(adsbygoogle = window.adsbygoogle || []).push({}); 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!

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# Convolution proof in R^n

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