Weighted norm inequalities for the Fourier transform

Abstract:

Given $p$ and $q$ satisfying $1 < p \leqslant q < \infty$, sufficient conditions on nonnegative pairs of functions $U,V$ are given to imply \[ {\left [ {\int _{{R^n}}^{} {|\hat f(x){|^q}U(x) dx}} \right ]^{1/q}} \leqslant c{\left [ {\int _{{R^n}}^{} {|f(x){|^p}V(x) dx}} \right ]^{1/p}},\] where $\hat f$ denotes the Fourier transform of $f$, and $c$ is independent of $f$. For the case $q = p’$ the sufficient condition is that for all positive $r$, \[ \left [ {\int _{U(x) > Br} {U(x)\;dx}} \right ]\left [ {\int _{V(x) < {r^{p - 1}}} {V{{(x)}^{- 1/(p - 1)}}\;dx}} \right ] \leqslant A,\] where $A$ and $B$ are positive and independent of $r$. For $q \ne p’$ the condition is more complicated but also is invariant under rearrangements of $U$ and $V$. In both cases the sufficient condition is shown to be necessary if the norm inequality holds for all rearrangements of $U$ and $V$. Examples are given to show that the sufficient condition is not necessary for a pair $U,V$ if the norm inequality is assumed only for that pair.