This book presents important analytic and geometric techniques to prove regularity estimates for solutions to second order elliptic equations, both in divergence and nondivergence form, and to nonlocal equations driven by the fractional Laplacian.

1. Introduction. 1.1. Divergence Form Equations. 1.2. Nondivergence Form Equations. 1.3. Nonlocal Equations: The Fractional Laplacian. Section I. The Laplacian. 2. Harmonic Functions. 2.1. Definition and Examples. 2.2. The Mean Value Property and Smoothness. 2.3. Consequences of the Mean Value Property. 3. The Schauder estimates for the Laplacian. 3.1. Review of Fourier Transform. 3.2. The Poisson Equation: Ideas of the Method. 3.3. The Classical Heat Semigroup. 3.4. The Fundamental Solution of the Laplacian. 3.5. Solvability of the Poisson Equation. 3.6. Schauder Estimates by Representation Formulas. 3.7. Schauder Estimates by the Method of Maximum Principle. 4. The Calderόn–Zygmund estimates for the Laplacian. 4.1. Solvability with Lp Right Hand Side. 4.2. L2 Estimate for Second Derivatives. 4.3. The Calderόn–Zygmund Theorem. 4.4. The BMO Space. 4.5. The John–Nirenberg Inequality. 4.6. Principal Value Representation of Second Derivatives. II. Divergence Form Equations. 5. The De Giorgi Theorem. 5.1. The De Giorgi Theorem. 5.2. L2 Implies L∞. 5.3. L∞ Implies Cα: De Giorgi’s Geometric Proof. 5.4. L∞ Implies Cα: Moser’s Critical Density Proof. 6. The Moser Theorem. 6.1. The Moser Theorem. 6.2. Upper and Lower Bounds. 6.3. Closing the Gap. 6.4. Harnack Inequality Implies Hölder Regularity. 7. Perturbation theory for Divergence Form Equations. 7.1. Schauder Estimates. 7.2. Calderόn–Zygmund Estimates. Section III. Nondivergence Form Equations. 8. Viscosity Solutions and the ABP Estimate. 8.1 Nondivergence Form Equations. 8.2 Viscosity Solutions. 8.3. The Alexandroff–Bakelman–Pucci Estimate. 9. The Krylov–Safonov Harnack Inequality. 9.1. The Krylov–Safonov Harnack Inequality. 9.2 The Weak-Lɛ Estimate for Supersolutions. 9.3. Subsolutions in Weak-Lɛ are Bounded and Conclusion. 10. Savin’s Method of Sliding Paraboloids. 10.1. Savin’s Sliding Paraboloids for Harnack Inequality. 10.2. The Point-To-Measure Estimate for Supersolutions. 10.3. The Localization Lemma. 10.4. The Covering Lemma. 10.5. Conclusion: Proof of the Harnack Inequality. 11. Perturbation Theory for Nondivergence Form Equations. 11.1. Schauder Estimates. 11.2. Calderόn–Zygmund Estimates. Section IV. The Fractional Laplacian. 12. Basic Properties of the Fractional Laplacian. 12.1. Method of Semigroups and Pointwise Formulas. 12.2. Pointwise Limits. 12.3. Maximum and Comparison Principles. 12.4. The Inverse Fractional Laplacian. 12.5. Weak Solutions and Fractional Sobolev Spaces. 12.6. An Explicit Example. 12.7. Viscosity and Pointwise Solutions, Hölder Regularity. 13. Hölder and Schauder Estimates. 13.1 Hölder Estimates. 13.2 Schauder Estimates. 13.3 Regularity Estimates via the Method of Semigroups. 14. The Caffarelli–Silvestre Extension Problem. 14.1. The Extension Problem for (−Δ)1/2 . 14.2. The Extension Problem for (−Δ)s . 14.3. A Detour to Degenerate Elliptic Equations. 14.4. Applications to Regularity Estimates.