Weighted norm inequalities for the Fourier transform on certain totally disconnected groups

Abstract:

Let $G$ be a locally compact totally disconnected Abelian group with dual group $\Gamma$. Let $U$ and $V$ be nonnegative measurable functions on $\Gamma$ and $G$, respectively. In this paper we give, in terms of $U$ and $V$, a necessary condition and some sufficient conditions for the inequality $||\hat fU|{|_q} \leq C||fV|{|_p}$ to hold for all $f$ in ${L_1}\left ( G \right )$, where $\hat f$ denotes the Fourier transform of $f$ and $1 < p \leq q < \infty$. If $U$ and $V$ are both radial, we give a necessary and sufficient condition for the above norm inequality to hold for all $f$ in ${L_1}\left ( G \right )$.