Sobolev Spaces on Metric Spaces

2001 ◽  
pp. 34-42
Author(s):  
Juha Heinonen
Keyword(s):  
Sobolev Spaces ◽  
Metric Spaces

Weighted higher order exponential type inequalities in metric spaces and applications

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Huiju Wang ◽  
Pengcheng Niu
Keyword(s):  
Homogeneous Space ◽  
Lie Groups ◽  
Sobolev Spaces ◽  
Exponential Type ◽  
Metric Spaces ◽  
Type Inequality ◽  
Higher Order ◽  
Geodesic Space ◽  
Stratified Lie Groups ◽  
Weighted Case

AbstractIn this paper, we establish weighted higher order exponential type inequalities in the geodesic space {({X,d,\mu})} by proposing an abstract higher order Poincaré inequality. These are also new in the non-weighted case. As applications, we obtain a weighted Trudinger’s theorem in the geodesic setting and weighted higher order exponential type estimates for functions in Folland–Stein type Sobolev spaces defined on stratified Lie groups. A higher order exponential type inequality in a connected homogeneous space is also given.

scholarly journals ON THE DEFINITIONS OF SOBOLEV AND BV SPACES INTO SINGULAR SPACES AND THE TRACE PROBLEM

2007 ◽  
Vol 09 (04) ◽  
pp. 473-513 ◽  
Author(s):  
DAVID CHIRON
Keyword(s):  
Sobolev Spaces ◽  
Metric Spaces ◽  
The Other ◽  
Dirichlet Energy ◽  
Singular Spaces ◽  
Other Hand ◽  
The One

The purpose of this paper is to relate two notions of Sobolev and BV spaces into metric spaces, due to Korevaar and Schoen on the one hand, and Jost on the other hand. We prove that these two notions coincide and define the same p-energies. We review also other definitions, due to Ambrosio (for BV maps into metric spaces), Reshetnyak and finally to the notion of Newtonian–Sobolev spaces. These last approaches define the same Sobolev (or BV) spaces, but with a different energy, which does not extend the standard Dirichlet energy. We also prove a characterization of Sobolev spaces in the spirit of Bourgain, Brezis and Mironescu in terms of "limit" of the space Ws,p as s → 1, 0 < s < 1, and finally following the approach proposed by Nguyen. We also establish the [Formula: see text] regularity of traces of maps in Ws,p (0 < s ≤ 1 < sp).

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.

Variable Hajłasz-Sobolev spaces on compact metric spaces

2017 ◽  
Vol 67 (1) ◽  
pp. 199-208
Author(s):  
Michał Gaczkowski ◽  
Przemysław Górka
Keyword(s):  
Sobolev Spaces ◽  
Variable Exponent ◽  
Metric Spaces ◽  
Hölder Continuity ◽  
Sobolev Type ◽  
Holder Continuity ◽  
Study Variable ◽  
Compact Metric Spaces ◽  
Variable Exponent Sobolev Spaces

Abstract We study variable exponent Sobolev spaces on compact metric spaces. Without the assumption of log-Hölder continuity of the exponent, the compact Sobolev-type embeddings theorems for these spaces are shown.

scholarly journals Isoperimetry and symmetrization for Sobolev spaces on metric spaces

2009 ◽  
Vol 347 (11-12) ◽  
pp. 627-630 ◽  
Author(s):  
Joaquim Martín ◽  
Mario Milman
Keyword(s):  
Sobolev Spaces ◽  
Metric Spaces

scholarly journals Real interpolation of Sobolev spaces

2009 ◽  
Vol 105 (2) ◽  
pp. 235 ◽  
Author(s):  
Nadine Badr
Keyword(s):  
Sobolev Spaces ◽  
Metric Spaces ◽  
Interpolation Space ◽  
Real Interpolation

We prove that $W^{1}_{p}$ is a real interpolation space between $W^{1}_{p_{1}}$ and $W^{1}_{p_{2}}$ for $p>q_{0}$ and $1\leq p_{1}<p<p_{2}\leq \infty$ on some classes of manifolds and general metric spaces, where $q_{0}$ depends on our hypotheses.

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