$\{v_n \} $ is bounded in $W_0^{1,q}$ for $ 1 \leq q < \frac{N}{N-1}$

Question

I have a encountered to a problem in reading article .

Can someone look at the page 9 in this article and give a hint that why $\{v_n \} $ is bounded in $W_0^{1,q}$ for $ 1 \leq q < \frac{N}{N-1}$ and also I am curious to know that $\frac{N}{N-1}$ comes from where ?

and also why $\{T_k(v_n) \} $ is bounded in $W_0^{1,2}$

I am so Thanked for helping .