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Twisted bilinear spherical maximal functions (2410.17583v1)
Published 23 Oct 2024 in math.CA
Abstract: We obtain $Lp-$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.