Twisted bilinear spherical maximal functions

We obtain $L^p-$estimates for the full and lacunary maximal functions associated to the twisted bilinear spherical averages given by $${\mathfrak{A}}_t(f_1,f_2)(x,y)={\int }_{{\mathbb{S}}^{2d-1}}{f}_1(x+t{z}_1,y)f_2(x,y+t{z}_2)d\sigma (z_1,z_2),t>0,$$ for all dimensions $d\ge 1$. We show that the estimates for such operators in dimensions $d\ge 2$ essentially relies on the method of slicing. The bounds for the lacunary maximal function in dimension one is more delicate and requires a trilinear smoothing inequality which is based on an appropriate sublevel set estimate in this context.