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No. Consider the positive part of a coordinate function on a large but thin annulus.

I have now figured out the question, so I'll record it here. Let $L=ih(\nabla/i)$ be a constant coefficient differential operator, where $h$ is a polynomial. Recall the Bourgain norm is defined as $ …

Meyers and Serrin's $H=W$ is well known, but how does it generalize when we add weights? Let's define $H^{m,p}(\mu_0,\dots,\mu_m)$ to be the completion of $C^\infty(\Omega)$ in the norm $$\|u\|_{m,p …

I'm reading the proof of Lemma 2.11 of that book, for which Tao has an errata showing that the case $b=b'$ is not obvious. But I can't quite understand his explanation on how to show that case. Could …

This is indeed true, and let me illustrate this in the special case $k=1$. We are given a function $u\in H^1(0, 2\pi)$ such that $u(0)=u(2\pi)$. First we note that this boundary condition indeed make …

Recently I saw an interesting lemma: For any $s>0$, the closed unit ball in $H^s$ is also closed in the $L^2$ norm. That is, suppose $u_j\in H^s$ and $\|u_j\|_{H^s}\le 1$. Suppose $u_j\to u$ in $L^2$ …

Sometimes one has to roll up his sleeves and get his hands dirty in analysis. So here is the estimate you need: $$ \|f\|_2^2=(\sum_{|\xi|\le R}+\sum_{|\xi|>R}) |\hat f(\xi)|^2\ll_n R^n\sup_{\xi} |\h …

This is not an answer, but an elaboration of my comments above. Let $(f,g)\in L^2(D)\times L^2(\partial D)$. Find $g_n\in C^\infty(\partial D)$ such that $g_n\to g$ in $L^2(\partial D)$. Let $h_n\in …