# Associative convolution on p-adic distribution

Let $\mathcal{D}(\mathbb{Q}_p)$ be the space of the locally constant functions with compact support and let $\mathcal{D}'(\mathbb{Q}_p)$ be the space of distributions: linear functionals on $\mathcal{D}(\mathbb{Q}_p)$. In the book of "p-adic Analysis and Mathematical Physics", a convolution between $f,g\in \mathcal{D}'(\mathbb{Q}_p)$, for all $\phi \in \mathcal{C}$, is defined as:

$$\langle f*g,\phi \rangle =\lim\limits_{k\to \infty} \langle f(x), \langle g(y),1_{B(0,p^k)}\phi(x+y) \rangle \rangle,$$

if the limit exists. It is commutative.

However I want to know if it is associative, if not, is there any convolution defined on the space of distribution which is associative? Thanks.

It is associative, if you assume the distributions have support in the ball $B_N$.
It is a rather reasonable assumption, since if $f,g\in \mathcal{D}'(\mathbb{Q}_p)$ and $\mathbb{supp}$ $g \subset B_N$, then the convolution always exists.