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

We prove that the solution map for a family of non-linear transport equations in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="double-struck upper R Superscript n"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="double-struck">R</mml:mi> </mml:mrow> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> <mml:annotation encoding="application/x-tex">{\mathbb {R}}^n</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="minus left-parenthesis n minus 1 right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo> − </mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo> − </mml:mo> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">-(n-1)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> 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.