This paper studies the regularity of solutions to the critical dissipative quasi-geostrophic equation and drift-diffusion equations with fractional diffusion. The authors prove that solutions to the quasi-geostrophic equation with $ L^2 $ initial data and critical diffusion $ (-\Delta)^{1/2} $ are locally smooth in any space dimension. They also show that drift-diffusion equations with $ L^2 $ initial data and minimal assumptions on the drift are locally Hölder continuous. The key results are: 1. **Theorem 1**: Solutions to the quasi-geostrophic equation with $ L^2 $ initial data and critical diffusion $ (-\Delta)^{1/2} $ are uniformly bounded in $ L^\infty $ for any time $ t > 0 $. 2. **Theorem 2**: Solutions to the quasi-geostrophic equation with $ L^2 $ initial data and critical diffusion $ (-\Delta)^{1/2} $ are locally Hölder continuous in any space dimension. 3. **Theorem 3**: Solutions to the quasi-geostrophic equation with $ L^2 $ initial data and critical diffusion $ (-\Delta)^{1/2} $ are locally Hölder continuous in any space dimension. The proofs rely on energy inequalities and the use of harmonic extensions and singular integral operators. The authors also show that the regularity of solutions to the quasi-geostrophic equation follows from these results. The paper also discusses the connection between the quasi-geostrophic equation and other models in fluid mechanics and provides a detailed analysis of the regularity properties of solutions.This paper studies the regularity of solutions to the critical dissipative quasi-geostrophic equation and drift-diffusion equations with fractional diffusion. The authors prove that solutions to the quasi-geostrophic equation with $ L^2 $ initial data and critical diffusion $ (-\Delta)^{1/2} $ are locally smooth in any space dimension. They also show that drift-diffusion equations with $ L^2 $ initial data and minimal assumptions on the drift are locally Hölder continuous. The key results are: 1. **Theorem 1**: Solutions to the quasi-geostrophic equation with $ L^2 $ initial data and critical diffusion $ (-\Delta)^{1/2} $ are uniformly bounded in $ L^\infty $ for any time $ t > 0 $. 2. **Theorem 2**: Solutions to the quasi-geostrophic equation with $ L^2 $ initial data and critical diffusion $ (-\Delta)^{1/2} $ are locally Hölder continuous in any space dimension. 3. **Theorem 3**: Solutions to the quasi-geostrophic equation with $ L^2 $ initial data and critical diffusion $ (-\Delta)^{1/2} $ are locally Hölder continuous in any space dimension. The proofs rely on energy inequalities and the use of harmonic extensions and singular integral operators. The authors also show that the regularity of solutions to the quasi-geostrophic equation follows from these results. The paper also discusses the connection between the quasi-geostrophic equation and other models in fluid mechanics and provides a detailed analysis of the regularity properties of solutions.