Convolution of distributions with a compactly supported factor ($u\ast v$)

If $v$ is a compactly supported distribution and $u$ is any distribution, define $⟨u\ast v,\varphi ⟩=⟨u_x,⟨v_y,\varphi (x+y)⟩⟩$. The inner function is smooth and supported in $\mathrm{supp}\varphi -\mathrm{supp}v$, hence is a test function for $u$. Finite-order estimates prove continuity, while the tensor product of distributions proves commutativity after inserting compact cutoffs. For every test function $\psi$, $(u\ast v)\ast \psi =u\ast (v\ast \psi )$; evaluating at zero against reflected test functions proves uniqueness. If both distributions have compact support, $\mathrm{supp}(u\ast v)\subset \mathrm{supp}u+\mathrm{supp}v$ and the convolution theorem gives $\widehat{u\ast v}=\hat{u}\hat{v}$. Without a compact factor or another support/growth condition, arbitrary distributional convolution is not generally defined.

If at least one of $u,v\in {\mathcal{D}}^{\prime }({\mathbb{R}}^n)$ is a compactly supported distribution, thenThe sums are Minkowski sums. The first rule follows from the pairing $⟨u_x\otimes v_y,\varphi (x+y)⟩$: a test function away from the support sum gives zero. For the second rule, localize the output to a small compact set avoiding the sum of the singular supports. The relevant pairs of input points form a compact set because one input support is compact. Cover it by finitely many product neighborhoods on each of which at least one input distribution is smooth; otherwise the output would meet the sum of the singular supports. Smooth cutoffs in these neighborhoods decompose the localized pairing into finitely many convolutions with a smooth compactly supported factor. Each is smooth by smoothing convolution with a test function. This proves the second rule locally.

The same argument works when addition is a proper map on the product of the two supports. Some condition defining the convolution is necessary; arbitrary pairs of distributions cannot always be convolved.

Target: Cauchy kernel fundamental solution. Parent: Fundamental solution of a linear differential operator. Intended file: distribution-theory.bigb.