This paper presents the properties of Gyrator potential operator G μ l and related L p -Sobolev spaces associated with Gyrator transform (GT). The Schwartz-type space S Δ ( R 2 ) is introduced. Various properties of the kernel of GT is studied including its continuity. Pseudo-differential operators A α , q associated with GT is defined and derived it's continuity on S Δ ( R 2 ) . The Gyrator potential operator G μ l is defined as a pseudo-differential operator associated with a precise symbol and obtained certain fruitful properties. The operator G μ l is extended to a space of distributions. Another integral representation of A α , q is obtained and its L p ( R 2 ) -boundedness result is derived. The spaces H α m , p ( R 2 ) and H Δ m , p ( R 2 ) are defined. It is shown that the operator G μ l is an isometry of H Δ m , p ( R 2 ) . An L p -boundedness result for the operator G μ l is proved. We conclude the article by applying some of the results to show that the analytical solution of certain non-homogeneous partial differential equations belong to these spaces.