Let $\mathcal{D}(\mathbb{Q}_p)$ be the space of the locally constant functions with compact support and let $\mathcal{D}'$ be the space of distributions: linear functionals on $\mathcal{D}$. In the book of "P-ADIC ANALYSIS AND MATHEMATICAL PHYSICS", it defined a convolution between $f,g\in \mathcal{D}'$, for all $\phi \in \mathcal{C}$, $$ \langle f*g,\phi \rangle =\lim\limits_{k\to \infty} \langle f(x), \langle g(y),1_{B(0,p^k)}\phi(x+y) \rangle \rangle, $$ 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.

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.