361 research outputs found

### Lebesgue points via the Poincar\'e inequality

In this article, we show that in a $Q$-doubling space $(X,d,\mu),$ $Q>1,$
which satisfies a chain condition, if we have a $Q$-Poincar\'e inequality for a
pair of functions $(u,g)$ where $g\in L^Q(X),$ then $u$ has Lebesgue points
$H^h$-a.e. for $h(t)=\log^{1-Q-\epsilon}(1/t).$ We also discuss how the
existence of Lebesgue points follows for $u\in W^{1,Q}(X)$ where $(X,d,\mu)$ is
a complete $Q$-doubling space supporting a $Q$-Poincar\'e inequality for $Q>1.$Comment: 16 page

### Geometry and Analysis of Dirichlet forms

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 on $X$, and that $X$ is compact,
satisfies a doubling property and supports a weak $(1, 2)$-Poincar\'e
inequality. We first discuss the (non-)coincidence of the intrinsic length
structure and the gradient structure. Under the further assumption that the
Ricci curvature of $X$ is bounded from below in the sense of
Lott-Sturm-Villani, the following are shown to be equivalent:
(i) the heat flow of $\mathscr E$ gives the unique gradient flow of $\mathscr
U_\infty$,
(ii) $\mathscr E$ satisfies the Newtonian property,
(iii) the intrinsic length structure coincides with the gradient structure.
Moreover, for the standard (resistance) Dirichlet form on the Sierpinski
gasket equipped with the Kusuoka measure, we identify the intrinsic length
structure with the measurable Riemannian and the gradient structures. We also
apply the above results to the (coarse) Ricci curvatures and asymptotics of the
gradient of the heat kernel.Comment: Advance in Mathematics, to appear,51p

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