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Abdelmalek Abdesselam
Abdelmalek Abdesselam
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Interesting question. I don't have an answer but the following could be of help. There is a complete description of distributions which are invariant under the restricted Lorentz group. For example, see this article by Rieckers and Güttinger. Morally, it amounts to a reduction to one-dimensional distributions in the variable $B(p)$. A modification of your question which one could look at as a preliminary step is as follows.

Let $V=\{p\ |\ B(p)\ge 0\}$. Are there Lorentz invariant distributions $T$ such that $T$ as well as $\widehat{T}$ have support contained in $V$?

Note that there is a rather vast literature on various versions of a "qualitative uncertainty principle" where you impose conditions on the support of a function as well as its Fourier transform. This new question is of this type. Finally, you might want to see if there is an explicit formula for the Rieckers-Güttinger spectral representation of the Fourier transform in terms of that of the original distribution. If so you might have a one-dimensional reduction of the problem, albeit perhaps with a strange 1d Fourier transform.

Interesting question. I don't have an answer but the following could be of help. There is a complete description of distributions which are invariant under the restricted Lorentz group. For example, see this article by Rieckers and Güttinger. Morally, it amounts to a reduction to one-dimensional distributions in the variable $B(p)$. A modification of your question which one could look at as a preliminary step is as follows.

Let $V=\{p\ |\ B(p)\ge 0\}$. Are there Lorentz invariant distributions $T$ such $T$ as well as $\widehat{T}$ have support contained in $V$?

Note that there is a rather vast literature on various versions of a "qualitative uncertainty principle" where you impose conditions on the support of a function as well as its Fourier transform. This new question is of this type. Finally, you might want to see if there is an explicit formula for the Rieckers-Güttinger spectral representation of the Fourier transform in terms of that of the original distribution. If so you might have a one-dimensional reduction of the problem, albeit perhaps with a strange 1d Fourier transform.

Interesting question. I don't have an answer but the following could be of help. There is a complete description of distributions which are invariant under the restricted Lorentz group. For example, see this article by Rieckers and Güttinger. Morally, it amounts to a reduction to one-dimensional distributions in the variable $B(p)$. A modification of your question which one could look at as a preliminary step is as follows.

Let $V=\{p\ |\ B(p)\ge 0\}$. Are there Lorentz invariant distributions $T$ such that $T$ as well as $\widehat{T}$ have support contained in $V$?

Note that there is a rather vast literature on various versions of a "qualitative uncertainty principle" where you impose conditions on the support of a function as well as its Fourier transform. This new question is of this type. Finally, you might want to see if there is an explicit formula for the Rieckers-Güttinger spectral representation of the Fourier transform in terms of that of the original distribution. If so you might have a one-dimensional reduction of the problem, albeit perhaps with a strange 1d Fourier transform.

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Abdelmalek Abdesselam
Abdelmalek Abdesselam
  • 23k
  • 1
  • 67
  • 127

Interesting question. I don't have an answer but the following could be of help. There is a complete description of distributions which are invariant under the restricted Lorentz group. For example, see this article by Rieckers and Güttinger. Morally, it amounts to a reduction to one-dimensional distributions in the variable $B(p)$. A modification of your question which one could look at as a preliminary step is as follows.

Let $V=\{p\ |\ B(p)\ge 0\}$. Are there Lorentz invariant distributions $T$ such $T$ as well as $\widehat{T}$ have support contained in $V$?

Note that there is a rather vast literature on various versions of a "qualitative uncertainty principle" where you impose conditions on the support of a function as well as its Fourier transform. This new question is of this type. Finally, you might want to see if there is an explicit formula for the Rieckers-Güttinger spectral representation of the Fourier transform in terms of that of the original distribution. If so you might have a one-dimensional reduction of the problem, albeit perhaps with a strange 1d Fourier transform.