Heat kernel and Lipschitz–Besov spaces

2015 ◽  
Vol 27 (6) ◽  
Author(s):  
Alexander Grigor'yan ◽  
Liguang Liu
Keyword(s):  
Heat Kernel ◽  
Besov Spaces ◽  
Measure Space ◽  
Metric Measure Space

AbstractOn a metric measure space (

scholarly journals Embeddings between Triebel-Lizorkin Spaces on Metric Spaces Associated with Operators

2020 ◽  
Vol 8 (1) ◽  
pp. 418-429
Author(s):  
Athanasios G. Georgiadis ◽  
George Kyriazis
Keyword(s):  
Heat Kernel ◽  
Riemannian Manifolds ◽  
Besov Spaces ◽  
Measure Space ◽  
General Framework ◽  
Metric Spaces ◽  
Adjoint Operator ◽  
Embedding Theorems ◽  
Euclidean Case ◽  
Volume Property

Abstract We consider the general framework of a metric measure space satisfying the doubling volume property, associated with a non-negative self-adjoint operator, whose heat kernel enjoys standard Gaussian localization. We prove embedding theorems between Triebel-Lizorkin spaces associated with operators. Embeddings for non-classical Triebel-Lizorkin and (both classical and non-classical) Besov spaces are proved as well. Our result generalize the Euclidean case and are new for many settings of independent interest such as the ball, the interval and Riemannian manifolds.

MAXIMAL HARDY SPACES ASSOCIATED TO NONNEGATIVE SELF-ADJOINT OPERATORS

2014 ◽  
Vol 91 (2) ◽  
pp. 286-302
Author(s):  
GUORONG HU
Keyword(s):  
Heat Kernel ◽  
Hardy Spaces ◽  
Upper Bound ◽  
Measure Space ◽  
Adjoint Operator ◽  
Maximal Functions ◽  
Metric Measure Space ◽  
Holder Continuity ◽  
Upper Bound Estimate ◽  
Adjoint Operators

AbstractLet $(X,d,{\it\mu})$ be a metric measure space satisfying the doubling, reverse doubling and noncollapsing conditions. Let $\mathscr{L}$ be a nonnegative self-adjoint operator on $L^{2}(X,d{\it\mu})$ satisfying a pointwise Gaussian upper bound estimate and Hölder continuity for its heat kernel. In this paper, we introduce the Hardy spaces $H_{\mathscr{L}}^{p}(X)$, $0<p\leq 1$, associated to $\mathscr{L}$ in terms of grand maximal functions and show that these spaces are equivalently characterised by radial and nontangential maximal functions.

scholarly journals Diffusion, Optimal Transport and Ricci Curvature for Metric Measure Space

2017 ◽  
Vol 2017-3 (103) ◽  
pp. 19-28
Author(s):  
Luigi Ambrosio ◽  
Nicola Gigli ◽  
Giuseppe Savaré
Keyword(s):  
Ricci Curvature ◽  
Measure Space ◽  
Optimal Transport ◽  
Metric Measure Space

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.

A note on fractional integral operators defined by weights and non-doubling measures

2010 ◽  
Vol 106 (2) ◽  
pp. 283 ◽  
Author(s):  
Oscar Blasco ◽  
Vicente Casanova ◽  
Joaquín Motos
Keyword(s):  
Measure Space ◽  
Fractional Integral ◽  
Integral Operators ◽  
Metric Measure Space ◽  
Integral Type ◽  
Lebesgue Spaces ◽  
Lipschitz Spaces ◽  
Fractional Integral Operators ◽  
Doubling Measures

Given a metric measure space $(X,d,\mu)$, a weight $w$ defined on $(0,\infty)$ and a kernel $k_w(x,y)$ satisfying the standard fractional integral type estimates, we study the boundedness of the operators $K_w f(x)=\int_X k_w(x,y)f(y)\,d\mu(y)$ and $\tilde K_w f(x)=\int_X (k_w(x,y)-k_w(x_0,y))f(y)\,d\mu(y)$ on Lebesgue spaces $L^p(\mu)$ and generalized Lipschitz spaces $\mathrm{Lip}_\phi$, respectively, for certain range of the parameters depending on the $n$-dimension of $\mu$ and some indices associated to the weight $w$.

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