Let $ n \in \mathbb{N} $ and $ p \in \mathbb{R}_{\geq 1} $. If $ f \in \mathscr{S}(\mathbb{R}^{n}) $ and $ g \in {L^{p}}(\mathbb{R}^{n}) $, then it is a well-known fact from real analysis that the convolution $ f \star g $ is defined almost everywhere on $ \mathbb{R}^{n} $ and that $$ \| f \star g \|_{p} \leq \| f \|_{1} \| g \|_{p}. \quad (\text{Young's Inequality}) $$ This implies that $ f \star g \in {L^{p}}(\mathbb{R}^{n}) $.

My question: Is it true that $ f \star g $ is a smooth $ L^{p} $-function?

Thank you very much for your help!

  • 4
    $\begingroup$ Yes, the convolution is smooth. To see that this is so, you should convince yourself that you can move the derivative of $f \ast g$ on to $f$ or $g$. Since $f$ is smooth, you can differentiate the convolution by moving the derivatives to $f$ and using the fact that it is differentiable. $\endgroup$ – Elchanan Solomon Nov 17 '13 at 6:30
  • $\begingroup$ @IsaacSolomon Dear Solomon, a related post is here, you may be interested, math.stackexchange.com/questions/1509579/… $\endgroup$ – user186225 Nov 2 '15 at 16:12

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