Sparse Bounds for Maximal Triangle and Bilinear Spherical Averaging Operators

Abstract We show that the method in the recent work by Roncal, Shrivastava, and Shuin can be adapted to show that certain $$L^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> -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 $$L^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> -improving bounds in the interior of the boundedness region for the single-scale maximal bilinear spherical averaging operator implies sparse bounds for the corresponding full maximal operator. More generally we show that the proof applies for bilinear convolutions with compactly supported finite Borel measures that satisfy appropriate $$L^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> -improving and continuity estimates. This shows that the method used by Roncal, Shrivastava, and Shuin can be adapted obtain sparse bounds for a general class of bilinear operators that are not of product type, for a certain range of $$L^p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mi>p</mml:mi> </mml:msup> </mml:math> exponents.