# Besov class via heat semigroup on Dirichlet spaces I: Sobolev type inequalities

@article{Ruiz2020BesovCV, title={Besov class via heat semigroup on Dirichlet spaces I: Sobolev type inequalities}, author={Patricia Alonso Ruiz and Fabrice Baudoin and Li Chen and Luke G. Rogers and Nageswari Shanmugalingam and Alexander Teplyaev}, journal={Journal of Functional Analysis}, year={2020} }

## 16 Citations

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

- MathematicsCalculus of Variations and Partial Differential Equations
- 2020

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…

Besov spaces associated with non-negative operators on Banach spaces

- Mathematics
- 2020

Motivated by a variety of representations of fractional powers of operators, we develop the theory of abstract Besov spaces $B^{ s, A }_{ q, X }$ for non-negative operators $A$ on Banach spaces $X$…

BV functions and fractional Laplacians on Dirichlet spaces

- Mathematics
- 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 Kernels and Besov Spaces on Metric Measure Spaces

- Mathematics
- 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…

Sobolev spaces and Poincar\'e inequalities on the Vicsek fractal

- Mathematics
- 2022

In this paper we prove that several natural approaches to Sobolev spaces coincide on the Vicsek fractal. More precisely, we show that the metric approach of Korevaar-Schoen, the approach by limit of…

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

- MathematicsJournal of Functional Analysis
- 2021

Yet another heat semigroup characterization of BV functions on Riemannian manifolds

- Mathematics
- 2020

This paper provides a characterization of functions of bounded variation (BV) in a compact Riemannian manifold in terms of the short time behavior of the heat semigroup. In particular, the main…

$L^p$-Poincar\'e inequalities on nested fractals.

- Mathematics
- 2020

We prove on some nested fractals scale invariant $L^p$-Poincar\'e inequalities on metric balls in the range $1 \le p \le 2$. Our proof is based on the development of the local $L^p$-theory of…

## References

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

- MathematicsCalculus of Variations and Partial Differential Equations
- 2020

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…

BV functions and fractional Laplacians on Dirichlet spaces

- Mathematics
- 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 and Lipschitz-Besov spaces

- Mathematics
- 2015

On a metric measure space (M, ρ, μ) we consider a family of the Lipschitz-Besov spaces Λ s,q that is defined only using the metric ρ and measure μ, and a family of Besov spaces B s p,q that is…

Heat kernel based decomposition of spaces of distributions in the framework of Dirichlet spaces

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Classical and nonclassical Besov and Triebel-Lizorkin spaces with complete range of indices are developed in the general setting of Dirichlet space with a doubling measure and local scale-invariant…

Heat kernel characterisation of Besov-Lipschitz spaces on metric measure spaces

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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…

Sobolev Spaces on Metric Measure Spaces: An Approach Based on Upper Gradients

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Preface 1. Introduction 2. Review of basic functional analysis 3. Lebesgue theory of Banach space-valued functions 4. Lipschitz functions and embeddings 5. Path integrals and modulus 6. Upper…

Sobolev Spaces and Calculus of Variations on Fractals

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The present note contains a review of $p$-energies and Sobolev spaces on metric measure spaces that carry a strongly local regular Dirichlet form. These Sobolev spaces are then used to generalize…

Characterizations of Sets of Finite Perimeter Using Heat Kernels in Metric Spaces

- Mathematics
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The overarching goal of this paper is to link the notion of sets of finite perimeter (a concept associated with N1,1-spaces) and the theory of heat semigroups (a concept related to N1,2-spaces) in…