Fractional Laplacian And Kato Inequality

The fractional Laplacian can be obtained by using method. From the harmon-ic extension problem to the upper half space. Luis Caffarelli-Luis Silvestre charac-terize the fractional Laplacian as the operator that maps the Dirichlet boundary condition to the Neumann condition. From those characterizations, they prove the equivalence of three definitions of the fractional Laplacian and establish the Hanarck inequality of the the fractional Laplacian. Ray Yang popularize the three definitions of the fractional Laplacian to the higher order. In order to study the properties of the solutions to the equation of the fractional Laplace, we verify the fundamental solution of (-△)s,s∈(1,2)The Kato’s inequalities have kinds of forms and are applied many places. Especially, The Kato’s inequalities play an important role in proving existence and uniqueness of solutions, equivalence of norm, self-adjointness of operators. In order to study the the essential self-adjointness of the Schrodinger operators with singular potentials, Tosis Kato establish the Kato’s inequality of the △ based on the theory of B.Silmon. In this article, we establish the Kato’s inequality of the (-△)s, s∈(0,1) on which we use basic tools and methods such as maximum principle, the structure function method.