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

Abstract

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