题 目: Self-improving properties of Poincare inequalities
报告人：Professor Juha Kalevi Kinnunen (University of Aalto, Finland)
时 间： 12月14日15:30至16:30
地 点: 后主楼1220室
摘 要: A doubling condition for the measure and a Poincare inequality are rather standard assumptions in the analysis on metric measure spaces. A Poincare inequality is a tool to transfer the infinitesimal information encoded in the gradient to an oscillation estimate for a function in larger scales. It relates the notion of gradient to the measure and, together with the doubling condition, implies Sobolev inequalities on metric measure spaces. A theorem of Keith and Zhong asserts that a Poincare inequality is self-improving in the sense that if it holds true for one exponent, it also holds true for a slightly better exponent. This result is of fundamental importance not only because of its theoretical interest, but also in applications, for example, to the regularity theory in the calculus of variations. We discuss a new point-of-view to this result with a special emphasis on the role of the underlying space and relevant maximal function estimates. We also consider the corresponding results in the two-measure case and show that the self-improving result holds true under a balance condition on the measures. Examples are constructed to illustrate that the optimality of the results.