Characterizations of Sobolev spaces in Euclidean spaces and Heisenberg groups

2013 ◽  
Vol 28 (4) ◽  
pp. 531-547 ◽  
Author(s):  
Xiao-yue Cui ◽  
Nguyen Lam ◽  
Guo-zhen Lu
Keyword(s):  
Sobolev Spaces ◽  
Heisenberg Groups ◽  
Euclidean Spaces

scholarly journals Regularities of Time-Fractional Derivatives of Semigroups Related to Schrodinger Operators with Application to Hardy-Sobolev Spaces on Heisenberg Groups

2020 ◽  
Vol 2020 ◽  
pp. 1-21
Author(s):  
Zhiyong Wang ◽  
Chuanhong Sun ◽  
Pengtao Li
Keyword(s):  
Sobolev Spaces ◽  
Equivalence Relation ◽  
Fractional Derivatives ◽  
Sobolev Type ◽  
Square Function ◽  
Heisenberg Groups ◽  
Square Functions ◽  
Regularity Properties ◽  
Reverse Hölder ◽  
Derivatives Of

In this paper, assume that L=−Δℍn+V is a Schrödinger operator on the Heisenberg group ℍn, where the nonnegative potential V belongs to the reverse Hölder class BQ/2. By the aid of the subordinate formula, we investigate the regularity properties of the time-fractional derivatives of semigroups e−tLt>0 and e−tLt>0, respectively. As applications, using fractional square functions, we characterize the Hardy-Sobolev type space HL1,αℍn associated with L. Moreover, the fractional square function characterizations indicate an equivalence relation of two classes of Hardy-Sobolev spaces related with L.

Best Constants for Moser-Trudinger Inequalities on High Dimensional Hyperbolic Spaces

2013 ◽  
Vol 13 (4) ◽  
Author(s):  
Guozhen Lu ◽  
Hanli Tang
Keyword(s):  
Riemannian Manifolds ◽  
Best Constants ◽  
Hyperbolic Spaces ◽  
High Dimensional ◽  
Sharp Constant ◽  
Heisenberg Groups ◽  
Euclidean Spaces ◽  
Sharp Constants ◽  
Trudinger Inequality

AbstractThough there have been extensive works on best constants for Moser-Trudinger inequalities in Euclidean spaces, Heisenberg groups or compact Riemannian manifolds, much less is known for sharp constants for the Moser-Trudinger inequalities on hyperbolic spaces. Earlier works only include the sharp constant for the Moser-Trudinger inequality on the twodimensional hyperbolic disc. In this paper, we establish best constants for several types of Moser-Trudinger inequalities on high dimensional hyperbolic spaces ℍ

scholarly journals Singular integrals on self-similar sets and removability for Lipschitz harmonic functions in Heisenberg groups

2014 ◽  
Vol 2014 (691) ◽  
Author(s):  
Vasilis Chousionis ◽  
Pertti Mattila
Keyword(s):  
Harmonic Functions ◽  
Singular Integrals ◽  
Measure Zero ◽  
General Setting ◽  
Heisenberg Groups ◽  
Euclidean Spaces ◽  
Self Similar ◽  
Metric Groups

Abstract.In this paper we study singular integrals on small (that is, measure zero and lower than full dimensional) subsets of metric groups. The main examples of the groups we have in mind are Euclidean spaces and Heisenberg groups. In addition to obtaining results in a very general setting, the purpose of this work is twofold; we shall extend some results in Euclidean spaces to more general kernels than previously considered, and we shall obtain in Heisenberg groups some applications to harmonic (in the Heisenberg sense) functions of some results known earlier in Euclidean spaces.

CONSTRUCTION OF GENERALIZED UNCERTAINTY PRINCIPLES AND WAVELETS IN ANISOTROPIC SOBOLEV SPACES

2005 ◽  
Vol 03 (02) ◽  
pp. 189-209 ◽  
Author(s):  
GERD TESCHKE
Keyword(s):  
Sobolev Spaces ◽  
Quadratic Forms ◽  
Uncertainty Relations ◽  
Uncertainty Principles ◽  
Heisenberg Groups ◽  
Anisotropic Sobolev Spaces

This paper deals with the construction of generalized uncertainty relations. Starting with a two-operator parabola ansatz, we derive a new set of uncertainties by extending the parabola ansatz to quadratic forms. This setting can be applied to the computation of uncertainties where more than two operators are involved. For affine Weyl–Heisenberg groups, some examples are explored.

Sobolev Spaces

2018 ◽  
Author(s):  
D. E. Edmunds ◽  
W. D. Evans
Keyword(s):  
Sobolev Spaces ◽  
Embedding Theorems ◽  
Approximation Numbers ◽  
Selection Of

This chapter presents a selection of some of the most important results in the theory of Sobolev spacesn. Special emphasis is placed on embedding theorems and the question as to whether an embedding map is compact or not. Some results concerning the k-set contraction nature of certain embedding maps are given, for both bounded and unbounded space domains: also the approximation numbers of embedding maps are estimated and these estimates used to classify the embeddings.

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