My Math Forum Lebesgue Points Question

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 March 10th, 2014, 03:48 AM #1 Member   Joined: Aug 2012 Posts: 32 Thanks: 0 Lebesgue Points Question Let $\Omega \subset \mathbb{R}^{n}$ be a bounded open set and take $p' = \frac{p}{(p-1)}$. Let $a: \Omega \times \mathbb{R} \times \mathbb{R}$ be a Caratheodory function satisfying $|a(x,s,\xi)| \leq k(x) + \beta(|s|^{p-1} + |\xi|^{p-1})$ for almost every $x \in \Omega$, for every $(s, \xi) \in \mathbb{R} \times \mathbb{R}^{n}$ and for some $k \in L^{p^{'}}(\Omega), \beta \geq 0, p > 1$. If we further assume that $x_{0}$ is a Lebesgue point of $a$ and let $s \in \mathbb{R}$, $\xi, \eta \in \mathbb{R}^{n}$ ($\xi \neq \eta$) be such that $\langle a(x_{0},s,\xi)-a(x_{0},s,\eta); \xi-\eta \rangle < 0$. Then how does it follow that since $x_{0}$ is a Lebesgue point of $a$ we have: $\lim\limits_{R \rightarrow 0}\{\frac{1}{\text{meas}H_{R}(x_{0})}}\int_{H_{R}( x_{0})}\langle a(x,s,\xi)-a(x,s,\eta); \xi-\eta \rangle dx \}< 0$, where $H_{R}(x_{0})$ is Hypercube of $\mathbb{R}^{n}$? Let me know if any further info is required. Thanks.

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