Estimates of functions with vanishing periodizations

Abstract

We prove that if a function $f\in L^p({\mathbb{R}}^d)$ has vanishing periodizations then $\Vert f{\Vert }_{p^{\prime }}\lesssim \Vert f{\Vert }_p$, provided $1\le p<\frac{2d}{d+2}$ and dimension $d\ge 3$.