论文标题
庞加莱的不平等和$ a_p $ thow on Bowies
Poincaré inequalities and $A_p$ weights on bow-ties
论文作者
论文摘要
如果可以写入$ x = x _ {+cup x _ { - } $，则指标空间$ x $称为a \ emph {bow-tie}，其中$ x _ {+cap x _ {+cap x _ { - } - { - } $ x $。我们表明，$ x $上的加倍量度$ $ $支持$（q，p）$ - poincaré$ x $的不平等时，仅当$ x $满足quasiconvexity-type状态，$μ$支持$（q，p）$ - POINCARUR在两种$ x $ x $ _ _+x $ _ {+xap and-pop} $} $ {+_ {+apap and-pop} $ { - { - _ { - - 条件保持。此容量条件反过来又以$ x_0 $为特征。 特别是，我们研究了$ x _ {\ mathbf {r}^n} $，由$ \ mathbf {r}^n $组成，配备了径向加倍的重量，并表征了\ p-poincaré的有效性，并在$ x _ _ mathbf^in} $ x _ {r} $ x^n}中表征了\ p-poincaré的有效性。对于这样的权重，我们还为Annuli围绕起源的能力提供了一般公式。
A metric space $X$ is called a \emph{bow-tie} if it can be written as $X=X_{+} \cup X_{-}$, where $X_{+} \cap X_{-}=\{x_0\}$ and $X_{\pm} \ne \{x_0\}$ are closed subsets of $X$. We show that a doubling measure $μ$ on $X$ supports a $(q,p)$--Poincaré inequality on $X$ if and only if $X$ satisfies a quasiconvexity-type condition, $μ$ supports a $(q,p)$-Poincaré inequality on both $X_{+}$ and $X_{-}$, and a variational \p-capacity condition holds. This capacity condition is in turn characterized by a sharp measure decay condition at $x_0$. In particular, we study the bow-tie $X_{\mathbf{R}^n}$ consisting of the positive and negative hyperquadrants in $\mathbf{R}^n$ equipped with a radial doubling weight and characterize the validity of the \p-Poincaré inequality on $X_{\mathbf{R}^n}$ in several ways. For such weights, we also give a general formula for the capacity of annuli around the origin.
