$L^{p}-$estimates for uncentered spherical averages and lacunary maximal functions

Abstract: The primary goal of this paper is to introduce bilinear analogues of uncentered spherical averages, Nikodym averages associated with spheres and the associated bilinear maximal functions. We obtain $L^p$-estimates for uncentered bilinear maximal functions for dimensions $d\geq2$. Moreover, we also discuss the one-dimensional case. In the process of developing these results, we also establish new and interesting results in the linear case. In particular, we will prove $L^p$-improving properties for single scale averaging operators and $L^p$-estimates for lacunary maximal functions in this context.