Suppose that support of $f:\mathbb R \to \mathbb R$ is compact set $K\subset \mathbb R.$ Assume that $ \int_{\mathbb R} |\widehat{f}|^q d\xi <\infty.$ ($\widehat{\cdot}$ denote the Fourier transform).
Question: Can we say $ \left\| \|\chi_{n+(-1/2, 1/2]} \widehat{f}\|_{L^p_{\xi}} \right\|_{\ell^q_n}<\infty$? (Here $1\leq p, q \le \infty$ and $p\neq q$)
Note: For $p=q$ it is just correct.
I believe that this is true for all values of $p\in[1,\infty]$. By Hölder, it will suffice to consider the case $p=\infty$; essentially, this will follow from local constancy/reverse Hölder. With the choice $p=\infty$, we may assume that $q<\infty$.
By enlarging $K$, we may as well take $K=[-N,N]$ for some $N>2$. Let $\eta$ be such that $\eta\equiv 1$ on $K$ and $\hat{\eta}$ is supported in $[-\frac{2}{N},\frac{2}{N}]$, $\eta\geq 0$, and such that $\int\eta\lesssim N$.
Then, for each $n$ and $\xi\in[n-\frac{1}{2},n+\frac{1}{2}]$, $\hat{f}(\xi)=\hat{f}*\hat{\eta}(\xi)$, so
$$|\hat{f}(\xi)|\leq\int_{-2/N}^{2/N}|\hat{f}(\xi-\omega)||\hat{\eta}(\omega)|d\omega\lesssim N\int_{-2/N}^{2/N}|\hat{f}(\xi-\omega)|d\omega.$$
Consequently,
$$\max_{\xi\in[n-\frac{1}{2},n+\frac{1}{2}]}|\hat{f}(\xi)|\lesssim N\int_{n-\frac{3}{2}}^{n+\frac{3}{2}}|\hat{f}(\omega)|d\omega\lesssim N(\int_{n-\frac{3}{2}}^{n+\frac{3}{2}}|\hat{f}(\omega)|^qd\omega)^{1/q},$$
so that
$$\Big\|\|\chi_{n+(-\frac{1}{2},\frac{1}{2}]}\hat{f}\|_{L_\xi^\infty}\Big\|_{\ell_n^q}\lesssim N\Big\|\|\chi_{n+(-\frac{1}{2},\frac{1}{2}]}\hat{f}\|_{L_\xi^q}\Big\|_{\ell_n^q}<+\infty$$
as claimed.
