On the sense of convergence in the dyadic representation theorem

The dyadic representation of any singular integral operator, as an average of dyadic model operators, has found many applications. While for many purposes it is enough to have such a representation for a “suitable class” of test functions, we show that, under quite general assumptions (essentially minimal ones to make sense of the formula), the representation is actually valid for all pairs $(f,g)\in L^p({\mathbb{R}}^d)\times L^{p^{\mathrm{\prime }}}({\mathbb{R}}^d)$, not just test functions.