Sparse bounds for maximal rough singular integrals via the fourier transform

We prove a quantified sparse bound for the maximal truncations of convolution-type singular integrals with suitable Fourier decay of the kernel. Our result extends the sparse domination principle by Conde-Alonso, Culiuc, Ou and the first author to the maximally truncated case, and covers the rough homogeneous singular integrals TΩ on Rd with bounded angular part Ω having vanishing integral on the sphere. Among several consequences, we obtain new quantitative weighted norm inequalities for the maximal truncation of TΩ, extending a result by Roncal, Tapiola and the second author. A convex-body valued version of the sparse bound is also deduced and employed towards novel matrix-weighted norm inequalities for the maximal truncations of TΩ. Our result is quantitative, but even the qualitative statement is new, and the present approach via sparse domination is the only one currently known for the matrix weighted bounds of this class of operators.