local moments of measures whose Fourier transform vanish in an interval

Question

Assume h is a measure whose Fourier transform vanishes in an interval $[-\Omega,\Omega]$. I'm interested in obtaining inequalities of the form \begin{equation*} \int_{-\delta}^{+\delta}|h|(dt)\le C(\delta,\Omega)\int_{-\delta}^{+\delta}t^2|h|(dt) \end{equation*}

Answer 1

If I understand this correctly--that you'd like an inequality like this to hold uniformly over all measures $dh$ whose Fourier transform is $0$ on $[-\Omega, \Omega]$--then you're unfortunately out of luck.

Here's one way of seeing that such an inequality can't be true. Let $g(\xi)$ be a $C^{\infty}$ function that's supported in the interval $[1, 2]$. Define $f(t)$ via $\hat{f} = g$. Note that $f$ is a Schwartz function, so for any $N > 0$ there's a constant $C_{N}$ such that $|f(t)| \leq C_{N} (1 + |t|)^{-N}$. Now introduce a rescaling parameter $r > 0$: let $g_{r}(\xi) = g(\xi / r)$, and define $f_{r}$ by $\hat{f}_{r} = g_{r}$, so that $f_{r}(t) = r f(rt)$. The measures (really just functions) I have in mind to examine are $dh_{r} = f_{r} \, dt$, letting $r \to \infty$.

Since $\widehat{dh}_{r} = g_{r}$ is supported in $[r, 2r]$, for all $r > \Omega$ the measure $dh_{r}$ will satisfy the criterion. The $dh_{r}$ all have the same mass, but as $r \to \infty$ the mass becomes concentrated closer and closer to the origin and so $$ \int_{-\delta}^{\delta} d|h_{r}| \to \| f \|_{1} $$ for any fixed $\delta$. On the other hand, $$ \int_{-\delta}^{\delta} t^{2} \, d|h_{r}| = \int_{-\delta}^{\delta} t^{2} r |f(rt)| \, dt \leq C_{N} \int_{-\delta}^{\delta} \frac{t^{2} r}{(1 + r|t|)^{N}} \, dt = C_{N} r^{-2} \int_{-r \delta}^{r \delta} \frac{s^{2}}{(1 + |s|)^{N}} \, ds, $$ from which we have $$ \int_{-\delta}^{\delta} t^{2} \, d|h_{r}| \to 0 $$ by taking $N > 3$.

Roughly speaking, for an inequality like the one you want to hold, you need some way to make sure the measures can't get concentrated near the origin, since the $t^{2}$ factor then causes problems. I haven't thought this through, but perhaps a condition that $\widehat{dh}$ have compact support, or perhaps at least some rate of decay, might give a result in this direction.