On asymptotic linearity of L-estimates

A theorem on asymptotic linearity of L-estimates is proved under general set of regularity conditions, allowing the sampled distribution to be non-integrable. The main result is the improvement in the order of the remainder term in the formula for asymptotic linearity of L-statistic. It is shown that in the case of the integral coefficients this term R _n = O _P (1/n) and the case of functional coefficients is also covered.     

General M-estimates for contaminated p th-order autoregressive processes: Consistency and asymptotic normality

Summary This work is concerned with simultaneous estimation of coefficients and a scale parameter of a p th-order autoregressive process (X _t ). The observations are Y _t =V _t Z _t +(1-V _t )X _t where (Z _t) is a contaminating process and (V _t ) represents the proportion of contamination. If (X _t ) or (Z _t ) have heavy tails both least squares estimates and ordinary M-estimates are seriously affected. Under general conditions we prove consistency and asymptotic normality of a general class of M-estimates which contains some M-estimates studied by Denby and Martin [6].     

On the rate of conveergence in the local limit theorem for densities

The aim of this paper is to improve the structure of the convergence rate estimates in the local limit theorem for densities and to give asymptotic estimates of the absolute constants involved in uniform, nonuniform, and L_p-estimates. Supported by the Russian Foundation for Fundamental Research (grant No. 96-01-01920). Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, Russia, 1996, Part II.     

Lp Estimates for the Solution of the Model Verigin Problem

In this paper, we consider a model free-boundary problem related to the Verigin problem. L _p-estimates of solutions are obtained with the help of results on Fourier multipliers. These estimates can be used to prove the solvability of the Verigin problem in Sobolev functional spaces. Bibliography: 14 titles.     

On the asymptotic distribution of multivariate M-estimates

The asymptotic distribution of multivariate M-estimates is studied. It is shown that, in general, consistency leads to asymptotic normality and a Law of the Iterated Logarithm. The results are used to compute via matrix derivatives the asymptotic distribution of a class of estimates due to Maronna.     

A new proof of certain littlewood-paley inequalities

The aim of this article is to give a new proof of the L^p-inequalities for the Littlewood-Paley g_*-function. Our main tool is a pointwise equality, relating a function f, and the associated functional g_*(f), which has the form f²=h(f)+g _* ² (f), where h(f) is an explicit function. We obtain this equality as a particular case of a more general one, which is reminiscent of a well-known identity in the stochastic calculus setting, namely the Itô formula. Once the above equality is proved, L^p-estimates for g_*(f) are obviously equivalent to L^p/2-estimates for h(f). We obtain these last estimates (more precisely, H^p/2-estimates for h(f) by using a slight extension of the Coifman-Meyer-Stein theorem relating the so-called tent-spaces and the Hardy spaces. We observe that our methods clearly show that the restriction p>2n/n+1 is closely related to cancellation and size properties of the gradient of the Poisson kernel.     

Estimates for Littlewood-Paley Functions and Extrapolation

We prove certain L ^p -estimates for Littlewood-Paley functions arising from rough kernels. The estimates are useful for extrapolation to prove L ^p -boundedness of the Littlewood-Paley functions under a sharp kernel condition.      

Symmetrization and initial boundary-value problems for certain classes of nonlinear second order parabolic equations

Initial boundary-value problems are considered for equations of porous medium andp-Laplace types. By using the Schwarz symmetrization method, theL _p-estimates for the solutions of the original problems are obtained in terms of the analogous estimates for the corresponding symmetric solutions.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 7, pp. 884–892, July, 1993.     

Decay estimates for strong solutions of the Navier-Stokes equations in exterior domains

Based on theL _p−L_q-estimates for the Stokes semigroup on exterior domains, proven by Iwashita and refined by Maremonti-Solonnikov and Shibata, the precise time-asymptotic behaviour of solutions of the Navier-Stokes equations is derived. The estimates are improved, if some additional information on the initial value is available. The results apply also to weak solutions in three and four dimensions.

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Sunto In questo lavoro forniamo l’andamento asintotico nel tempo preciso di soluzioni delle equazioni di Navier-Stokes in un dominio esterno, basandoci sulle stimeL _p−L _q per il semigruppo di Stokes fornite, per primo, da Iwashita e quindi rifinite da Maremonti-Solonnikov e Shibata. Tali stime, se sono note ulteriori informazioni sui dati iniziali, sono migliorate. Il risultato si applica anche a soluzioni deboli in dimensione tre e quattro.

     

Estimates for solutions of the nonstationary stokes problem in anisotropic Sobolev spaces with mixed norm

A new method is given to establish L_q,r-estimates for solutions of the nonstationary Stokes problem. The method is based on estimates for heat potentials in these spaces, and it is not connected with investigation of the resolvent for the Stokes operator. It is expected that the method is applicable to a wide class of parabolic initial boundary-value problems. Bibliography:11 titles. Translated fromZapiski Nauchnykh Seminarow POMI, Vol. 222, 1994, pp. 124–151.     

Lp-Estimates for a Solution to the Nonstationary Stokes Equations

L _p-estimates for the nonstationary Stokes problem are established. Bibliography: 14 titles.     

OnL p -convergence rates for statistical functions with application toL-estimates

Summary A parameter which may be represented as a functionalT(F) of a distribution functionF may be estimated by the “statistical function”T(F _n ), whereF _n is the empirical distribution function. Recently, Boos and Serfling (1979, Florida State University Statistics Report No. M 499) obtained sufficient conditions for the Berry-Esseen theorem to hold forT(F _n )-T(F) and applied the results to derive rates of convergence inL _∞ forL-estimates. The present note complements their work by obtaining theL _p -rates of convergence, 1≦p<∞ forT(F _n )-T(F) and its application toL-estimates.     

Asymptotic structure, -estimates of sequences, and compactness of multilinear mappings

We relate the moduli of asymptotic uniform smoothness and convexity of a Banach space with the existence of upper and lower ℓ_p-estimates of sequences in the space. To this end, we introduce two properties which are related to the (m_p)-property defined by Kalton and Werner. In this way we obtain a connection between the moduli of asymptotic uniform smoothness and convexity, and compactness or weak-sequential continuity of multilinear mappings. Finally, we give some applications to the existence of analytic and asymptotically flat norms on a Banach space.     

Square-like Functions Generated by a Composite Wavelet Transform

The classical square functions play important role in Harmonic Analysis and have a very direct connection with L ₂-estimates and Littlewood-Paley theory. In this paper we introduce a new variant of square-like functions generated by some composite wavelet transform. We establish Calderón-type reproducing formula and then prove L ₂-boundedness of newly defined square-like functions.     

Direct approach to <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> estimates in homogenization theory

We derive interior L ^p -estimates for solutions of linear elliptic systems with oscillatory coefficients. The estimates are independent of ε, the small length scale of the rapid oscillations. So far, such results are based on potential theory and restricted to periodic coefficients. Our approach relies on BMO-estimates and an interpolation argument, gradients are treated with the help of finite differences. This allows to treat coefficients that depend on a fast and a slow variable. The estimates imply an L ^p -corrector result for approximate solutions.      

On the mean value of the function $$ {\bar{S}_k}(n) $$

An asymptotic formula is constructed for the mean value of the function [`(S)]_k(n) {\bar{S}_k}(n) dual to the Smarandache function S _k (n). The O- and Ω-estimates for the second moment of the remainder are obtained.     

Some equilibrium finite element methods for two-dimensional elasticity problems

Summary We consider some equilibrium finite element methods for two-dimensional elasticity problems. The stresses and the displacements are approximated by using piecewise linear functions. We establishL ₂-estimates of orderO(h ²) for both stresses and displacements.     

On theL p-L q-estimates for the solutions of certain hyperbolic equations

We establish conditions, which should be imposed on the exponentsp andq and the dimensionalityn of a space in order that theL _p-L _q-estimates should hold for the solutions of the Cauchy problem for a second-order hyperbolic equation with constant coefficients.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 7, pp. 992–1008, July, 1993.     

Error estimates for the finite element approximation to a maxwell-type boundary value problem

Maxwell's boundary value problem for the time-harmonic case in a smooth, bounded domain G of R ² is considered. The optimal asymptotic L₂(G) and H₁(G)-error estimates 0(h²) and 0(h) resp, are derived for a piecewise linear finite element solution.     

Local polynomial fitting under association

We consider the estimation of multivariate regression functions r(x₁,…,x_d) and their partial derivatives up to a total order p1 using high-order local polynomial fitting. The processes {Y_i,X_i} are assumed to be (jointly) associated. Joint asymptotic normality is established for the estimates of the regression function r and all its partial derivatives up to the total order p. Expressions for the bias and variance/covariance matrix (of the asymptotic distribution) are given.