Georgia Institute of Technology College of Sciences

For \( c\in(1,2) \) we consider the following operators
\[
\mathcal{C}_{c}f(x) \colon = \sup_{\lambda \in [-1/2,1/2)}
\bigg| \sum_{n \neq 0} f(x-n) \frac{e^{2\pi i\lambda \lfloor |n|^{c} \rfloor}}{n} \bigg|\text{,}
\]
\[
\mathcal{C}^{\mathsf{sgn}}_{c}f(x) \colon = \sup_{\lambda \in [-1/2,1/2)}
\bigg| \sum_{n \neq 0} f(x-n) \frac{e^{2\pi i\lambda \mathsf{sign}(n) \lfloor |n|^{c} \rfloor}}{n} \bigg| \text{,}
\]
and prove that both extend boundedly on \( \ell^p(\mathbb{Z}) \), \( p\in(1,\infty) \). 

The second main result is establishing almost everywhere pointwise convergence for the following ergodic averages
\[
A_Nf(x)\colon =\frac{1}{N}\sum_{n=1}^N f(T^n S^{\lfloor n^c\rfloor} x) \text{,}
\]
where $T,S\colon X\to X$ are commuting measure-preserving transformations on a  $\sigma$-finite measure space $(X,\mu)$, and $f\in L_{\mu}^p(X), p\in(1,\infty)$. 

The point of departure for both proofs is the study of exponential sums with phases  $\xi_2 \lfloor |n^c|\rfloor+ \xi_1n$ through the use of a simple variant of the circle method.

This talk is based on joint work with Leonidas Daskalakis.
 

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 $E\subseteq [1,2]$.