Survey on Bilinear Spherical Averages and Associated Maximal Operators

Abstract In this survey, we collect recent progress in the understanding 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> bounds for bilinear spherical averages and some associated maximal functions like the bilinear spherical maximal function and its lacunary counterpart. We describe necessary conditions satisfied by triples in the $$L^{p}$$ <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 region of a bilinear spherical averaging operator and the localized bilinear spherical maximal function, as well as describe the best-known boundedness regions to date. We state some open questions along the way to motivate future research on this topic, and we exploit some possible generalizations.