I'm considering the following estimate in something that I'm reading:
Consider $z \in B(0,R)$ and the following expression: $$\int_{|y|<2R}u(y)e^{-|z-y|^2/2t}dy + \int_{|y|\geq2R}u(y)e^{-|z-y|^2/2t}dy$$
The text wishes to come up with an estimate for the exponents. So it writes:
If $|y| \geq 2$ we get $|z-y|^2 \geq (|y|-|z|)^2 \geq |y|^2/4$
The first inequality is reverse triangle ineqality. I'm lost at how the next inequality comes about. Thoughts?
I'm assuming it's for the second term (are you missing a "$R$" in the "If $\lvert y\rvert \geq 2$"?). In that one, we have $\lvert y \rvert \geq 2R$; since $\lvert z\rvert < R$ (as $z\in B(0,r)$), we get $$ \lvert y\rvert-\lvert z\rvert \geq \frac{\lvert y\rvert}{2} $$ and squaring gives what you want.