Timeline for Decay of the Fourier transform of a surface area measure
Current License: CC BY-SA 3.0
14 events
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| Jun 26, 2016 at 16:58 | comment | added | Asaf | 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) | |
| Jun 26, 2016 at 16:55 | comment | added | Asaf | @AlanWatts, first of all your formula have extra square. Secondly, in all the relevant cases (suitably curved and so) such an estimate would follow from a stationary phase technique. You ought to understand this technique before asking anything further, look for example in Sogge's book or Wolff's notes, or the classic books by Hormander or Stein. This estimate would give best decay but also generic decay (as long as your measure is ac wrt the surface area measure), simply because of your statement "This decay appears to hold for any hyper-surface with non-zero Gaussian curvature." | |
| Jun 26, 2016 at 11:58 | comment | added | Alan Watts | @Asaf so given $\mu$, a surface area measure of a manifold, how can one prove that $|\hat\mu(\xi)|^2\le |\xi|^{-\dim_HM/2}$? $\dim_FM=n$ gives only "the best decay" and not the answer for an explicit measure as far as I understood. | |
| Jun 25, 2016 at 18:27 | comment | added | Asaf | The easiest way is to show that $F$-dim is dominated (by $H$-dim, or Minkowski-dimension (probably Minkowski's should be easier, as it is a volume-packing estimate, in other cases you might need Frostman's lemma or so). Now you should go into some lengths showing that for nice manifolds, the topological dimension (Lebesgue) is indeed the $H$-dim or so. You can go slightly better than the dimension (for example, in the sphere case you get extra $1/2$ in the bound over $n-1$), but that happens in concrete geometrical cases for known reasons, but it won't be something substantial. | |
| Jun 25, 2016 at 15:00 | comment | added | Alan Watts | @Asaf how can we deduce that the Fourier dimension of the set is less or equal $\dim_H M$? For instance here (mathoverflow.net/questions/44192/…) the OP says it's known but doesn't give any reference. | |
| Jun 25, 2016 at 14:58 | comment | added | Alan Watts | @WillieWong but the estimates are only proven for manifolds which their last coordinates are homogeneous and smooth, isn't it? | |
| Jun 24, 2016 at 13:02 | comment | added | Willie Wong | @AlanWatts: restriction estimates are dual to decay estimates for surface measures. The paper I linked to, as Dirk remarked, actually proves the restriction estimates through the decay estimates. | |
| Jun 24, 2016 at 11:23 | comment | added | Asaf | 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. | |
| Jun 24, 2016 at 11:19 | comment | added | Asaf | Basically such an estimate correspond to the Fourier dimension of your set. As your set is nice (and not some fractal), what you can and should expect is decay in the rate of the dimension (half of, as it is $L^{2}$ normalized). In certain "geometric" you can slightly beat it (for example in the ball/sphere case, where you can analyze the Bessel functions occurring there a bit more carefully). If you have a bit of a nicer situation (for example an ensemble of manifolds, where transversality considerations enter), you may do better. | |
| Jun 24, 2016 at 11:06 | comment | added | Dirk | 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. | |
| Jun 24, 2016 at 10:31 | history | edited | Alan Watts | CC BY-SA 3.0 |
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| Jun 24, 2016 at 10:31 | comment | added | Alan Watts | @WillieWong thanks but this article deals with $f\in\mathcal{S}(\mathbb{R}^{n+1})$ where I'm concerned with $f=\hat{d\mu}\in\mathcal{S}^\prime(\mathbb{R}^{n+1})$. | |
| Jun 23, 2016 at 13:42 | comment | added | Willie Wong | googling "higher codimension restriction estimates" bring up this paper. Its corresponding MathSciNet entry is here. Maybe you can get some answers/references from it. | |
| Jun 23, 2016 at 7:36 | history | asked | Alan Watts | CC BY-SA 3.0 |