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awarded  Nice Question 
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revised 
Dual of Banachvalued $L^p$
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2d

asked  Dual of Banachvalued $L^p$ 
Jan 27 
comment 
Surjectivity of curl
@Khavkine I think that Hachino has settled the question. Thanks for caring about this question. 
Jan 27 
comment 
Surjectivity of curl
Thanks for this very nice counterexample. The vanishing of the mean (i.e. $\hat \nu(0)=0$) that I required in the question above is certainly necessary, but as you have just pointed out is not sufficient. All moments should vanish, this is an elegant addendum to Poincaré Lemma. 
Jan 18 
answered  Decompose the Laplacian 
Jan 18 
comment 
Surjectivity of curl
Thanks for your answer. However, I doubt that $v$ in the Schwartz space implies that $w$ is also in the Schwartz space. With $\mathbb P$ standing for the Leray projector, $\mathbb P$ is the matrix Fourier multiplier $curl^2(\Delta)^{1}$. What you wrote is $v=Pv=curl(curl(\Delta)^{1})v$ and you take $w=curl(\Delta)^{1})v$. Now there is a priori a singularity at $\xi=0$, that should be gotten rid of. 
Jan 17 
asked  Surjectivity of curl 
Jan 15 
answered  Witten index nontrivial in the context of Quantum Mechanics? 
Jan 6 
answered  Transforming a recurrence to the product of two other recurrences 
Jan 6 
answered  Examples of potentials for which Schrödinger equation lacks discrete points in continuous spectrum 
Dec 22 
answered  Application of Egorov's Theorem for Pseudodifferential Operators 
Dec 14 
answered  Morse lemma with least amount of regularity. 
Dec 10 
answered  Solution of a second order nonlinear ode 
Dec 10 
answered  Local fractional Sobolev inequality 
Dec 8 
comment 
Extension of solutions of PDEs with constant coefficients
I should have made my answer more precise, since it deals indeed with real analyticity (I have modified my answer). Of course, ellipticity alone does not imply the sought property globally in $\mathbb C^n$: take for instance for $n=1$, $\frac{\partial}{\partial z}$ and the elliptic equation $\frac{\partial u}{\partial z}=0$, which has no (nontrivial) holomorphic function as a solution. 
Dec 8 
revised 
Extension of solutions of PDEs with constant coefficients
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Dec 8 
answered  Extension of solutions of PDEs with constant coefficients 
Dec 8 
comment 
Is there a compactly supported function that its Fourier transfrom vanishes at given n real points?
I agree with your answer, but it could not be generalized to an infinite number of zeroes, a natural extension of the question. The Weierstrass factorization theorem allows to construct entire functions with prescribed zeroes and in some case these functions could be of exponential type, thus with compactly supported Fourier transforms, e.g $\sin z$. 
Dec 6 
revised 
Is there a compactly supported function that its Fourier transfrom vanishes at given n real points?
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