6
$\begingroup$

If $m(\xi)$ satisfies $$D^{\alpha}m(\xi)\leq \frac{C}{(1+|\xi|)^{|\alpha|+1}}$$ then is $m$ a Fourier transform of a $L^{1}$ function? (Note that the Bernstein theorem can't be applied here, since $m(\xi)$ may not be in $H^{s}$, where $s>\frac{n}{2}$.)

Generally, are there some simple ways to make sure that a given function belongs to $\mathcal{F}L^{1}$?

  • 0
    Are you assuming the inequality for all $\alpha$? Wouldn't that imply that $m$ is smooth and all its derivatives are integrable?2012-09-04
  • 0
    @timur:I think it should be $\alpha\leq [\frac{n}{2}]$2012-09-05

1 Answers 1