  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.
