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Alan Watts
Alan Watts
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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 $$|\mu(\xi)|^2\le |\xi|^{\frac 1k}$$$$|\hat\mu(\xi)|^2\le |\xi|^{\frac 1k}$$ 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 $M$.

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 $$|\mu(\xi)|^2\le |\xi|^{\frac 1k}$$ 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 $M$.

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 $M$.

Source Link
Alan Watts
Alan Watts
  • 43
  • 5

Decay of the Fourier transform of a surface area measure

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 $$|\mu(\xi)|^2\le |\xi|^{\frac 1k}$$ 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 $M$.