In this talk, we explore the continuity properties of the solution map, in Hölder and Zygmund spaces, to a class of nonlinear transport equations in R^n. The velocity field in these equations is given by the convolution of the density with a kernel that is homogeneous of degree -(n-1) and smooth away from the origin. This setting encompasses significant models, including the 2D Euler equations and the 3D surface quasi-geostrophic (SQG) equations.