Lp Estimates Research Articles

Lp estimates and asymptotic behavior of extremal function to Hardy-Sobolev type inequality on the H-type group*

This paper is devoted to discuss the regularity of the weak solution to a class of non-linear equations corresponding to Hardy-Sobolev type inequality on the H-type group. Combining the Serrin's idea and the Moser's iteration, L p estimates of the weak solution are obtained, which generalize the results of Garofalo and Vassilev in [6, 14]. As an application, asymptotic behavior of the weak solution has been discussed. Finally, doubling property and unique continuation of the weak solution are given.

Generalizations of the Carleson-Hunt theorem I. The classical singularity case

In this article, we prove Lp estimates for a general maximal operator, which extend both the classical Coifman-Meyer and Carleson-Hunt theorems in harmonic analysis.

Semiclassical L p Estimates

The purpose of this paper is to use semiclassical analysis to unify and generalize Lp estimates on high energy eigenfunctions and spectral clusters. In our approach these estimates do not depend on ellipticity and order, and apply to operators which are selfadjoint only at the principal level. They are estimates on weakly approximate solutions to semiclassical pseudodifferential equations. The revision corrects an exponent in the main theorems.

Learning and approximation by Gaussians on Riemannian manifolds

We confirm by the multi-Gaussian support vector machine (SVM) classification that the information of the intrinsic dimension of Riemannian manifolds can be used to illustrate the efficiency (learning rates) of learning algorithms. We study an approximation scheme realized by convolution operators involving the Gaussian kernels with flexible variances. The essential analysis lies in the study of its approximation order in Lp (1 ≤ p < ∞) norm as the variance of the Gaussian tends to zero. It is different from the analysis for approximation in C(X) since pointwise estimations do not work any more. The Lp approximation arises from the SVM case where the approximated function is the Bayes rule and is not continuous, in general. The approximation error is estimated by imposing a regularity condition that the approximated function lies in some interpolation spaces. Then, the learning rates for multi-Gaussian regularized classifiers with general classification loss functions are derived, and the rates depend on the intrinsic dimension of the Riemannian manifold instead of the dimension of the underlying Euclidean space. Here, the input space is assumed to be a connected compact C∞ Riemannian submanifold of ℝn. The uniform normal neighborhoods of the Riemannian manifold and the radial basis form of Gaussian kernels play an important role.

Resolvent estimates in W−1,p related to strongly coupled‐linear parabolic systems with coupled nonsmooth capacities

AbstractWe investigate linear parabolic systems with coupled nonsmooth capacities and mixed boundary conditions. We prove generalized resolvent estimates in W−1, p spaces. The method is an appropriate modification of a technique introduced by Agmon to obtain Lp estimates for resolvents of elliptic differential operators in the case of smooth boundary conditions. Moreover, we establish an existence and uniqueness result. Copyright © 2007 John Wiley &amp; Sons, Ltd.

A Class of Maximal Functions with Oscillating Kernels

The author studies the Lp mapping properties of a class of maximal functions that are related to oscillatory singular integral operators. Lp estimates, as well as the corresponding weighted estimates of such maximal functions, are obtained. Moreover, several applications of our results are highlighted.

Well-posedness theory for geometry-compatible hyperbolic conservation laws on manifolds

Motivated by many applications (geophysical flows, general relativity), we attempt to set the foundations for a study of entropy solutions to non-linear hyperbolic conservation laws posed on a (Riemannian or Lorentzian) manifold. The flux of the conservation laws is viewed as a vector-field on the manifold and depends on the unknown function as a parameter. We introduce notions of entropy solutions in the class of bounded measurable functions and in the class of measure-valued mappings. We establish the well-posedness theory for conservation laws on a manifold, by generalizing both Kruzkov's and DiPerna's theories originally developed in the Euclidian setting. The class of geometry-compatible (as we call it) conservation laws is singled out as an important case of interest, which leads to robust Lp estimates independent of the geometry of the manifold. On the other hand, general conservation laws solely enjoy the L1 contraction property and leads to a unique contractive semi-group of entropy solutions. Our framework allows us to construct entropy solutions on a manifold via the vanishing diffusion method or the finite volume method.

On the Lp estimation of a quantity from a set of observations

This paper considers the estimation of a quantity from a set of repeated measurements {xi} by the value of t minimizing ∑i |xi − t|p, as in Pennecchi and Callegaro (2006 Metrologia 43 213–19). The estimate with p = 2 is the familiar sample mean. Assumptions that are relevant when preferring an estimate with p < 2 are identified and examined. Theoretical considerations and results of simulations weaken the case for choosing an estimate with p < 2. The simulation results support the regular use of the sample mean or an estimate that involves the value p = 1.5 when the kurtosis of the dataset is sufficiently large.

Resolvent Estimates for the Linearized Compressible Navier-Stokes Equation in an Infinite Layer

The resolvent problem of the linearized compressible Navier-Stokes equation around a given constant state is considered in an infinite layer Rn-1 × (0,a), n ≥ 2, under the no slip boundary condition for the momentum. It is proved that the linearized operator is sectorial in W1,p × Lp for 1 < p < ∞. The Lp estimates for the resolvent are established for all 1 ≤ p ≤ ∞. The estimates for the high frequency part of the resolvent are also derived, which lead to the exponential decay of the corresponding part of the semigroup.

Fundamental solution and sharp Lp estimates for Laplace operators in the contact complex of Heisenberg groups

In this paper we construct a fundamental solution for the Laplace operator on the contact complex in Heisenberg groups ${\mathbb H}^{n}$ (Rumin’s complex) relying on the notion of currents in ${\mathbb H}^{n}$ given recently by Franchi, Serapioni and Serra Cassano. This operator is of order 2 on k intrinsic forms for k≠ n, but is of order 4 on n intrinsic forms. As an application, we prove sharp L p a priori estimates for horizontal derivatives. Keywords: Heisenberg groups, Differential forms, Currents, Laplace operators, Fundamental solution Mathematics Subject Classification (2000): 43A80, 58A10, 58A25, 35A08

Anisotropically weighted Poincaré-type inequalities; Application to the Oseen problem

The aim of this article is to prove Poincare-type inequalities concerning functions in Sobolev spaces with anisotropic weights that appear in the investigation of the Oseen equations. The inequalities are a generalization and an extension of inequalities we established in a previous work. As an application, we derive existence result and anisotropically weighted Lp estimates for the pressure of the Oseen problem in ℝ3. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Lp Estimates for Multilinear Operators of Strongly Singular Integral Operators

AbstractIn this paper, the authors get the Lp estimates for the commutators generated by strongly singular integral operators and BMO functions and the corresponding multilinear operators by the scale changing method introduced by Carleson and Sjölin.

Between the mean and the median: the Lp estimator

This paper presents the class of the Lp (least power) estimators and their main properties. Particular estimators belonging to such a class are well known in metrology, like the sample median, the sample mean and the mid-range (that is the Lp estimator with p = 1, 2, ∞, respectively), whereas the Lp estimator with generic p is not yet applied within the metrological context. Thanks to robustness and efficiency properties intermediate between those of the sample mean and the sample median, the Lp estimator with 1 < p < 2 (in particular, with p = 1.5) appears suitable for the statistical processing of experimental results. Application examples of the Lp estimator to international intercomparison data processing are reported.

The bi-Carleson operator

We prove Lp estimates (Theorem 1.3) for the bi-Carleson operator defined below. The methods used are essentially based on the treatment of the Walsh analogue of the operator in the prequel [MTT4] of this paper, but with additional technicalities due to the fact that in the Fourier model one cannot obtain perfect localization in both space and frequency.

Shape Optimization and Supremal Minimization Approaches in Landslides Modeling

The steady-state unidirectional (anti-plane) flow for a Bingham fluid is considered. We take into account the inhomogeneous yield limit of the fluid, which is well adjusted to the description of landslides. The blocking property is analyzed and we introduce the safety factor which is connected to two optimization problems in terms of velocities and stresses. Concerning the velocity analysis the minimum problem in BV(Ω) is equivalent to a shape-optimization problem. The optimal set is the part of the land which slides whenever the loading parameter becomes greater than the safety factor. This is proved in the one-dimensional case and conjectured for the two-dimensional flow. For the stress-optimization problem we give a stream function formulation in order to deduce a minimum problem in W1,ź(Ω) and we prove the existence of a minimizer. The Lp(Ω) approximation technique is used to get a sequence of minimum problems for smooth functionals. We propose two numerical approaches following the two analysis presented before. First, we describe a numerical method to compute the safety factor through equivalence with the shape-optimization problem. Then the finite-element approach and a Newton method is used to obtain a numerical scheme for the stress formulation. Some numerical results are given in order to compare the two methods. The shape-optimization method is sharp in detecting the sliding zones but the convergence is very sensitive to the choice of the parameters. The stress-optimization method is more robust, gives precise safety factors but the results cannot be easily compiled to obtain the sliding zone.

L p estimates on a time-inhomogeneous diffusion process

In this paper we consider the diffusion process X determined by the one-variable time-dependent Fokker-Planck equation, (∂∕∂t)P(y,t)=−g(t)(∂∕∂y)yP(y,t)+12f(t)2(∂2∕∂y2)P(y,t), where f,g:R+→R+ are two continuous functions, i.e., X satisfies Itô stochastic differential equation, dXt=f(t)dBt−g(t)Xtdt, where B is a standard Brownian motion starting at zero. We obtain Lp estimates on the process, and we show that ∥log(1+Jτ)∥p and ∥sup0⩽t⩽τ∣a(t)Xt∣∥p are equivalent for all stopping times τ of B, where Jt=∫0ta2(s)f2(s)ds and a:R+→R+ the solution to the equation (da∕dt)−g(t)a=−a3f2(t),a(0)=1.

Polynomials of Least Deviation from Zero

A new family of polynomials of least deviation from zero is defined on the unit disk B. Lower bounds for best approximations in the space Lp(B), p ≥ 1, are given.

An Overview of Asymptotic Properties of Lp Regression Under General Classes of Error Distributions

We survey the asymptotic properties of regression Lp estimators under general classes of error distributions. It is found that the asymptotic distributions of Lp estimators depend crucially on p and the shape of the error distribution near the origin. A number of important features arise as a result, among which are (a) use of a small p may yield accelerated convergence rates for Lp estimators under certain classes of error distributions; (b) for p < 1, Lp regression should, under some circumstances, be undertaken by locally maximizing, rather than minimizing, the sum of the pth powers of the absolute deviations; and (c) consistent estimation of the sampling distributions of the Lp estimators can be achieved by the m out of n bootstrap in general. Numerical examples are provided to illustrate our theoretical findings, and a computational algorithm is suggested for local maximization as may sometimes be required by the Lp procedure.

L p estimates for the uniform norm of solutions of quasilinear SPDE's

In this paper we prove Lp estimates (p≥2) for the uniform norm of the paths of solutions of quasilinear stochastic partial differential equations (SPDE) of parabolic type. Our method is based on a version of Moser's iteration scheme developed by Aronson and Serrin in the context of non-linear parabolic PDE.

A Review of Robust Regression and Diagnostic Procedures in Linear Regression

The concern over outliers is old since Bernoulli (see [12]), reviewed historically by [11] and updated with [10] in their encyclopedia textbook. James et al.[46] used simulation technique to compare some recent published outlier detection procedures. The history of adept and diagnosis of outliers is traced from old and presence comments. Theil-type or Rank, Brown-Mood, L p , M, adaptive M, GM, and Trimmed-Winsorization estimators are the most popular estimators that we will review in this paper as an application to outlier accommodation. We will review and compare the most numerical and graphical displays based on residuals to flag outliers.