Uniform Sobolev estimates on $R^N$ and compact manifolds and theire some applications

      In this talk, I would like to review some results related to uniform Sobolev estimates of the Laplacian on $R^n$ and compact manifolds. On $R^n$, this kind of estimates are closely to classical Sobolev/Hardy-Littlewood-Sobolev estimates, as well Stein-Tomas theorem. On compact manifolds, the estimates display a different uniformity due to the discrete distribution of spectrum. Besides, I shall discuss some applications to spectral theorem and PDE by using uniform Sobolev estimates