Rotational smoothing is a phenomenon consisting in a gain of regularity by means of averaging over rotations. This phenomenon is present in operators that regularize only in certain directions, in contrast to operators regularizing in all directions.
The gain of regularity is the result of rotating the directions where the corresponding operator performs the smoothing effect. In this paper we carry out a systematic study of the rotational smoothing for a class of operators that includes $k$-vector-space Riesz potentials in $mathbb{R}^n$ with $k < n$, and the convolution with fundamental solutions of elliptic constant-coefficient differential operators acting on $k$-dimensional linear subspaces. Examples of the latter type of operators are the planar Cauchy transform in $mathbb{R}^n$, or a solution operator for the transport equation in $mathbb{R}^n$. The analysis of rotational smoothing is motivated by the resolution of some inverse problems under low-regularity assumptions.
We study the inverse scattering problem of determining a magnetic field and electric potential from scattering measurements corresponding to finitely many plane waves. The main result shows that the coefficients are uniquely determined by $2n$ measur
ements up to a natural gauge. We also show that one can recover the full first order term for a related equation having no gauge invariance, and that it is possible to reduce the number of measurements if the coefficients have certain symmetries. This work extends the fixed angle scattering results of Rakesh and M. Salo to Hamiltonians with first order perturbations, and it is based on wave equation methods and Carleman estimates.
