This paper by Luis Caffarelli and Luis Silvestre explores the fractional Laplacian and other integro-differential operators through extension problems. The authors derive characterizations for these operators by solving extension problems in the upper half-space, relating them to harmonic functions in higher dimensions. They show that the fractional Laplacian can be expressed as the limit of a sequence of derivatives of solutions to these extension problems. The paper also discusses the properties of the resulting partial differential equations (PDEs), including harmonic functions in \(n+1+a\) dimensions, fundamental solutions, and conjugate equations. Additionally, it presents Poisson formulas and proves Harnack and boundary Harnack inequalities for the fractional Laplacian using these PDEs. The authors further extend their results to other integro-differential operators, suggesting that many can be understood in a similar manner. The paper concludes with a discussion on the realizability of various symbols as pseudodifferential operators.This paper by Luis Caffarelli and Luis Silvestre explores the fractional Laplacian and other integro-differential operators through extension problems. The authors derive characterizations for these operators by solving extension problems in the upper half-space, relating them to harmonic functions in higher dimensions. They show that the fractional Laplacian can be expressed as the limit of a sequence of derivatives of solutions to these extension problems. The paper also discusses the properties of the resulting partial differential equations (PDEs), including harmonic functions in \(n+1+a\) dimensions, fundamental solutions, and conjugate equations. Additionally, it presents Poisson formulas and proves Harnack and boundary Harnack inequalities for the fractional Laplacian using these PDEs. The authors further extend their results to other integro-differential operators, suggesting that many can be understood in a similar manner. The paper concludes with a discussion on the realizability of various symbols as pseudodifferential operators.