{\Omega \subset \mathbb R^N}. We prove that if N 1 2m{N \ne 2m} and ò₀¹ s^-1 (meas\nolimits {x ? W: |a(x)| ￡ s })^q ds < ￥, q = min(\frac2mN,1){\int_0^1 s^{-1} (\mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \})^\theta {\rm d}s < \infty,\ \theta=\min\left(\frac{2m}N,1\right)}, then the solution u vanishes in a finite time. When N = 2m, the same property holds if ${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}${\int_0^1 s^{-1} \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) \ln \left( \mathop{\rm meas}\nolimits \{x \in \Omega : |a(x)| \leq s \} \right) {\rm d}s > - \infty}.     

Bulk universality holds in measure for compactly supported measures

Let μ be a measure with compact support, with orthonormal polynomials {p _n } and associated reproducing kernels {K _n }. We show that bulk universality holds in measure in {ξ: μ′(ξ) > 0}. More precisely, given ɛ, r > 0, the linear Lebesgue measure of the set {ξ: μ′(ξ) > 0} and for which

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 $$V(\partial G) = \sup \left\{ {\left\| {r'} \right\|_{L_1 (\partial G)} :r(z) = a/(z - b),{\mathbf{ }}\left\| r \right\|_{G(\partial G)} \leqq 1} \right\}.$$      

Applications of weighted estimates of Calderon’s commutators

The paper introduces singular integral operators of a new type defined in the space L ^p with the weight function on the complex plane. For these operators, norm estimates are derived. Namely, if V is a complex-valued function on the complex plane satisfying the condition |V(z) ? V(??)| ?? w|z ? ??| and F is an entire function, then we put $$P_F^* f(z) = \mathop {\sup }\limits_{\varepsilon > 0} \left| {\int\limits_{\left| {\zeta - z} \right| > \varepsilon } {F\left( {\frac{{V(\zeta ) - V(z)}} {{\zeta - z}}} \right)\frac{{f(\zeta )}} {{\left( {\zeta - z} \right)^2 }}d\sigma (\zeta )} } \right|.$$ It is shown that if the weight function ?? is a Muckenhoupt A _p weight for 1 < p < ??, then $$\left\| {P_F^* f} \right\|_{p,\omega } \leqslant C(F,w,p)\left\| f \right\|_{p,\omega } .$$ .