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Let $f$ and $g$ be distributions on $\mathbb{R}$ with compact support. Do we have

$\inf (\textrm{supp}(f*g)) = \inf (\textrm{supp}(f)) + \inf (\textrm{supp}(g))$

Where 'supp' denotes de the support of a distribution ?

The left term is obviously greater than the right one. But the other inequality seems trickier to me (I guess it's always easier to prove that things are equal to zero). Any help in finding either a counterexample, proof, or reference for that statement would be appreciated. Thanks.

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In what sense is the infimum? –  Glen Wheeler May 6 '11 at 6:37
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@Glen: What is ambigious about the infimum of a compact subset of $\mathbb R$? –  Rasmus May 6 '11 at 7:05
    
@Rasmus One might have interpreted this to be the infimum of a set of sets. Anyway, I guess that is not meant, and my question is irrelevant. –  Glen Wheeler May 6 '11 at 12:34
    
@Rasmus : That is indeed what I meant. –  Joel Cohen May 6 '11 at 14:29
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1 Answer

up vote 5 down vote accepted

You might wish to take a look at the Titchmarsh Convolution Theorem and its generalization due to Lions, which states that your equation holds. A proof can be found in Hörmander's Analysis of linear partial differential operators I, section 4.3.

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Thanks, this is exactly what I was looking for. –  Joel Cohen May 6 '11 at 14:31
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