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 \[\mathfrak{A}_t(f_1,f_2)(x,y)=\int_{\mathbb S^{2d-1}}f_1(x+tz_1,y)f_2(x,y+tz_2)\;d\sigma(z_1,z_2),\;t>0,\] for all dimensions \(d\geq1\). We show that the estimates for such operators in dimensions \(d\geq2\) 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.