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Given a hypersurface $S\subset \mathbb{R}^{2d}$, we study the bilinearaveraging operator that averages a pair of functions over $S$, as well as moregeneral bilinear multipliers of limited decay and various maximal analogs. Ofparticular interest are bilinear maximal operators associated to a fractaldilation set $E\subset [1,2]$; in this case, the boundedness region of themaximal operator is associated to the geometry of the hypersurface and variousnotions of the dimension of the dilation set. In particular, we determineSobolev smoothing estimates at the exponent $L^2 \times L^2 \rightarrow L^2$using Fourier-analytic methods, which allow us to deduce additional $L^p$improving bounds for the operators and sparse bounds and their weightedcorollaries for the associated multi-scale maximal functions. We also extendthe method to study analogues of these questions for the triangle averagingoperator and biparameter averaging operators. In addition, some necessaryconditions for boundedness of these operators are obtained.

Sobolev smoothing estimates for bilinear maximal operators with fractal dilation sets