This talk is about an application of complex function theory to inverse spectral problems for differential operators. We consider the Schroedinger operator on a finite interval with an L^1-potential. Borg's two spectra theorem says that the potential can be uniquely recovered from two spectra. By another classical result of Marchenko, the potential can be uniquely recovered from the spectral measure or Weyl m-function.
Back in the year 2000, Christ and Kiselev introduced a useful "maximal trick" in their study of spectral properties of Schro edinger operators.
The trick was completely abstract and only at the level of basic functional analysis and measure theory. Over the years it was reproven,
generalized, and reused by many authors. We will present its recent application in the theory of restriction of the Fourier transform to
surfaces in the Euclidean space.
Let $f$ be defined on $\mathbb{Z}$. Let $A_N f$ be the average of $f$ along the square integers.
Brascamp-Lieb inequalities are estimates for certain multilinear forms on functions on Euclidean spaces. They generalize several classical inequalities, such as Hoelder's inequality or Young's convolution inequality. In this talk we consider singular Brascamp-Lieb inequalities, which arise when one of the functions in the Brascamp-Lieb inequality is replaced by a singular integral kernel. Examples include multilinear singular integral forms such as paraproducts or the multilinear Hilbert transform. We survey some results in the area.
Fine properties of spherical averages in the continuous setting include
$L^p$ improving estimates
and sparse bounds, interesting in the settings of a fixed radius, lacunary sets of radii, and the
full set of radii. There is a parallel theory in the setting of discrete spherical averages, as studied
by Elias Stein, Akos Magyar, and Stephen Wainger. We recall the continuous case, outline the
discrete case, and illustrate a unifying proof technique. Joint work with Robert Kesler, and
Dario Mena Arias.
Mathematicians have long been trying to understand which domains admit an orthogonal (or, sometimes, not) basis of exponentials of the form , for some set of frequencies (this is the spectrum of the domain). It is well known that we can do so for the cube, for instance (just take ), but can we find such a basis for the ball? The answer is no, if we demand orthogonality, but this problem is still open when, instead of orthogonality, we demand just a Riesz basis of exponentials.
If $f$ is a function supported on a truncated paraboloid, what can we say about $Ef$, the Fourier transform of f? Stein conjectured in the 1960s that for any $p>3$, $\|Ef\|_{L^p(R^3)} \lesssim \|f\|_{L^{\infty}}$.
We make a small progress toward this conjecture and show that it holds for $p> 3+3/13\approx 3.23$. In the proof, we combine polynomial partitioning techniques introduced by Guth and the two ends argument introduced by Wolff and Tao.
