This paper is devoted to the study of fractional(q, p)-Sobolev-Poincar′e inequalities in irregular domains. In particular, the author establishes(essentially) sharp fractional(q, p)-Sobolev-Poincar′e inequalities in...This paper is devoted to the study of fractional(q, p)-Sobolev-Poincar′e inequalities in irregular domains. In particular, the author establishes(essentially) sharp fractional(q, p)-Sobolev-Poincar′e inequalities in s-John domains and in domains satisfying the quasihyperbolic boundary conditions. When the order of the fractional derivative tends to 1, our results tend to the results for the usual derivatives. Furthermore, the author verifies that those domains which support the fractional(q, p)-Sobolev-Poincar′e inequalities together with a separation property are s-diam John domains for certain s, depending only on the associated data. An inaccurate statement in [Buckley, S. and Koskela, P.,Sobolev-Poincar′e implies John, Math. Res. Lett., 2(5), 1995, 577–593] is also pointed out.展开更多
The weighted Poincare inequalities in weighted Sobolev spaces are discussed, and the necessary and sufficient conditions for them to hold are given. That is, the Poincare inequalities hold if, and only if, the ball me...The weighted Poincare inequalities in weighted Sobolev spaces are discussed, and the necessary and sufficient conditions for them to hold are given. That is, the Poincare inequalities hold if, and only if, the ball measure of non-compactness of the natural embedding of weighted Sobolev spaces is less than 1.展开更多
Recently, many new features of Sobolev spaces W k,p ？RN ？ were studied in [4-6, 32]. This paper is devoted to giving a brief review of some known characterizations of Sobolev spaces in Euclidean spaces and describin...Recently, many new features of Sobolev spaces W k,p ？RN ？ were studied in [4-6, 32]. This paper is devoted to giving a brief review of some known characterizations of Sobolev spaces in Euclidean spaces and describing our recent study of new characterizations of Sobolev spaces on both Heisenberg groups and Euclidean spaces obtained in [12] and [13] and outlining their proofs. Our results extend those characterizations of first order Sobolev spaces in [32] to the Heisenberg group setting. Moreover, our theorems also provide diff erent characterizations for the second order Sobolev spaces in Euclidean spaces from those in [4, 5].展开更多
In this paper, we prove Poincaré and Sobolev inequalities for differential forms in L^1(R^n). The singular integral estimates that it is possible to use for L^p, p > 1, are replaced here with inequalities whic...In this paper, we prove Poincaré and Sobolev inequalities for differential forms in L^1(R^n). The singular integral estimates that it is possible to use for L^p, p > 1, are replaced here with inequalities which go back to Bourgain and Brezis(2007).展开更多
Motivated from the study on logarithmic Sobolev. Nash and other functional inequalities, the variational formulas for Poincare inequalities are extended to a large class of Banach (Orlicz) spaces of functions on the l...Motivated from the study on logarithmic Sobolev. Nash and other functional inequalities, the variational formulas for Poincare inequalities are extended to a large class of Banach (Orlicz) spaces of functions on the line. Explicit criteria for the inequalities to hold and explicit estimates for the optimal constants in the inequalities are presented. As a typical application, the logarithmic Sobolev constant is carefully examined.展开更多
Weak log-Sobolev and Lp weak Poincare inequalities for general symmetric forms are investigated by using newly defined Cheeger＇s isoperimetric constants. Some concrete examples of ergodic birth-death processes are al...Weak log-Sobolev and Lp weak Poincare inequalities for general symmetric forms are investigated by using newly defined Cheeger＇s isoperimetric constants. Some concrete examples of ergodic birth-death processes are also presented to illustrate the results.展开更多
文摘This paper is devoted to the study of fractional(q, p)-Sobolev-Poincar′e inequalities in irregular domains. In particular, the author establishes(essentially) sharp fractional(q, p)-Sobolev-Poincar′e inequalities in s-John domains and in domains satisfying the quasihyperbolic boundary conditions. When the order of the fractional derivative tends to 1, our results tend to the results for the usual derivatives. Furthermore, the author verifies that those domains which support the fractional(q, p)-Sobolev-Poincar′e inequalities together with a separation property are s-diam John domains for certain s, depending only on the associated data. An inaccurate statement in [Buckley, S. and Koskela, P.,Sobolev-Poincar′e implies John, Math. Res. Lett., 2(5), 1995, 577–593] is also pointed out.
基金Project supported by the National Natural Science Foundation of China(Nos.10261004 and 10461006)the Visiting Scholar Foundation of Key Laboratory of University and the Natural Science Foundation of the Inner Mongolia Autonomous Region of China(No.200408020104)
文摘The weighted Poincare inequalities in weighted Sobolev spaces are discussed, and the necessary and sufficient conditions for them to hold are given. That is, the Poincare inequalities hold if, and only if, the ball measure of non-compactness of the natural embedding of weighted Sobolev spaces is less than 1.
文摘Recently, many new features of Sobolev spaces W k,p ？RN ？ were studied in [4-6, 32]. This paper is devoted to giving a brief review of some known characterizations of Sobolev spaces in Euclidean spaces and describing our recent study of new characterizations of Sobolev spaces on both Heisenberg groups and Euclidean spaces obtained in [12] and [13] and outlining their proofs. Our results extend those characterizations of first order Sobolev spaces in [32] to the Heisenberg group setting. Moreover, our theorems also provide diff erent characterizations for the second order Sobolev spaces in Euclidean spaces from those in [4, 5].
基金Supported by Institution of Higher Education Scientific Research Project in Ningxia(NGY2017011)Natural Science Foundations of Ningxia(NZ15055)+1 种基金Natural Science Foundations of China(1156105511461053)
基金supported by Funds for Selected Research Topics from the University of BolognaMAnET Marie Curie Initial Training Network+3 种基金GNAMPA (Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM (Istituto Nazionale di Alta Matematica "F. Severi"), ItalyPRIN (Progetti di Ricerca di Rilevante Interesse Nazionale) of the MIUR (Ministero dell’Istruzione dell’Università e della Ricerca), Italysupported by MAnET Marie Curie Initial Training Network, Agence Nationale de la Recherche (Grant Nos. ANR-10-BLAN 116-01 GGAA and ANR-15-CE40-0018 SRGI)the hospitality of Isaac Newton Institute, of EPSRC (Engineering and Physical Sciences Research Council) (Grant No. EP/K032208/1) and Simons Foundation
文摘In this paper, we prove Poincaré and Sobolev inequalities for differential forms in L^1(R^n). The singular integral estimates that it is possible to use for L^p, p > 1, are replaced here with inequalities which go back to Bourgain and Brezis(2007).
基金Research supported in part by NSFC (No. 10121101)973 ProjectRFDP
文摘Motivated from the study on logarithmic Sobolev. Nash and other functional inequalities, the variational formulas for Poincare inequalities are extended to a large class of Banach (Orlicz) spaces of functions on the line. Explicit criteria for the inequalities to hold and explicit estimates for the optimal constants in the inequalities are presented. As a typical application, the logarithmic Sobolev constant is carefully examined.
文摘Weak log-Sobolev and Lp weak Poincare inequalities for general symmetric forms are investigated by using newly defined Cheeger＇s isoperimetric constants. Some concrete examples of ergodic birth-death processes are also presented to illustrate the results.
