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Gradient estimates for heat kernels and harmonic functions
Journal of Functional Analysis ( IF 1.7 ) Pub Date : 2020-05-01 , DOI: 10.1016/j.jfa.2019.108398
Thierry Coulhon , Renjin Jiang , Pekka Koskela , Adam Sikora

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 inequality. In this article, we study the following properties of harmonic functions, heat kernels and Riesz transforms for $p\in (2,\infty]$: (i) $(G_p)$: $L^p$-estimate for the gradient of the associated heat semigroup; (ii) $(RH_p)$: $L^p$-reverse Holder inequality for the gradients of harmonic functions; (iii) $(R_p)$: $L^p$-boundedness of the Riesz transform ($p<\infty$); (iv) $(GBE)$: a generalised Bakry-Emery condition. We show that, for $p\in (2,\infty)$, (i), (ii) (iii) are equivalent, while for $p=\infty$, (i), (ii), (iv) are equivalent. Moreover, some of these equivalences still hold under weaker conditions than the $L^2$-Poincare inequality. Our result gives a characterisation of Li-Yau's gradient estimate of heat kernels for $p=\infty$, while for $p\in (2,\infty)$ it is a substantial improvement as well as a generalisation of earlier results by Auscher-Coulhon-Duong-Hofmann [7] and Auscher-Coulhon [6]. Applications to isoperimetric inequalities and Sobolev inequalities are given. Our results apply to Riemannian and sub-Riemannian manifolds as well as to non-smooth spaces, and to degenerate elliptic/parabolic equations in these settings.

中文翻译:

热核和调和函数的梯度估计

令 $(X,d,\mu)$ 是一个倍增度量空间,赋予一个 Dirichlet 形式 $\E$ 派生自“carre du champ”。假设 $(X,d,\mu,\E)$ 支持尺度不变的 $L^2$-Poincare 不等式。在本文中,我们研究了 $p\in (2,\infty]$ 的调和函数、热核和 Riesz 变换的以下性质: (i) $(G_p)$: $L^p$-梯度估计相关的热半群;(ii) $(RH_p)$:$L^p$-调和函数梯度的反向Holder 不等式;(iii) $(R_p)$:$L^p$-Riesz 的有界性transform ($p<\infty$); (iv) $(GBE)$: 一个广义的 Bakry-Emery 条件。我们证明,对于 $p\in (2,\infty)$, (i), (ii) (iii) 是等价的,而对于 $p=\infty$,(i), (ii), (iv) 是等价的。此外,其中一些等价在比 $L^2$-Poincare 不等式更弱的条件下仍然成立。我们的结果给出了 Li-Yau 对 $p=\infty$ 的热核梯度估计的特征,而对于 $p\in (2,\infty)$,这是一个实质性的改进,也是对 Auscher 早期结果的概括-Coulhon-Duong-Hofmann [7] 和 Auscher-Coulhon [6]。给出了等周不等式和 Sobolev 不等式的应用。我们的结果适用于黎曼和亚黎曼流形以及非光滑空间,并适用于这些设置中的退化椭圆/抛物线方程。\infty)$ 这是一个实质性的改进,也是对 Auscher-Coulhon-Duong-Hofmann [7] 和 Auscher-Coulhon [6] 早期结果的概括。给出了等周不等式和 Sobolev 不等式的应用。我们的结果适用于黎曼和亚黎曼流形以及非光滑空间,并适用于这些设置中的退化椭圆/抛物线方程。\infty)$ 这是一个实质性的改进,也是对 Auscher-Coulhon-Duong-Hofmann [7] 和 Auscher-Coulhon [6] 早期结果的概括。给出了等周不等式和 Sobolev 不等式的应用。我们的结果适用于黎曼和亚黎曼流形以及非光滑空间,并适用于这些设置中的退化椭圆/抛物线方程。
更新日期:2020-05-01
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