Laplace operator and the heat equation in $\mathbb {R}̂n$ Function spaces in $\mathbb {R}̂n$ Laplace operator on a Riemannian manifold Laplace operator and heat equation in $L̂{2}(M)$ Weak maximum principle and related topics Regularity theory in $\mathbb {R}̂n$ The heat kernel on a manifold Positive solutions Heat kernel as a fundamental solution Spectral properties Distance function and completeness Gaussian estimates in the integrated form Green function and Green operator Ultracontractive estimates and eigenvalues Pointwise Gaussian estimates I Pointwise Gaussian estimates II Appendix A. Reference material

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Electronic reproduction. Providence, Rhode Island : American Mathematical Society. 2012