Sparse Bounds for Maximal Triangle Averaging Operators

We show that the method in recent work of Roncal, Shrivastava, and Shuin can be adapted to show that $L^p$ bounds in the interior of the boundedness region for the triangle averaging operator imply sparse bounds for the lacunary maximal triangle averaging operator, and that $L^p$ bounds in the interior of the boundedness region for the single-scale maximal triangle averaging operator imply sparse bounds for the full maximal triangle averaging operator. The proof uses simultaneous $L^p$-continuity estimates for the triangle averaging operator and its single-scale maximal variant. This shows that the method of Roncal, Shrivastava, and Shuin can be adapted to operators that are not of product type.