The notation W^k,p refers to a specific type of function space known as a Sobolev space. These spaces are used in mathematical analysis, particularly in the study of partial differential equations. The parameters k and p indicate the order of derivatives and the integrability condition, respectively. Functions in W^k,p have derivatives up to order k that are p -integrable, meaning their p -th power is integrable over a given domain. Sobolev spaces, including W^k,p , are essential in various fields such as functional analysis and numerical analysis. They provide a framework for understanding the behavior of functions and their derivatives, allowing for the treatment of weak solutions to differential equations. This makes W^k,p crucial for both theoretical studies and practical applications in physics and engineering.