Georgia Institute of Technology College of Sciences

Let $S^{2d-1}$ be the unit sphere in $\mathbb{R}^{2d}$, and $\sigma_{2d-1}$ the normalized spherical measure in $S^{2d-1}$. The (scale t) bilinear spherical average is given by 
$$\mathcal{A}_{t}(f,g)(x):=\int_{S^{2d-1}}f(x-ty)g(x-tz)\,d\sigma_{2d-1}(y,z).$$
There are geometric motivations to study bounds for such bilinear spherical averages, in connection to the study of some Falconer distance problem variants. Sobolev smoothing bounds for the operator 
$$\mathcal{M}_{[1,2]}(f,g)(x)=\sup_{t\in [1,2]}|\mathcal{A}_{t}(f,g)(x)|$$
 are also relevant to get bounds for the bilinear spherical maximal function
$$\mathcal{M}(f,g)(x):=\sup_{t>0} |\mathcal{A}_{t}(f,g)(x)|.$$
In a joint work with B. Foster and Y. Ou, we put that in a general framework where $S^{2d-1}$ can be replaced by more general smooth surfaces in $\mathbb{R}^{2d}$, and one can allow more general dilation sets in the maximal functions: instead of supremum over $t>0$, the supremum can be taken over $t\in \tilde{E}$ where $\tilde{E}$ is the set of all scales obtained by dyadic dilation of fixed set of scales $E\subseteq [1,2]$.