Multipliers Between Sobolev Spaces

1991 ◽  
Vol 34 (4) ◽  
pp. 465-473
Author(s):  
R. C. Fabec
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Higher Dimensions ◽  
Sufficient Condition

AbstractA sufficient condition for the boundedness of a multiplier from a Sobolev space of index t > 1 / 4 to one of opposite index — t is obtained. The condition relates the indices of the Sobolev spaces to which the multiplier belongs to the pairs of Sobolev spaces between which the multiplier is bounded. The result is applied to homogeneous multipliers and a description of these multipliers in this setting is presesented. Extensions to higher dimensions are indicated.

A CONSTRUCTION OF CONVERGENT CASCADE ALGORITHMS IN SOBOLEV SPACES

2007 ◽  
Vol 05 (05) ◽  
pp. 685-697 ◽  
Author(s):  
DI-RONG CHEN
Keyword(s):  
Computer Graphics ◽  
Sobolev Space ◽  
Wavelet Analysis ◽  
Sobolev Spaces ◽  
Mild Condition ◽  
Time Domain ◽  
Sufficient Condition ◽  
Cascade Algorithms ◽  
The Time Domain

Cascade algorithms play an important role in wavelet analysis and computer graphics. The paper considers the convergence of cascade algorithms in Sobolev spaces. With the help of the factorization of matrix masks, we give a sufficient condition for the convergence. The condition is expressed in the time domain. More importantly, an algorithm for the construction of convergent cascade algorithms in Sobolev space starting from any matrix mask satisfying a mild condition is presented. Examples are given to illustrate our theorems.

scholarly journals Reproducing kernels of Sobolev spaces on ℝd and applications to embedding constants and tractability

2018 ◽  
Vol 16 (05) ◽  
pp. 693-715 ◽  
Author(s):  
Erich Novak ◽  
Mario Ullrich ◽  
Henryk Woźniakowski ◽  
Shun Zhang
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Reproducing Kernel ◽  
Absolute Error ◽  
Reproducing Kernels ◽  
Worst Case ◽  
Reproducing Kernel Formula ◽  
Positive Integers ◽  
Integration Errors ◽  
Exponentially Small

The standard Sobolev space [Formula: see text], with arbitrary positive integers [Formula: see text] and [Formula: see text] for which [Formula: see text], has the reproducing kernel [Formula: see text] for all [Formula: see text], where [Formula: see text] are components of [Formula: see text]-variate [Formula: see text], and [Formula: see text] with non-negative integers [Formula: see text]. We obtain a more explicit form for the reproducing kernel [Formula: see text] and find a closed form for the kernel [Formula: see text]. Knowing the form of [Formula: see text], we present applications on the best embedding constants between the Sobolev space [Formula: see text] and [Formula: see text], and on strong polynomial tractability of integration with an arbitrary probability density. We prove that the best embedding constants are exponentially small in [Formula: see text], whereas worst case integration errors of algorithms using [Formula: see text] function values are also exponentially small in [Formula: see text] and decay at least like [Formula: see text]. This yields strong polynomial tractability in the worst case setting for the absolute error criterion.

scholarly journals Characterizations of Orlicz-Sobolev Spaces by Means of Generalized Orlicz-Poincaré Inequalities

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Toni Heikkinen
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Measure Space ◽  
Poincaré Inequality ◽  
Metric Measure Space ◽  
Poincare Inequality ◽  
Poincaré Inequalities ◽  
Space Setting ◽  
Poincare Inequalities

Let Φ be anN-function. We show that a functionu∈LΦ(ℝn)belongs to the Orlicz-Sobolev spaceW1,Φ(ℝn)if and only if it satisfies the (generalized) Φ-Poincaré inequality. Under more restrictive assumptions on Φ, an analog of the result holds in a general metric measure space setting.

Existence results for a non-homogeneous Neumann problem through Orlicz–Sobolev spaces

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Shapour Heidarkhani ◽  
Giuseppe Caristi ◽  
Ghasem A. Afrouzi ◽  
Shahin Moradi
Keyword(s):  
Weak Solution ◽  
Variational Principle ◽  
Sobolev Space ◽  
Banach Spaces ◽  
Neumann Problem ◽  
Sobolev Spaces ◽  
Existence Results ◽  
Reflexive Banach Spaces

Abstract Based on a variational principle for smooth functionals defined on reflexive Banach spaces, the existence of at least one weak solution for a non-homogeneous Neumann problem in an appropriate Orlicz–Sobolev space is discussed.

scholarly journals Traces of Besov, Triebel-Lizorkin and Sobolev Spaces on Metric Spaces

2017 ◽  
Vol 5 (1) ◽  
pp. 98-115 ◽  
Author(s):  
Eero Saksman ◽  
Tomás Soto
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Metric Space ◽  
Besov Spaces ◽  
Metric Spaces ◽  
Function Spaces ◽  
First Order ◽  
Trace Theorems ◽  
Euclidean Setting ◽  
Regular Subset

Abstract We establish trace theorems for function spaces defined on general Ahlfors regular metric spaces Z. The results cover the Triebel-Lizorkin spaces and the Besov spaces for smoothness indices s < 1, as well as the first order Hajłasz-Sobolev space M1,p(Z). They generalize the classical results from the Euclidean setting, since the traces of these function spaces onto any closed Ahlfors regular subset F ⊂ Z are Besov spaces defined intrinsically on F. Our method employs the definitions of the function spaces via hyperbolic fillings of the underlying metric space.

scholarly journals On the boundary values of the solutions of linear elliptic equations

1983 ◽  
Vol 27 (1) ◽  
pp. 1-30 ◽  
Author(s):  
J. Chabrowski ◽  
H.B. Thompson
Keyword(s):  
Elliptic Equation ◽  
Sobolev Space ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Sufficient Condition ◽  
Boundary Values ◽  
Linear Elliptic Equation

The purpose of this article is to investigate the traces of weak solutions of a linear elliptic equation. In particular, we obtain a sufficient condition for a solution belonging to the Sobolev space to have an L2-trace on the boundar.

scholarly journals QKSpaces on the Unit Circle

2014 ◽  
Vol 2014 ◽  
pp. 1-9
Author(s):  
Jizhen Zhou
Keyword(s):  
Sobolev Space ◽  
Unit Circle ◽  
Measurable Function ◽  
Weight Functions ◽  
Sufficient Condition ◽  
General Criterion ◽  
Necessary And Sufficient Condition ◽  
New Space ◽  
Necessary And Sufficient

We introduce a new spaceQK(∂D)of Lebesgue measurable functions on the unit circle connecting closely with the Sobolev space. We obtain a necessary and sufficient condition onKsuch thatQK(∂D)=BMO(∂D), as well as a general criterion on weight functionsK1andK2,K1≤K2, such thatQK1(∂D)QK2(∂D). We also prove that a measurable function belongs toQK(∂D)if and only if it is Möbius bounded in the Sobolev spaceLK2(∂D). Finally, we obtain a dyadic characterization of functions inQK(∂D)spaces in terms of dyadic arcs on the unit circle.

scholarly journals Generalization of Poincar ´e inequality in a Sobolev Space with exponent constant to the case of Sobolev space with a variable exponent

2021 ◽  
Vol 10 (2) ◽  
pp. 31-37
Author(s):  
Moulay Rchid Sidi Ammi ◽  
Ibrahim Dahi
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Variable Exponent ◽  
Poincaré Inequality ◽  
Poincare Inequality ◽  
Equivalent Norms

In this work, we study the Poincare inequality in Sobolev spaces with variable exponent. As a consequence of this ´ result we show the equivalent norms over such cones. The approach we adopt in this work avoids the difficulty arising from the possible lack of density of the space C∞ 0 (Ω).

scholarly journals The boundedness of a class of semiclassical Fourier integral operators on Sobolev space $H^{s}$

2021 ◽  
Vol 56 (1) ◽  
pp. 61-66
Author(s):  
O. F. Aid ◽  
A. Senoussaoui
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Integral Operators ◽  
Fourier Integral ◽  
Schwartz Space ◽  
Background Information ◽  
Fourier Integral Operators ◽  
Phase Functions

We introduce the relevant background information thatwill be used throughout the paper.Following that, we will go over some fundamental concepts from thetheory of a particular class of semiclassical Fourier integraloperators (symbols and phase functions), which will serve as thestarting point for our main goal. Furthermore, these integral operators turn out to be bounded on$S\left(\mathbb{R}^{n}\right)$ the space of rapidly decreasingfunctions (or Schwartz space) and its dual$S^{\prime}\left(\mathbb{R}^{n}\right)$ the space of temperatedistributions. Moreover, we will give a brief introduction about$H^s(\mathbb{R}^n)$ Sobolev space (with $s\in\mathbb{R}$).Results about the composition of semiclassical Fourier integraloperators with its $L^{2}$-adjoint are proved. These allow to obtainresults about the boundedness on the Sobolev spaces$H^s(\mathbb{R}^n)$.

Poincaré and Friedrichs inequalities in abstract Sobolev spaces II

1994 ◽  
Vol 115 (1) ◽  
pp. 159-173 ◽  
Author(s):  
D. E. Edmunds ◽  
B. Opic ◽  
J. Rákosník
Keyword(s):  
Banach Space ◽  
Sobolev Space ◽  
Weight Function ◽  
Sobolev Spaces ◽  
Positive Constant ◽  
Banach Function Spaces ◽  
Distributional Derivative ◽  
Friedrichs Inequality ◽  
Image Position ◽  
Anisotropic Spaces

This paper is a continuation of [4]; its aim is to extend the results of that paper to include abstract Sobolev spaces of higher order and even anisotropic spaces. Let Ω be a domain in ℝN, let X = X(Ω) and Y = Y(Ω) be Banach function spaces in the sense of Luxemburg (see [4] for details), and let W(X, Y) be the abstract Sobolev space consisting of all f ∈ X such that for each i ∈ {l, …, N} the distributional derivative belongs to Y; equipped with the normW(X, Y) is a Banach space. Given any weight function w on Ω, the triple [w, X, Y] is said to support the Poincaré inequality on Ω. if there is a positive constant K such that for all u ∈ W(X, Y)the pair [X, Y] is said to support the Friedrichs inequality if there is a positive constant K such that for all u ∈ W0(X, Y) (the closure of in W(X, Y))

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