Title: Weighted inequalities for oscillatory integrals
Abstract: Motivated by seemingly difficult weighted conjectures for the disc multiplier and Fourier extension operators (originating in work of Córdoba, Fefferman and Stein from the 1970s) we investigate general-weighted inequalities of the form
$$
\int|Tf|^2w\lesssim\int|f|^2\mathcal{M}w
$$
for various oscillatory integrals $T$ and maximal averaging operators $\mathcal{M}$ on $\mathbb{R}^n$. Naturally the controlling maximal operators $\mathcal{M}$ are highly non-local in oscillatory contexts, involving averages over tubes, or wide approach regions. Our main results are joint work with David Beltran.