Is it possible to show that all the $L^p(\mathbb{R}^n)$ norms of a given Littlewood-Paley block are equivalent or else find a counterexample? One inequality follows from the Bernstein inequalities if working on the spatial side, whereas working on the frequency side an analogous inequality follows simply from Holder's inequality.

By Littlewood-Paley blocks I mean functions whose Fourier transform is supported in an annulus. They're typically denoted $\widehat{\Delta_jf}(\xi)=\varphi(\xi/2^j)\hat{f}(\xi)$ for a block supported in $\{2^{j-1}\leq|\xi|\leq 2^{j+1}\}$.

This is true on the torus, i.e. periodic functions. In this case, the assertion follows from the fact that $\ell^p$ norms of sequences with finitely many terms are all equivalent, i.e. if $(a_j)=(\dots,a_{-n},\dots,a_0,\dots,a_n,\dots)$, then $c_n||(a_j)||_{\ell^p}\leq||(a_j)||_{\ell^q}\leq C_N||(a_j)||_{\ell^p}$.

So it would seem surprising if it weren't true in $\mathbb{R}^n$, though I can't seem to supply a proof for it or otherwise.