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On page 92 of these notes, there is a discussion on how to find the fundamental solution to the D'Alembertian operator. It is firstly proposed that $c_n(t^2 - |x|^2 )^{-(n-1)/2}$ may be a good candidate. Then the author says that we can use the homogeneous family of distributions $j_a$ to get a homogeneous distribution that agrees with the above function in $t>|x|$.

However, I am unable to find the definition of $j_a.$

Could anyone give me a reference/clue on what result we are using here?

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  • $\begingroup$ These are Riesz distributions. They are mentioned in these answers, with references. The main one is Gelfand-Shilov Generalized Functions I. $\endgroup$
    – Igor Khavkine
    Commented May 3, 2021 at 1:10

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In sergiu's notes that you referred to, $j_a$ is defined in Definition 3.2 on Page 65. See equation (134).

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  • $\begingroup$ Thank you very much for pointing out! Somehow I have overlooked this. $\endgroup$
    – Ma Joad
    Commented May 3, 2021 at 7:25
  • $\begingroup$ No worries; that particular missing cross reference was already a small "bug" when I was a student 15 years ago. $\endgroup$
    – Willie Wong
    Commented May 3, 2021 at 13:47

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