In this paper, we study the ${{L}^{p}}$ mapping properties of a class of singular integral operators with rough kernels belonging to certain block spaces. We prove that our operators are bounded on ${{L}^{p}}$ provided that their kernels satisfy a size condition much weaker than that for the classical Calderón–Zygmund singular integral operators. Moreover, we present an example showing that our size condition is optimal. As a consequence of our results, we substantially improve a previously known result on certain maximal functions.

We prove a weak-type (1,1) inequality for square functions of non-commutative martingales that are simultaneously bounded in $L^2$ and $L^1$. More precisely, the following non-commutative analogue of a classical result of Burkholder holds: there exists an absolute constant $K > 0$ such that if $\mathcal{M}$ is a semi-finite von Neumann algebra and $( \mathcal{M}_n )^{ \infty }_{n = 1}$ is an increasing filtration of von Neumann subalgebras of $\mathcal{M}$ then for any given martingale $x = ( x_n )^{\infty}_{n = 1}$ that is bounded in $L^2 ( \mathcal{M} ) \cap L^1 ( \mathcal{M} )$, adapted to $( \mathcal{M}_n )^{\infty}_{n = 1}$, there exist two martingale difference sequences, $a = ( a_n )_{n = 1}^\infty$ and $b = ( b_n )_{n = 1}^\infty$, with $dx_n = a_n + b_n$ for every $n \geq 1$,

\| ( \sum^\infty_{n = 1} a_n^* a_n )^{1/2} \|_{2} + \| ( \sum^\infty_{n = 1} b_n b_n^* )^{1/2} \|_{2} \leq 2 \| x \|_2,

and

\| ( \sum^\infty_{n = 1} a_n^* a_n )^{1/2} \|_{1, \infty} + \| ( \sum^\infty_{n = 1} b_n b_n^* )^{1/2} \|_{1, \infty} \leq K \| x \|_1.

As an application, we obtain the optimal orders of growth for the constants involved in the Pisier–Xu non-commutative analogue of the classical Burkholder–Gundy inequalities.