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Convolution proof in R^n

  1. Jan 21, 2012 #1
    1. The problem statement, all variables and given/known data

    Prove the following: If [tex]\delta\in L_1(\mathbb{R}^n)[/tex] and [tex]f\in L_p(\mathbb{R}^n)[/tex] then the convolution [tex]\delta * f\in L_p(\mathbb{R}^n)[/tex] with [tex]\lVert \delta * f\rVert_p\leq\lVert\delta\rVert_1\lVert f\rVert_p[/tex].

    2. Relevant equations

    We use the natural isometry (or isometric isomorphism, if you like) [tex]h\mapsto\lambda_h[/tex] between [tex]L_q[/tex] and [tex]L_p^*[/tex], where [tex]\frac{1}{p}+\frac{1}{q}=1[/tex], and where we define each [tex]\lambda_h[/tex] by [tex]\lambda_h(f)=\int fh[/tex].

    3. The attempt at a solution

    Well I can show that [tex]\lVert (\delta * f)h\rVert_1\leq\lVert\delta\rVert_1 \lVert f\rVert_p\lVert h\rVert_q[/tex]. Supposedly I can use this along with the natural isometry between [tex]L_q[/tex] and [tex]L_p^*[/tex] to finish the proof. But I don't see how that natural isometry is applicable.

    Any help would be much appreciated. Thanks!
     
  2. jcsd
  3. Jan 22, 2012 #2

    MathematicalPhysicist

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    Gold Member

    I think you need to use here Minkowski inequality theorem, and the fact that dirac-delta measure is invriant under raising its power.

    http://www.math.duke.edu/~wka/math204/conv.12.4.pdf
    it's in page 2, but there's a misprint, and the LHS, the aboslute value integrand should be raised to power of p.
     
  4. Jan 22, 2012 #3
    Thanks, I appreciate the link, and indeed that is a very nice proof of the theorem in question. However, I'm looking to finish the particular approach I was given.

    Basically, I have to show, using the fact that for all [tex]h\in L_q[/tex] we have

    [tex]\lVert (\delta * f)h\rVert_1\leq\lVert\delta\rVert_1 \lVert f\rVert_p\lVert h\rVert_q[/tex]

    and also using the natural isometric isomorphism between Lp* and Lq, that the [tex]\delta *f\in L_p[/tex].
     
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