Sparse bounds for maximal triangle and bilinear spherical averaging operators

We show that the method in recent work of Roncal, Shrivastava, and Shuin can be adapted to show that certain $L^p$-improving bounds in the interior of the boundedness region for the bilinear spherical or triangle averaging operator imply sparse bounds for the corresponding lacunary maximal operator, and that $L^p$-improving bounds in the interior of the boundedness region for the corresponding single-scale maximal operators imply sparse bounds for the correpsonding full maximal operators. More generally we show that the framework applies for bilinear convolutions with compactly supported finite Borel measures that satisfy appropriate $L^p$-improving and continuity estimates. This shows that the method used by Roncal, Shrivastava, and Shuin can be adapted to obtain sparse bounds for a general class of bilinear operators that are not of product type, for a certain range of $L^p$ exponents.