Analysis On The Singular Space

The dissertation is devoted to the analysis on singular spaces, which is separated into two parts:the analysis on Alexandrov spaces and the analysis on algebraic vari-eties.The first part includes five chapters. By the estimates of singularities of Alexan-drov spaces, we prove the uniform Poincare inequality, and hence derive the global properties of harmonic functions such as the Liouville theorem, the finite dimension theorem of polynomial growth harmonic functions follow.In Chapter One, the analysis on singular spaces is reviewed and the author's new results are introduced.In Chapter Two, basic definitions and properties of Alexandrov spaces, in partic-ular their local structures, are shown here. After investigating the properties of sin-gularities and the almost differential structures, the definitions of Sobolev spaces and harmonic functions are given.In Chapter Three, on the Alexandrov space with curvature bounded from below, the uniform Poincare inequality is obtained by the Bishop-Gromov volume comparison. The Liouville theorem of harmonic functions is derived by the Nash-Moser iteration, which also implies the Holder regularity and the Holder growth at infinity for harmonic functions.In Chapter Four, by the previous Poincare inequality, the polynomial growth har-monic function theorem follows by Colding-Minicozzi's argument. A simplified proof by Li also works by means of the mean value inequality for subharmonic functions.In Chapter Five, on the Alexandrov space with generalized nonnegative Ricci cur-vature in the sense of Sturm-Lott-Villani, the uniform Poincare inequality is proved and the properties of harmonic functions follow immediately.The second part is the Chapter Six, which deals with the analysis on algebraic va-rieties. The Sobolev-Gagliardo-Nirenberg and Poincare inequalities are proved on such spaces. It is given the sufficient and necessary condition in which the Poincare inequal-ity holds. In this way, an implicit lower bound for the first eigenvalue of Laplacian operator is obtained.