Geometric measure theory

  1. the most natural problem in harmonic analysis may be: investigate for what pair $latex (p,q)$ we have : $latex L^p(R^n)\longrightarrow L^q(R^n)$ $latex \hat f(x)=\int_{R^n}e^{-2\pi ix\xi}f(\xi)d\xi$ is strong-$latex (p,q)$ bounded. obvious we have the paserval identity:$latex ||\hat f||_{2}=||f||_2$,and we have $latex ||\hat f||_{\infty}\leq||f||_{1}$. so by the Riesz-Thorin inteplotation theorem we have the Hausdorff-Young inequality: $latex \forall 1\leq p\leq 2,\frac{1}{p}+\frac{1}{…

  1. 基本性质，例子 1.1. 例子和基本性质 在这一章的第一节引入了我们的研究对象，一般是一个紧的度量空间$X$装备上了一个同胚 Basic setting: Let $ (M,g)$ be a compact $ C^\infty$ Riemannian manifold of dimension $ n$, let $ \phi_{\lambda}$ be an $ L^2$- normalized eigenfunction of the Laplacian: $ \Delta \phi_{\lambda} = −\lambda^2 \phi_{\lambda}\$ and let:$ N \phi_{\lambda} =\{x:\phi_{\lambda}(x)=0\}$ be its nodal hypersurface. Let $ H^{n−1}(N\phi_{\lambda} )$ denote its $ (n-1)$-dimensional Riemannian hypersurface measure. In this …