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It is a classical result in harmonic analysis that $$ \|\|P_kf\|_{\ell^2_k}\|_{L^p_x}\approx\|f\|_{L^p} $$ for $p\in(1,\infty)$, where $P_k$ is the Littlewood-Paley decomposition onto frquency $\app …

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

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 $L^2$ …

Bourgain and Demeter's proof of the $L^2$ decoupling conjecture decouples $\|f\|_{L^p}$ into an $L^2$ sum of $\|f_\theta\|_{L^p}$, where $\hat f$ is supported on a curved hypersurface $S$, where $\the …

In the equation (1.1.17) in Proposition 1.1.6 (ii) in Alazard and Delort's Sobolev Estimates for Two Dimensional Water Waves, there appears a norm named $C^{-1}$, but in Chapter 6 (Appendix) of the sa …

I'm just asking if there is a name for the space of functions on $\mathbb R^n$ whose norm is defined by $$ \|f\|=\|\hat f\|_{L^p} $$ for $p\in [1,\infty]$. I find it handy to give it a name when dis …

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 easier when passed to some sort of weak formulation. By Lebesgue differentiation theorem, for almost every x, $\lim_{r\to0} \frac{1}{|B_r|} \int_{x+B_r} f=f(x)$. Replace each f(x) by the left …

Consider the inhomogeneous linear heat equation $$\partial_tu-\Delta u=F$$ on $\mathbb R^n\times [0,1]$ (say) with zero initial data. Assume $F$ is very nice (say Schwarz), so that we have a nice so …

You can define the distributionial Fourier transform of a tempered distribution using all the abstract machinery established by Schwartz, and the thing you want to check is that it agrees with the int …

For those who don't have the book (or have the wrong version), here is the proof that the topological vector space of holomorphic functions on the unit disk is not normable (i.e. whose topology is not …

I'm going through the last steps of Bourgain and Demeter's proof of the $l^2$ decoupling conjecture, but I'm unable to see how the first inequality in (43) goes through. I'll water down the question a …

Integrating by parts, using a cutoff function if necessary, we have $$ \int_{B(r)} |\nabla u|^2\ll \int_{B(r)} a_{ij}\partial_iu\partial_ju=\int_{B(R)} uLu+O(u|\nabla u|)\ll \int_{B(R)} |u\nabla u|. …

For all $a\in(0,K)$, the function $\hat 1_{[0,a]}$ is $K$-frequency localized, yet the function itself decays like $1/x$ as $x\to\infty$ (as easily seen by integration by parts), so it is not spatiall …