The passage below is from a fixed point argument in Wang's paper enter link description here pg 566. At the end of the $I_1$ calculation, it somehow makes the following estimate $$C\|\phi\|_X (e^{-|x|^2/32t} +w^{-1}(x)) \leq C(p,n,l)\|\phi\|_X w^{-1}(x)$$ assuming according to the author that $T>1$. Here $w^{-1}(x) = (1+|x|)^{-l/(p-1)}$. My question is whether this passage is really correct? If $T<1$ I think it works, because $e^{-|x|^2/32t}\leq e^{-|x|^2/32}$ so using that $w(x)e^{-|x|^2/32}<C_1$, $C_1$ constant, $$ (e^{-|x|^2/32t} +w^{-1}(x)) \leq (w^{-1}(x)w(x)e^{-|x|^2/32} +w^{-1}(x)) $$

But the same passage for $T>1$ I can't see how it can be done. Is it something obvious? I appreciate any clarifications.