In this article, we prove a stability estimate going from the Radon transform of a function with limited angle-distance data to the $L^p$ norm of the function itself, under some conditions on the support of the function. We apply this theorem to obta
in stability estimates for an inverse boundary value problem with partial data.
The direct detection of continuous gravitational waves from pulsars is a much anticipated discovery in the emerging field of multi-messenger gravitational wave (GW) astronomy. Because putative pulsar signals are exceedingly weak large amounts of data
need to be integrated to achieve desired sensitivity. Contemporary searches use ingenious ad-hoc methods to reduce computational complexity. In this paper we provide analytical expressions for the Fourier transform of realistic pulsar signals. This provides description of the manifold of pulsar signals in the Fourier domain, used by many search methods. We analyze the shape of the Fourier transform and provide explicit formulas for location and size of peaks resulting from stationary frequencies. We apply our formulas to analysis of recently identified outlier at 1891.76 Hz.
We obtain new local smoothing estimates for the Euclidean wave equation on $mathbb{R}^{n}$, by replacing the space of initial data by a Hardy space for Fourier integral operators. This improves the bounds in the local smoothing conjecture for $pgeq 2
(n+1)/(n-1)$, and complements them for $2<p<2(n+1)/(n-1)$. These estimates are invariant under application of Fourier integral operators.
We study the Cauchy problem for the 3D Gross-Pitaevskii equation. The global well-posedness in the natural energy space was proved by Gerard cite{Gerard}. In this paper we prove scattering for small data in the same space with some additional angular
regularity, and in particular in the radial case we obtain small energy scattering.