= Convolution of Sobolev distributions
{title2=$u*v=\mathcal F^{-1}(\widehat u\widehat v)$}
= Convolution of elements of Sobolev spaces
{synonym}
For arbitrary real $s,t$, the <convolution> of $u\in H^s(\mathbb R)$ and $v\in H^t(\mathbb R)$ has a canonical definition as a <tempered distribution>. With $\langle\xi\rangle=(1+\xi^2)^{1/2}$, write $A=\langle\xi\rangle^s\widehat u$ and $B=\langle\xi\rangle^t\widehat v$, both in the <L2 space>. Then $AB\in L^1$ by <Cauchy-Schwarz inequality>, and
$$
|\langle\widehat u\widehat v,\phi\rangle|
\leq\|u\|_{H^s}\|v\|_{H^t}\sup_\xi\langle\xi\rangle^{-s-t}|\phi(\xi)|.
$$
Thus the product defines a <tempered distribution>, and the displayed <inverse Fourier transform> defines $u*v$. This extends ordinary <convolution> of <Schwartz functions> by the <convolution theorem> and density in the <Sobolev norm>.
The result need not belong to any <Sobolev space>, even when both inputs belong to every <Sobolev space>. For $\widehat u=\widehat v=\xi^{-1/3}\mathbf1_{(0,1)}(\xi)$, each weighted squared input is integrable, but the squared product is $\xi^{-4/3}\mathbf1_{(0,1)}(\xi)$, which is not locally integrable at zero. Every weight $(1+\xi^2)^r$ is bounded above and below there, proving the failure for all real $r$.
Back to article page
