Associative convolution on p-adic distribution

Question

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.

Answer 1

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.

You can fin a somewhat detailed discussion of this in the book "Theory of p-adic distributions: linear and nonlinear models", by Albeverio, Khrennikov and Shelkovich (2010), particularly chapter 4.