scholarly journals Poincaré Inequality for Weighted First Order Sobolev Spaces on Loop Spaces

2001 ◽  
Vol 185 (2) ◽  
pp. 527-563 ◽  
Author(s):  
Fuzhou Gong ◽  
Michael Röckner ◽  
Wu Liming
Keyword(s):  
Sobolev Spaces ◽  
Poincaré Inequality ◽  
Loop Spaces ◽  
Poincare Inequality ◽  
First Order

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.

scholarly journals A Poincaré inequality on loop spaces

2010 ◽  
Vol 259 (6) ◽  
pp. 1421-1442 ◽  
Author(s):  
Xin Chen ◽  
Xue-Mei Li ◽  
Bo Wu
Keyword(s):  
Poincaré Inequality ◽  
Loop Spaces ◽  
Poincare Inequality

scholarly journals Poincaré inequality and Hajłasz–Sobolev spaces on nested fractals

2013 ◽  
Vol 218 (1) ◽  
pp. 1-26 ◽  
Author(s):  
Katarzyna Pietruska-Pałuba ◽  
Andrzej Stós
Keyword(s):  
Sobolev Spaces ◽  
Poincaré Inequality ◽  
Poincare Inequality

Extension Theorems on Weighted Sobolev Spaces and Some Applications

2006 ◽  
Vol 58 (3) ◽  
pp. 492-528 ◽  
Author(s):  
Seng-Kee Chua
Keyword(s):  
Sobolev Spaces ◽  
Weighted Sobolev Spaces ◽  
Poincaré Inequality ◽  
Poincare Inequality ◽  
Extension Theorems ◽  
Doubling Weight ◽  
Interpolation Inequalities

AbstractWe extend the extension theorems to weighted Sobolev spaces on (ε, δ) domains with doubling weight w that satisfies a Poincaré inequality and such that w–1/p is locally Lp′. We also make use of the main theorem to improve weighted Sobolev interpolation inequalities.

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 Sobolev spaces and hyperbolic fillings

2018 ◽  
Vol 2018 (737) ◽  
pp. 161-187 ◽  
Author(s):  
Mario Bonk ◽  
Eero Saksman
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Measure Space ◽  
Weak Type ◽  
Poincaré Inequality ◽  
Metric Measure Space ◽  
Poincare Inequality ◽  
Fractional Sobolev Space ◽  
Gradient Condition

AbstractLetZbe an AhlforsQ-regular compact metric measure space, where{Q>0}. For{p>1}we introduce a new (fractional) Sobolev space{A^{p}(Z)}consisting of functions whose extensions to the hyperbolic filling ofZsatisfy a weak-type gradient condition. IfZsupports aQ-Poincaré inequality with{Q>1}, then{A^{Q}(Z)}coincides with the familiar (homogeneous) Hajłasz–Sobolev space.

On Weighted Sobolev Spaces

1996 ◽  
Vol 48 (3) ◽  
pp. 527-541 ◽  
Author(s):  
Seng-Kee Chua
Keyword(s):  
Sobolev Spaces ◽  
Weighted Sobolev Spaces ◽  
Poincaré Inequality ◽  
Poincare Inequality ◽  
Extension Theorems ◽  
Doubling Weight ◽  
Weighted Poincaré Inequality

AbstractWe study density and extension problems for weighted Sobolev spaces on bounded (ε, δ) domains𝓓when a doubling weight w satisfies the weighted Poincaré inequality on cubes near the boundary of 𝓓 and when it is in the MuckenhouptApclass locally in 𝓓. Moreover, when the weightswi(x) are of the form dist(x,Mi)αi,αi∈ ℝ,Mi⊂ 𝓓that are doubling, we are able to obtain some extension theorems on (ε, ∞) domains.

scholarly journals Poincaré inequalities for Sobolev spaces with matrix-valued weights and applications to degenerate partial differential equations

2018 ◽  
Vol 149 (1) ◽  
pp. 61-100 ◽  
Author(s):  
Dario D. Monticelli ◽  
Kevin R. Payne ◽  
Fabio Punzo
Keyword(s):  
Dirichlet Problem ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Sobolev Spaces ◽  
Poincaré Inequality ◽  
Second Order ◽  
Poincare Inequality ◽  
Partial Differential ◽  
The Matrix ◽  
Degenerate Elliptic

For bounded domains Ω, we prove that the Lp-norm of a regular function with compact support is controlled by weighted Lp-norms of its gradient, where the weight belongs to a class of symmetric non-negative definite matrix-valued functions. The class of weights is defined by regularity assumptions and structural conditions on the degeneracy set, where the determinant vanishes. In particular, the weight A is assumed to have rank at least 1 when restricted to the normal bundle of the degeneracy set S. This generalization of the classical Poincaré inequality is then applied to develop a robust theory of first-order Lp-based Sobolev spaces with matrix-valued weight A. The Poincaré inequality and these Sobolev spaces are then applied to produce various results on existence, uniqueness and qualitative properties of weak solutions to boundary-value problems for degenerate elliptic, degenerate parabolic and degenerate hyperbolic partial differential equations (PDEs) of second order written in divergence form, where A is calibrated to the matrix of coefficients of the second-order spatial derivatives. The notion of weak solution is variational: the spatial states belong to the matrix-weighted Sobolev spaces with p = 2. For the degenerate elliptic PDEs, the Dirichlet problem is treated by the use of the Poincaré inequality and Lax–Milgram theorem, while the treatment of the Cauchy–Dirichlet problem for the degenerate evolution equations relies only on the Poincaré inequality and the parabolic and hyperbolic counterparts of the Lax–Milgram theorem.

scholarly journals Functional Inequalities for Two-Level Concentration

2021 ◽  
Author(s):  
Franck Barthe ◽  
Michał Strzelecki
Keyword(s):  
Sobolev Inequality ◽  
Poincaré Inequality ◽  
Logarithmic Sobolev Inequality ◽  
Probability Measures ◽  
Exponential Rate ◽  
Functional Inequalities ◽  
Concentration Inequality ◽  
Poincare Inequality ◽  
Free Concentration ◽  
Concentration Property

AbstractProbability measures satisfying a Poincaré inequality are known to enjoy a dimension-free concentration inequality with exponential rate. A celebrated result of Bobkov and Ledoux shows that a Poincaré inequality automatically implies a modified logarithmic Sobolev inequality. As a consequence the Poincaré inequality ensures a stronger dimension-free concentration property, known as two-level concentration. We show that a similar phenomenon occurs for the Latała–Oleszkiewicz inequalities, which were devised to uncover dimension-free concentration with rate between exponential and Gaussian. Motivated by the search for counterexamples to related questions, we also develop analytic techniques to study functional inequalities for probability measures on the line with wild potentials.

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