The Hilbert-Schmidt Property for Embedding Maps between Sobolev Spaces

1966 ◽  
Vol 18 ◽  
pp. 1079-1084 ◽  
Author(s):  
Colin Clark
Keyword(s):  
Euclidean Space ◽  
Sobolev Space ◽  
Sobolev Spaces ◽  
Dimensional Euclidean Space ◽  
Generalized Derivatives ◽  
Bounded Region ◽  
Complete Continuity ◽  
Completely Continuous ◽  
Natural Embedding ◽  
Derivatives Of

Let H0m(Ω) denote the so-called Sobolev space consisting of functions denned on a region Ω in n-dimensional Euclidean space, which together with their generalized derivatives of all orders ⩽m belong to , and which vanish in a certain sense on the boundary ∂Ω. (Precise definitions are given in the next section.) For each pair m, k of non-negative integers the inclusion H0m+k(Ω) ⊂ H0m(Ω) defines a natural “embedding” map. For the case of a bounded region Ω it is well known that these maps are completely continuous, and even, for sufficiently large k, of Hilbert-Schmidt type. We have discussed complete continuity in the case of unbounded regions in an earlier paper; here we consider conditions on Ω which imply the Hilbert-Schmidt property for embeddings.

Some Imbedding Theorems for Sobolev Spaces

1971 ◽  
Vol 23 (3) ◽  
pp. 517-530 ◽  
Author(s):  
R. A. Adams ◽  
John Fournier
Keyword(s):  
Banach Space ◽  
Positive Integer ◽  
Euclidean Space ◽  
Sobolev Space ◽  
Dimensional Euclidean Space ◽  
Dense Subspace ◽  
Bounded Subset ◽  
Partial Derivatives ◽  
Image Position ◽  
Derivatives Of

We shall be concerned throughout this paper with the Sobolev space Wm,p(G) and the existence and compactness (or lack of it) of its imbeddings (i.e. continuous inclusions) into various LP spaces over G, where G is an open, not necessarily bounded subset of n-dimensional Euclidean space En. For each positive integer m and each real p ≧ 1 the space Wm,p(G) consists of all u in LP(G) whose distributional partial derivatives of all orders up to and including m are also in LP(G). With respect to the norm1.1Wm,p(G) is a Banach space. It has been shown by Meyers and Serrin [9] that the set of functions in Cm(G) which, together with their partial derivatives of orders up to and including m, are in LP(G) forms a dense subspace of Wm,p(G).

scholarly journals Critical probability of percolation over bounded region in N-dimensional Euclidean space

2016 ◽  
Vol 2016 (3) ◽  
pp. 033306 ◽  
Author(s):  
Emmanuel Roubin ◽  
Jean-Baptiste Colliat
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Bounded Region ◽  
Critical Probability

scholarly journals Sobolev boundedness and continuity for commutators of the local Hardy–Littlewood maximal function

2021 ◽  
Vol 47 (1) ◽  
pp. 203-235
Author(s):  
Feng Liu ◽  
Qingying Xue ◽  
Kôzô Yabuta
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Maximal Function ◽  
Pointwise Estimates ◽  
Boundary Values ◽  
First Order ◽  
Littlewood Maximal Function ◽  
Weak Derivatives ◽  
Derivatives Of

Let \(\Omega\) be a subdomain in \(\mathbb{R}^n\) and \(M_\Omega\) be the local Hardy-Littlewood maximal function. In this paper, we show that both the commutator and the maximal commutator of \(M_\Omega\) are bounded and continuous from the first order Sobolev spaces \(W^{1,p_1}(\Omega)\) to \(W^{1,p}(\Omega)\) provided that \(b\in W^{1,p_2}(\Omega)\), \(1<p_1,p_2,p<\infty\) and \(1/p=1/p_1+1/p_2\). These are done by establishing several new pointwise estimates for the weak derivatives of the above commutators. As applications, the bounds of these operators on the Sobolev space with zero boundary values are obtained.

scholarly journals Weighted composition operators on Orlicz-Sobolev spaces

2007 ◽  
Vol 83 (3) ◽  
pp. 327-334
Author(s):  
Subhash C. Arora ◽  
Gopal Datt ◽  
Satish Verma
Keyword(s):  
Euclidean Space ◽  
Sobolev Space ◽  
Composition Operators ◽  
Weighted Composition Operator ◽  
Weighted Composition Operators ◽  
Sufficient Condition ◽  
First Order ◽  
Singular Transformation ◽  
Weighted Composition ◽  
Derivatives Of

AbstractFor an open subset Ω of the Euclidean space Rn, a measurable non-singular transformation T: Ω → Ω and a real-valued measurable function u on Rn, we study the weighted composition operator uCτ: f ↦ u · (f º T) on the Orlicz-Sobolev space W1·Ψ (Ω) consxsisting of those functions of the Orlicz space LΨ (Ω) whose distributional derivatives of the first order belong to LΨ (Ω). We also discuss a sufficient condition under which uCτ is compact.

Properties of Equivalent Capacities

1971 ◽  
Vol 14 (1) ◽  
pp. 5-11 ◽  
Author(s):  
R. A. Adams
Keyword(s):  
Partial Differential Equations ◽  
Euclidean Space ◽  
Sobolev Space ◽  
Differential Equations ◽  
Spectral Theory ◽  
Sobolev Spaces ◽  
Elliptic Partial Differential Equations ◽  
Partial Differential ◽  
Closed Set ◽  
Shed Light

Various definitions of capacity of a subset of a domain in Euclidean space have been used in recent times to shed light on the solvability and spectral theory of elliptic partial differential equations and to establish properties of the Sobolev spaces in which these equations are studied. In this paper we consider two definitions of the capacity of a closed set E in a domain G. One of these capacities measures, roughly speaking, the amount by which the set of function in C∞(G) which vanish near E fails to be dense in the Sobolev space Wm, p(G).

On Vitali's Theorem for Groups of Motions of Euclidean Space

1999 ◽  
Vol 6 (4) ◽  
pp. 323-334
Author(s):  
A. Kharazishvili
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Finite Dimensional

Abstract We give a characterization of all those groups of isometric transformations of a finite-dimensional Euclidean space, for which an analogue of the classical Vitali theorem [Sul problema della misura dei gruppi di punti di una retta, 1905] holds true. This characterization is formulated in purely geometrical terms.

Some properties of Wigner coefficients and hyperspherical harmonics

1956 ◽  
Vol 52 (3) ◽  
pp. 424-430 ◽  
Author(s):  
A. P. Stone
Keyword(s):  
Angular Momentum ◽  
Euclidean Space ◽  
Rotation Group ◽  
Dimensional Euclidean Space ◽  
Hyperspherical Harmonics ◽  
Shift Operators ◽  
Analytical Relations

ABSTRACTGeneral shift operators for angular momentum are obtained and applied to find closed expressions for some Wigner coefficients occurring in a transformation between two equivalent representations of the four-dimensional rotation group. The transformation gives rise to analytical relations between hyperspherical harmonics in a four-dimensional Euclidean space.

On inequalities of the Tchebychev type

1963 ◽  
Vol 59 (1) ◽  
pp. 135-146 ◽  
Author(s):  
J. F. C. Kingman
Keyword(s):  
Probability Theory ◽  
Euclidean Space ◽  
Random Variable ◽  
Dimensional Euclidean Space ◽  
Real Random Variable ◽  
Finite Dimensional

1. A type of problem which frequently occurs in probability theory and statistics can be formulated in the following way. We are given real-valued functions f(x), gi(x) (i = 1, 2, …, k) on a space (typically finite-dimensional Euclidean space). Then the problem is to set bounds for Ef(X), where X is a random variable taking values in , about which all we know is the values of Egi(X). For example, we might wish to set bounds for P(X > a), where X is a real random variable with some of its moments given.

Curve following in n—dimensional Euclidean space

SIMULATION ◽  
1973 ◽  
Vol 21 (5) ◽  
pp. 145-149 ◽  
Author(s):  
John Rees Jones
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space
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