Sovine, Sean Russell2022-05-132022-05-132022-05-12vt_gsexam:34618http://hdl.handle.net/10919/110073This thesis contains work from the author's papers Palsson and Sovine (2020); Iosevich, Palsson, and Sovine (2022); and Palsson and Sovine (2022) with coauthors Eyvindur Palsson and Alex Iosevich. These works establish new $L^p$-improving, quasi-Banach, and sparse bounds for several bilinear and multilinear operators that generalize the linear spherical average to the multilinear setting, and maximal variants of these operators, with an emphasis on the triangle averaging operator and the bilinear spherical averaging operator.ETDenIn Copyrightharmonic analysismultilinear operatorsgeometric averagesDissertation