Weighted norm estimates for the Fourier transform with a pair of weights

Abstract:

We prove weighted norm inequalities of the form \[ {\left \| {\hat f} \right \|_{L_u^q}} \leq C{\left \| f \right \|_{H_\upsilon ^p}},\quad 0 < p \leq q < \infty ,\] for the Fourier transform on ${{\mathbf {R}}^n}$. For some weight functions $\upsilon$, the Hardy space $H_\upsilon ^p$ on the right can be replaced by $L_\upsilon ^p$. The proof depends on making an atomic decomposition of $f$ and using cancellation properties of the atoms.