Fourier Inequalities with Nonradial Weights

Abstract

Let $\mathcal {F}\;f(\gamma ) = {\smallint _{{\mathbb {R}^n}}}({e^{ - 2i\pi \gamma \bullet x}} - 1)f(x) dx,n > 1$, and $u$, $v$ be nonnegative functions. Sufficient conditions are found in order that $\left \| \mathcal {F}\;f\right \| _{q,u} \leq C\left \| f\right \| _{p,v}$ for all $f \in L_v^p({\mathbb {R}^n})$. Pointwise and norm approximations of $\mathcal {F}\;f$ are derived. Similar results are obtained when $u$ is replaced by a measure weight. In the case $v(x) = |x{|^{n(p - 1)}}$, a counterexample is given which shows that no Fourier inequality can hold for all $f$ in $L_{c,0}^\infty$. Spherical restriction theorems are established. Further conditions for the boundedness of $\mathcal {F}$ are discussed.