Eigenvalue And Heat Kernel Estimates For The Weighted Laplacian On A Riemannian Manifold

Over the course of his distinguished career, Robert Strichartz (1943-2021) had a substantial impact on the field of analysis with his deep, original results in classical harmonic, functional, and spectral analysis, and in the newly developed analysis on fractals. This is the first volume of a tribute to his work and legacy, featuring chapters that reflect his mathematical interests, written by his colleagues and friends. An introductory chapter summarizes his broad and varied mathematical work and highlights his profound contributions as a mathematical mentor. The remaining articles are grouped into three sections – functional and harmonic analysis on Euclidean spaces, analysis on manifolds, and analysis on fractals – and explore Strichartz’ contributions to these areas, as well as some of the latest developments.

The international workshop on which this proceedings volume is based on brought together leading researchers in the field of elliptic and parabolic equations. Particular emphasis was put on the interaction between well-established scientists and emerging young mathematicians, as well as on exploring new connections between pure and applied mathematics. The volume contains material derived after the workshop taking up the impetus to continue collaboration and to incorporate additional new results and insights.

The third of three parts comprising Volume 54, the proceedings of the Summer Research Institute on Differential Geometry, held at the University of California, Los Angeles, July 1990 (ISBN for the set is 0-8218-1493-1). Part 3 begins with an overview by R.E. Greene of some recent trends in Riemannia

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 $L^{2}(M)$ 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 Reference material Bibliography Some notation Index