  In this note, we address the local well-posedness for the active scalar
equation $\partial_t \theta + u\cdot \nabla \theta =0$, where $u = -
\nabla^\perp(-\Delta)^{-1+\beta/2}\theta$. The local existence of solutions in
the Sobolev class $H^{1+\beta+\epsilon}$, where $\epsilon>0$ and $\beta \in
(1,2)$, has been recently addressed in \cite{HKZ}. The critical case $\epsilon
=0$ has remained open. Using a different technique, we prove the local
well-posedness in the Besov space $B^{1+\beta}_{2,1}$, where $\beta \in (1,2)$.
The proof is based on log-Lipschitz estimates for the transport equation.
