Wm,p Compactness of Stationary Kinetic Equation

We study stationary kinetic equations and derive compactness results in spaces. More precisely, we show that if a solution sequence is locally relatively compact in the velocity variable , then it is also locally relatively compact in the ‐space, where denotes the spatial variable. To prove this result, we established a Fourier‐frequency‐based characterization of compactness within the framework, combining it with Sobolev space techniques, refined estimates, approximation arguments, and the Fourier multiplier theorem.