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I need some help with this problem:
let $\lambda>0$ and $w(x)=Ce^{-|x|}$ a function defined on $\mathbb{R}^{N}$, where $C>0$ is a constant such that $\int_{\mathbb{R}^{N}}w(x)dx=1$.
Then we define a new function as $w_{\lambda}(x)=C\lambda^{N}e^{-\lambda|x|}$.
Moreover, let $V\in L^{q}(\mathbb{R}^{N})$, with $q\geq\frac{N}{2}$, such that $V^{-}\not\equiv0$ on a set of positive measure and let $0$ a Lebesgue point of V belonging to this set (note that with $V^{-}$ I state the negative part of $V$).
I shall prove that $$\int_{\mathbb{R}^{N}}V(x)w_{\lambda}(x)dx\xrightarrow[\lambda\rightarrow+\infty]{}V(0).$$
I took the ball $B_{\lambda^{-1}}(0)$ and I wrote $$\bigg|\int_{\mathbb{R}^{N}}V(x)w_{\lambda}(x)dx-V(0)\bigg|\leq C\lambda^{N}\int_{\mathbb{R}^{N}}|V(x)-V(0)|e^{-\lambda|x|}dx$$$$=C\lambda^{N}\int_{B_{\lambda^{-1}(0)}}|V(x)-V(0)|e^{-\lambda|x|}dx+C\lambda^{N}\int_{\mathbb{R}^{N}\setminus B_{\lambda^{-1}(0)}}|V(x)-V(0)|e^{-\lambda|x|}dx$$ and I already proved the convergence to $0$ of the first integral, so I need help with the second one. I assume I should use the Hoelder Inequality in some way, but I really don't understand how.
I hope someone could help me and I already thank you.

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  • $\begingroup$ I'm sorry I made a mistake, the function $w(x)=Ce^{-|x|}$ without $\lambda$. $\endgroup$ – Francesca May 25 at 10:15
  • $\begingroup$ You can fix errors like that with the "edit" link. You meant $\int w=1$, not $C\int w=1$. $\endgroup$ – David C. Ullrich May 25 at 10:23

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