Given a sequence of functions $\{f_n\}_{n\in\mathbb N}\subset \mathcal S(\mathbb R^n)$, whose Fourier support are pairwise disjoint, i.e. $$\text{supp}(\hat f_m)\cap \text{supp}(\hat f_n)=\emptyset,\ \ \ m\neq n$$ where $\mathcal S(\mathbb R^n)$ denotes the Schwartz functions.
From Plancherel's theorem we know that for the case $p=2$, $$\sum_{i=1}^N||f_n||_{L^2}^2=||\sum_{i=1}^Nf_n||_{L^2}^2$$ But whether similar results hold for the case $1<p<\infty$, we would expect the form of $$c\sum_{i=1}^N||f_n||_{L^p}^p\leq||\sum_{i=1}^Nf_n||_{L^p}^p\leq C\sum_{i=1}^N||f_n||_{L^p}^p$$ I could construct counter examples for the left inequality of the case $2<p<\infty$,and the right inequality of the case $1<p<2$, so it remains to prove or disprove the following two inequalitys: $$c\sum_{i=1}^N||f_n||_{L^p}^p\leq||\sum_{i=1}^Nf_n||_{L^p}^p,\ \ \ 1<p<2$$ $$\left\Vert\sum_{i=1}^Nf_n\right\Vert_{L^p}^p\leq C\sum_{i=1}^N||f_n||_{L^p}^p\ \ \ 2<p<\infty$$ Any advise would be appreciated.