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Let $\mu$ be a surface area measure of a manifold $M\hookrightarrow\mathbb{R}^{n+1}$. If $M$ is the unit sphere $S^n$, it's known that surprisingly the Fourier transform of $\mu$ decays: $$|\hat\mu(\xi)|\le\ |\xi|^{(1-n)/2}.$$

This decay appears to hold for any hyper-surface with non-zero Gaussian curvature.

Let us now consider a manifold $M$ of dimension $1\le \dim M < n+1$. I know that there is a result that in that case $$|\hat\mu(\xi)|^2\le |\xi|^{\frac 1k}$$ where $k$ is the type of the manifold. Are there any other ("better") results about the decay of such measures that is when $M$ is not a hyper-surface? In particular I'm concerned with results that connect the decay with the dimension of $M$.

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  • $\begingroup$ googling "higher codimension restriction estimates" bring up this paper. Its corresponding MathSciNet entry is here. Maybe you can get some answers/references from it. $\endgroup$
    – Willie Wong
    Jun 23, 2016 at 13:42
  • $\begingroup$ @WillieWong thanks but this article deals with $f\in\mathcal{S}(\mathbb{R}^{n+1})$ where I'm concerned with $f=\hat{d\mu}\in\mathcal{S}^\prime(\mathbb{R}^{n+1})$. $\endgroup$
    – Alan Watts
    Jun 24, 2016 at 10:31
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    $\begingroup$ Actually, I think that the article liked by Willie is helpful. Section is called "The decay of the Fourier transform of the surface carried measure" which should address your question. $\endgroup$
    – Dirk
    Jun 24, 2016 at 11:06
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    $\begingroup$ Just to clarify something in my comment, you require curvature because otherwise you can take you favorite piece of a plane, and it will decay marvelously in many directions, but you will not get any decay in the direction perpendicular to the plane. This is basically dealt with the $N$-curvature condition in the reference given by Willie. If I'm not mistaken people call those things pencils nowadays, for some geometric reason I don't understand. $\endgroup$
    – Asaf
    Jun 24, 2016 at 11:23
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    $\begingroup$ Notice that as-long as you're not interested in going over the dimension (which as I've mention, only happens in very certain "symmetric" cases, such as the full sphere), you can localize your measure in any form you like (as long as your density is say Schwartz wrt to the induced measure) and get a local estimate (this is a feature in the stationary phase estimate, that you can multiply the exponential by a dampening function which would play a very "soft" role in your decay estimate) $\endgroup$
    – Asaf
    Jun 26, 2016 at 16:58

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