This paper studies the fractional Laplacian and its relation to extension problems in the upper half space. The operator $ (-\triangle)^{1/2} $ can be obtained from the harmonic extension problem, where the Dirichlet boundary condition maps to the Neumann condition. The authors generalize this to fractional powers of the Laplacian and other integro-differential operators. They show that these operators can be characterized through extension problems, and derive properties of these equations using local arguments. The fractional Laplacian $ (-\triangle)^s $ is defined as a pseudo-differential operator, and can also be expressed as an integral operator. The paper relates the fractional Laplacian to solutions of an extension problem in $ n+1+a $ dimensions. The equation $ \triangle_x u + \frac{a}{y}u_y + u_{yy} = 0 $ is shown to be equivalent to the fractional Laplacian for $ s = \frac{1-a}{2} $. The authors derive a Poisson formula for the extension problem, and show that the solution to the extension problem corresponds to the fractional Laplacian. They also prove a Harnack inequality and a boundary Harnack inequality for the fractional Laplacian, using local PDE methods. These inequalities are derived from the Harnack inequality for singular elliptic equations. The paper also discusses monotonicity formulas for solutions of the extension problem, and shows that Almgren's frequency formula applies to these solutions. The authors also consider other integro-differential operators and show that they can be obtained from extension problems with appropriate boundary conditions. The paper concludes with a discussion of the relationship between the fractional Laplacian and other operators, and the possibility of extending the results to more general classes of equations. The authors also mention the importance of the regularity of solutions and the role of the Harnack inequality in understanding the behavior of solutions to these equations.This paper studies the fractional Laplacian and its relation to extension problems in the upper half space. The operator $ (-\triangle)^{1/2} $ can be obtained from the harmonic extension problem, where the Dirichlet boundary condition maps to the Neumann condition. The authors generalize this to fractional powers of the Laplacian and other integro-differential operators. They show that these operators can be characterized through extension problems, and derive properties of these equations using local arguments. The fractional Laplacian $ (-\triangle)^s $ is defined as a pseudo-differential operator, and can also be expressed as an integral operator. The paper relates the fractional Laplacian to solutions of an extension problem in $ n+1+a $ dimensions. The equation $ \triangle_x u + \frac{a}{y}u_y + u_{yy} = 0 $ is shown to be equivalent to the fractional Laplacian for $ s = \frac{1-a}{2} $. The authors derive a Poisson formula for the extension problem, and show that the solution to the extension problem corresponds to the fractional Laplacian. They also prove a Harnack inequality and a boundary Harnack inequality for the fractional Laplacian, using local PDE methods. These inequalities are derived from the Harnack inequality for singular elliptic equations. The paper also discusses monotonicity formulas for solutions of the extension problem, and shows that Almgren's frequency formula applies to these solutions. The authors also consider other integro-differential operators and show that they can be obtained from extension problems with appropriate boundary conditions. The paper concludes with a discussion of the relationship between the fractional Laplacian and other operators, and the possibility of extending the results to more general classes of equations. The authors also mention the importance of the regularity of solutions and the role of the Harnack inequality in understanding the behavior of solutions to these equations.