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Math Help - Elliptic Regularity Theorem

  1. #1
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    Elliptic Regularity Theorem

    Hi there,

    I am following a proof of the elliptic regularity theorem, and I am having a few troubles in understanding where some of the workings come from.

    I want to first prove that sing supp (u * v) is a subset of sing supp u + sing supp v.

    The proof goes as follows:

    Choose functions p (infinitely differentiable) and psi (infinitely differentiable with compact support) such that p = 1 on a neighbourhood of sing supp u and psi = 1 on a neighbourhood of sing supp v. Then:

    (u * v) = (pu + (1 - p)u) * (psiv + (1 - psi)v) = pu * psiv + pu * (1 - psi)v + (1 - p)u * psiv + (1 - p)u * (1 - psi)v.

    Each of the convolutions other than pu * psiv has at least one infinitely differentiable factor, so is infinitely differentiable.

    Then, apparently, it follows that:

    sing supp (u * v) is a subset of supp pu + supp psiv which is a subset of supp p + supp psi.

    How does this follow? I cannot see it.
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  2. #2
    Super Member Rebesques's Avatar
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    Re: Elliptic Regularity Theorem

    Test functions (like p, smooth and compactly supported) belong to the Schwartz space.
    Last edited by Rebesques; February 12th 2015 at 02:31 PM.
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