Im reading the Appendix B in Terrence Tao's Textbook on Nonlinear Dispersive Equations, more precisely the proof of existence of a non-zero maximizer in $H^1(R^d)$ of the Weinstein-Functional $W(u)=\frac{||u||_{p+1}^{p+1}}{||u||_2^{2-\frac{(d-2)(p-1)}{2}}||\nabla u||^{\frac{d(p-1)}{2}}}$.
Taking a minimizing sequence $Q_n$ s.t. $\lim_{n\rightarrow\infty}W(Q_n)=W_{max}$ and normalizing s.t. $$||Q_n||_2^2=||\nabla u||_2^2=1, \ \lim_{n\rightarrow\infty}||u||_{p+1}^{p+1}=W_{max}$$ we get $$0<W_{max}^{\frac{1}{p+1}}=\limsup_{n\rightarrow\infty}||Q_n||_{p+1}\leq\limsup_{n\rightarrow\infty}\sum_{N}||P_NQ_n||_{p+1}$$ where $P_N$ is the Littlewood-Paley Projector with $N$ in the dyadic numbers. Since by Bernsteins inequality, absolute convergence of the sum and dominated convergence one can interchange sum and lim sup, there must be a diadic number $N_0$ s.t. $\limsup_{n\rightarrow\infty}||P_{N_0}Q_n||_{p+1}>0$.
Here the claim ist, that by Hölder and L2-normalization of $Q_n$, it follows that $\limsup_{n\rightarrow\infty}||P_{N_0}Q_n||_\infty>0$
Could somebody explain why?
Next, by translating each $P_{N_0}Q_n$ seperately we may assume $\limsup_{n\rightarrow\infty}|P_{N_0}Q_n(0)|>0$, hence by passing to a subsequence $|P_{N_0}Q_n(0)|>c>0 \ \forall n$.
Here it is claimed that by expanding the kernel of $P_{N_0}$, there must be some $\eta,R_0>0$ s.t. $$\int_{|x|\leq R_0}|Q_n|^{p+1}\geq\eta$$ Also this I do not see.
Any hints are highly appreciated!