## Probabilistic characterisation of Besov-Lipschitz spaces on metric measure spaces (2008)

### BibTeX

@MISC{Pietruska-Pałuba08probabilisticcharacterisation,

author = {Katarzyna Pietruska-Pałuba},

title = {Probabilistic characterisation of Besov-Lipschitz spaces on metric measure spaces },

year = {2008}

}

### OpenURL

### Abstract

We give a probabilistic characterisation of the Besov-Lipschitz spaces Lip(α, p, q)(X) on domains which support a Markovian kernel with appropriate exponential bounds. This extends former results of [11, 14, 15, 7] which were valid for α = dw 2, p = 2, q = ∞, where dw is the walk dimension of the space X.

### Citations

45 | Brownian motion and harmonic analysis on Sierpinski carpets
- Barlow, Bass
- 1999
(Show Context)
Citation Context ...ρ(x,y) t1/dw )dw/(dw−1)), see [12], • the Brownian motion on p.c.f. self similar sets and on the Sierpiński carpets, where we have an estimate analogous to that on simple nested fractals (see [9] and =-=[2]-=-). 3 The main theorem In a series of papers ([11, 14, 15, 7]) it has been proven that the domain of the Dirichlet form associated with the Markovian kernel satisfying (1)-(6) is equal to the space Lip... |

18 |
Transition density estimates for diffusion processes on post critically finite self-similar fractals
- Hambly, Kumagai
- 1999
(Show Context)
Citation Context ... exp(−c(ρ(x,y) t1/dw )dw/(dw−1)), see [12], • the Brownian motion on p.c.f. self similar sets and on the Sierpiński carpets, where we have an estimate analogous to that on simple nested fractals (see =-=[9]-=- and [2]). 3 The main theorem In a series of papers ([11, 14, 15, 7]) it has been proven that the domain of the Dirichlet form associated with the Markovian kernel satisfying (1)-(6) is equal to the s... |

12 | Heat kernels on metric-measure spaces and an application to semi-linear elliptic equations
- Grigor’yan, Hu, et al.
(Show Context)
Citation Context ... We give a probabilistic characterisation of the Besov-Lipschitz spaces Lip(α, p, q)(X) on domains which support a Markovian kernel with appropriate exponential bounds. This extends former results of =-=[11, 14, 15, 7]-=- which were valid for α = dw 2 , p = 2, q = ∞, where dw is the walk dimension of the space X. 1 Introduction There are several definitions of Besov-Lipschitz spaces on measure spaces. In this paper we... |

9 |
Heat kernels and function theory on metric measure spaces, in: “Heat kernels and analysis on manifolds, graphs, and metric spaces
- Grigor’yan
(Show Context)
Citation Context ... paper dealt with the Sierpiński gasket embedded in R d only, but did not really use the embedding itself, and so this particular definition can be extended to general metric measure spaces (see e.g. =-=[6]-=-, [13]). The most convenient to analyse are those spaces on which there exists a complete, symmetric Markovian kernel with appropriate exponential bounds. There are several results concerning such spa... |

4 |
Symmetric stable processes on d-sets
- Stós
- 2000
(Show Context)
Citation Context ... 46E30, secondary 60J35 † Supported by a KBN grant no. 1-PO3A-008-29. 1so it can be described using the kernel p(·, ·, ·). Also, the spaces Lip(α,2,2) allow for a probabilistic characterisation (see =-=[18]-=-). In present work we extend these results and provide a characterisation of the spaces Lip(α,p,q)(X) in terms of the Markovian kernel whose existence we are assuming, for α > 0, p,q ≥ 1 (nox excludin... |

3 |
Bessel potentials, and Lipschitz spaces
- Flett, Temperatures
(Show Context)
Citation Context ...e class of metric measure spaces, were also introduced bu Hu and Zähle – in a different way – in their paper [10]. The way they are defined owes to the classical characterisation of Besov spaces from =-=[5]-=-, [17]. Those spaces will be denoted by B p,q β with B p,q β (X); We will see that our characterisation of Lip(α,p,q)(X) is consistent (X) for some range of parameters (see Section 4.2). In the case o... |

2 |
Diffusion on fractals, Lectures on Probability and Statistics, Ecole d’Eté de Prob. de St
- Barlow
- 1998
(Show Context)
Citation Context ...re are several results concerning such spaces, see e.g. [7], [15], [16], [10]. The existence of a Markovian kernel on X of this type is equivalent to the existence of a fractional diffusion on X (see =-=[1]-=- for the definition). Its generator, often called ‘the Laplacian’ on a general metric space, serves as a substitute for the bona fide differentiation operator, even though the differential itself is n... |

2 |
On function spaces related to fractional diffusions on dsets
- Pietruska-Paluba
(Show Context)
Citation Context ... We give a probabilistic characterisation of the Besov-Lipschitz spaces Lip(α, p, q)(X) on domains which support a Markovian kernel with appropriate exponential bounds. This extends former results of =-=[11, 14, 15, 7]-=- which were valid for α = dw 2 , p = 2, q = ∞, where dw is the walk dimension of the space X. 1 Introduction There are several definitions of Besov-Lipschitz spaces on measure spaces. In this paper we... |

2 |
Heat kernels on metric spaces and a characterisation of constant functions
- PIETRUSKA-PA̷LUBA
- 2004
(Show Context)
Citation Context ...e spaces Lip(α,p,p)(X). It is clear that for f ∈ Lp (X) we have ‖f‖α,p,p ≍ ∫ X ∫ X |f(x) − f(y)| p ρ(x,y) d+pα dµ(x)dµ(y) + ‖f‖p Lp (3.12) (when diam X < ∞, then the term ‖f‖ p Lp can be omitted). In =-=[16]-=- it has been proven that the finiteness of the integral in (3.12), when p = 2 and α ≥ dw dw 2 implies that f = const (and for α < 2 we get dense subspaces of L2 (X), which are domains of the stable pr... |

1 |
A Discrete Characterisation of Lipschitz Spaces on Fractals
- BODIN
(Show Context)
Citation Context ...he definition uses a discrete approximation of simple fractals, and so it is mandatory that the functions concerned be continuous. This is not necessarily true for small values of α. It is known (see =-=[3]-=-) that the Strichartz spaces and the Jonsson spaces agree for certain range of parameters. Therefore our characterisation remains valid for Strichartz spaces as well (see Section 4.1). 2 Preliminaries... |

1 |
How to recognize constant functions
- BREZIS
(Show Context)
Citation Context ... α = 2 is critical when p > 2, and where should the threshold be placed when 1 ≤ p < 2. For open subsets of the Euclidean space it is known that α = 1(= dw(Rd ) 2 ) works for all values of p ≥ 1, see =-=[4]-=-. We do not expect this to hold in general. 124 Links with other definitions of Besov-Lipschitz spaces 4.1 Strichartz Besov spaces on simple fractals Strichartz in [19] introduced the definition of t... |

1 | On the dichotomy of the heat kernel two sidedestimates’, Analysis on Graphs and its Applications - GRIGOR’YAN, KUMAGAI |

1 |
Brownian motion on fractals and function
- JONSSON
- 1996
(Show Context)
Citation Context ... We give a probabilistic characterisation of the Besov-Lipschitz spaces Lip(α, p, q)(X) on domains which support a Markovian kernel with appropriate exponential bounds. This extends former results of =-=[11, 14, 15, 7]-=- which were valid for α = dw 2 , p = 2, q = ∞, where dw is the walk dimension of the space X. 1 Introduction There are several definitions of Besov-Lipschitz spaces on measure spaces. In this paper we... |

1 | Recent developments of analysis on fractals - KUMAGAI - 2008 |

1 | Some function spaces related to the Brownian motion on simple nested fractals
- PIETRUSKA-PA̷LUBA
- 1999
(Show Context)
Citation Context |

1 |
Brownian motion characterisation on some BesovLipschitz spaces on domains
- SIKIČ
(Show Context)
Citation Context ...ss of metric measure spaces, were also introduced bu Hu and Zähle – in a different way – in their paper [10]. The way they are defined owes to the classical characterisation of Besov spaces from [5], =-=[17]-=-. Those spaces will be denoted by B p,q β with B p,q β (X); We will see that our characterisation of Lip(α,p,q)(X) is consistent (X) for some range of parameters (see Section 4.2). In the case of simp... |

1 |
Function spaces on fractals’, J.Funct. Anal
- STRICHARTZ
- 2003
(Show Context)
Citation Context ...) for some range of parameters (see Section 4.2). In the case of simple fractals (and the Sierpiński gasket in particular), yet another definition of Besov-Lipschitz spaces was given by Strichartz in =-=[19]-=-. This definition uses a discrete approximation of the space X. The Strichartz spaces we think of are the spaces (Λ p,q α ) 1 (X) (in [19], one can find other spaces as well, corresponding to large va... |