Continuity of the solution map of some active scalar equations in Hölder and Zygmund spaces

We prove that the solution map for a family of non-linear transport equations in ${\mathbb{R}}^n$, with a velocity field given by the convolution of the density with a kernel that is smooth away from the origin and homogeneous of degree $-(n-1)$, is continuous in both the little Hölder class and the little Zygmund class. For particular choices of the kernel, one recovers well-known equations such as the 2D Euler or the 3D quasi-geostrophic equations.