On well-posedness of $\alpha$-SQG equations in the half-plane

We investigate the well-posedness of $\alpha$-SQG equations in the half-plane, where $\alpha=0$ and $\alpha=1$ correspond to the 2D Euler and SQG equations respectively. For $0<\alpha \le 1/2$, we prove local well-posedness in certain weighted anisotropic H\"older spaces. We also show that such a well-posedness result is sharp: for any $0<\alpha \le 1$, we prove nonexistence of H\"older regular solutions (with the H\"older regularity depending on $\alpha$) for initial data smooth up to the boundary.