Harmonic analysis

  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}{…

Last year I read a nice blog articles Recent progress on the Kakeya conjecture and have several questions with this article. follows the proof strategy called Multiscale analysis,although we can use the estimate with large $ \delta_1$ to get estimate with small $ \delta_2$,(may be loss some $ \delta^c$ in the inequality in this way),but the main difficult is we should proof the new tubes with scales $ \delta_2$ is contains in the the olders.as soon as we proof this ,to obtain a lower b…

这是来自mathoverflow的一个问题： 原始的问题是简单的，陈述如下： 问题1：$ f\in C_{c}^{\infty}(B)$，且满足对任意满足：$ \exists n\in N^*$,$ \Delta^n g=0$的函数$ g$,有$ \int_{B}fg=0$,那么$ f=0$。 证明只需注意到所有多项式都满足条件以及weierstrass逼近定理。 人们自然想到将算子$ \Delta$推广到算子 $ \Delta_f=f\Delta$，这将导致如下问题。 问题2$ f\in C_{c}^{\infty}(B)$，且满足对任意满足：$ \exists n\in N^*$,$ (\Delta_f)^n g=0$的函数$ g$,有$ \int_{B}fg=0$,那么$ f=0$。   问题1在分布意义下是这样的： 问题3：$ T$是一个分布，满足对任意$ \exists n\in N^*,\Delta^n g=0$的测试函数空间中的函数$ g\in D(B)$,成立有$ T(g)=0$，则$ T=0$。   同样的问题2也能放入分布的框架下： 问题4$ T…