# Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates

@article{AlonsoRuiz2020BesovCV,
title={Besov class via heat semigroup on Dirichlet spaces II: BV functions and Gaussian heat kernel estimates},
author={Patricia Alonso-Ruiz and Fabrice Baudoin and Li Chen and Luke G. Rogers and Nageswari Shanmugalingam and Alexander Teplyaev},
journal={Calculus of Variations and Partial Differential Equations},
year={2020},
volume={59},
pages={1-32}
}
We introduce the class of bounded variation (BV) functions in a general framework of strictly local Dirichlet spaces with doubling measure. Under the 2-Poincaré inequality and a weak Bakry–Émery curvature type condition, this BV class is identified with the heat semigroup based Besov class $$\mathbf {B}^{1,1/2}(X)$$ B 1 , 1 / 2 ( X ) that was introduced in our previous paper. Assuming furthermore a quasi Bakry–Émery curvature type condition, we identify the Sobolev class $$W^{1,p}(X)$$ W 1 , p…

## Figures from this paper

Besov class via heat semigroup on Dirichlet spaces I: Sobolev type inequalities
• Mathematics, Physics
Journal of Functional Analysis
• 2020
We introduce heat semigroup-based Besov classes in the general framework of Dirichlet spaces. General properties of those classes are studied and quantitative regularization estimates for the heat
Equivalence of Besov spaces on p.c.f. self-similar sets
• Mathematics
• 2020
On p.c.f. self-similar sets, of which the walk dimensions of heat kernels are in general larger than 2, we find a sharp region where two classes of Besov spaces, the heat Besov spaces
Sobolev spaces on p.c.f. self-similar sets I: critical orders and atomic decompositions
• Mathematics
Journal of Functional Analysis
• 2021
We consider the Sobolev type spaces $H^\sigma(K)$ with $\sigma\geq 0$, where $K$ is a post-critically finite self-similar set with the natural boundary. Firstly, we compare different classes of
Heat Kernels and Besov Spaces on Metric Measure Spaces
• 2019
Let (M, ρ, μ) be a metric measure space satisfying the volume doubling condition. Assume also that ( M, ρ, μ) supports a heat kernel satisfying the upper and lower Gaussian bounds. We study the
RCD*(K,N) Spaces and the Geometry of Multi-Particle Schrödinger Semigroups
With $(X,\mathfrak{d},\mathfrak{m})$ an $\mathrm{RCD}(K,N)$ space for some $K\in\mathbf{R}$, $N\in [1,\infty)$, let $H$ be the self-adjoint Laplacian induced by the underlying Cheeger form. Given
Gagliardo-Nirenberg, Trudinger-Moser and Morrey inequalities on Dirichlet spaces
• Mathematics
• 2020
With a view towards Riemannian or sub-Riemannian manifolds, RCD metric spaces and specially fractals, this paper makes a step further in the development of a theory of heat semigroup based $(1,p)$
BV functions and fractional Laplacians on Dirichlet spaces
• Mathematics, Physics
• 2019
We study $L^p$ Besov critical exponents and isoperimetric and Sobolev inequalities associated with fractional Laplacians on metric measure spaces. The main tool is the theory of heat semigroup based
Heat kernel analysis on diamond fractals
This paper presents a detailed analysis of the heat kernel on an $(\mathbb{N}\times\mathbb{N})$-parameter family of compact metric measure spaces, which do not satisfy the volume doubling property.
On (non-)singularity of energy measures under full off-diagonal heat kernel estimates
• Mathematics
• 2019
We show that for a strongly local, regular symmetric Dirichlet form over a complete, locally compact geodesic metric space, full off-diagonal heat kernel estimates with walk dimension strictly larger
BV Functions and Sets of Finite Perimeter on Configuration Spaces
• Mathematics
• 2021
This paper contributes to foundations of the geometric measure theory in the infinite dimensional setting of the configuration space over the Euclidean space R equipped with the Poisson measure π. We

## References

SHOWING 1-10 OF 220 REFERENCES
Besov class via heat semigroup on Dirichlet spaces I: Sobolev type inequalities
• Mathematics, Physics
Journal of Functional Analysis
• 2020
We introduce heat semigroup-based Besov classes in the general framework of Dirichlet spaces. General properties of those classes are studied and quantitative regularization estimates for the heat
Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below
• Mathematics
• 2014
This paper is devoted to a deeper understanding of the heat flow and to the refinement of calculus tools on metric measure spaces $(X,\mathsf {d},\mathfrak {m})$. Our main results are: A general
Gradient estimates for heat kernels and harmonic functions
• Mathematics
• 2017
Let $(X,d,\mu)$ be a doubling metric measure space endowed with a Dirichlet form $\E$ deriving from a "carre du champ". Assume that $(X,d,\mu,\E)$ supports a scale-invariant $L^2$-Poincare
Heat kernel characterisation of Besov-Lipschitz spaces on metric measure spaces
We give a heat-kernel 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
Heat kernels on metric measure spaces and an application to semilinear elliptic equations
• Mathematics
• 2003
We consider a metric measure space (M, d, p) and a heat kernel p t (x,v) on M satisfying certain upper and lower estimates, which depend on two parameters a and β. We show that under additional mild
Geometry and Analysis of Dirichlet forms
• Mathematics
• 2012
Let $\mathscr E$ be a regular, strongly local Dirichlet form on $L^2(X, m)$ and $d$ the associated intrinsic distance. Assume that the topology induced by $d$ coincides with the original topology
Interpolation properties of Besov spaces defined on metric spaces
• Mathematics
• 2010
Let X = (X, d, μ)be a doubling metric measure space. For 0 < α < 1, 1 ≤p, q < ∞, we define semi-norms When q = ∞ the usual change from integral to supremum is made in the definition. The
A sub-Riemannian curvature-dimension inequality, volume doubling property and the Poincaré inequality
• Mathematics
• 2010
Let $$\mathbb M$$M be a smooth connected manifold endowed with a smooth measure $$\mu$$μ and a smooth locally subelliptic diffusion operator $$L$$L satisfying $$L1=0$$L1=0, and which is symmetric
Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and Lp-Liouville properties.
The basic object for the sequel is a fixed regul r Dirichlet form S with domain Q) ( } on a real Hubert space H = L(X, m). The underlying topological space X is a locally compact separable Hausdorff
Geometry and analysis of Dirichlet forms (II)
• Mathematics
• 2014
Abstract Given a regular, strongly local Dirichlet form E , under assumption that the lower bound of the Ricci curvature of Bakry–Emery, the local doubling and local Poincare inequalities are