Bazin
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Member for 12 years, 9 months
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Hörmander's hypoellipticity theorem for complex coefficients
Yes of course, Chapter 27 in ALPDO is studying subelliptic pseudodifferential operators which are special cases of hypoelliptic operators.
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Gevrey estimate of derivatives
Yes, you are right, I have to replace my $n$ by $n-k$ then it gives your formula. Nice job.
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Gevrey estimate of derivatives
About the penultimate equality: I believe that the sum should be replaced by $\sum_{1\le k\le n} 2^k\binom{n+k-1}{k-1}$, which increases geometrically wrt $n$ anyway.
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Gevrey estimate of derivatives
@Abdelmalek Abdesselam I tried Faa di Bruno's formula but it got messy.
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Gevrey estimate of derivatives
@Icv I did formulate a question in a new version.
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Gevrey estimate of derivatives
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Norm of a singular integral operator
Some more explanations on one of the assertions.
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If $\mathcal R_j f\in L^1$ then $\widehat{\mathcal R_j f}=-i\frac{\xi_j}{|\xi|}\widehat{f}(\xi)$
@Mizar If you consider the two brackets of duality in my comment above, when $f\in L^1$, the second one is a true integral; now using that and the proof that you have for the convergence in the Lorentz space, don't you obtain what you wish?
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If $\mathcal R_j f\in L^1$ then $\widehat{\mathcal R_j f}=-i\frac{\xi_j}{|\xi|}\widehat{f}(\xi)$
@Mizar Yes because the formula holds true in a weak sense, by the definition of my answer; you have $\mathcal R_jf=\text{Fourier}^{-1}(\tau_j \hat f)$, which means that the bracket of duality $\langle \mathcal R_jf,\phi\rangle$ is equal to $\langle \tau_j\hat f,\text{Fourier}^{-1}\phi\rangle$, for any function $\phi\in \mathscr S(\mathbb R^n)$.
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Largest known zero of the Riemann zeta function
My last comments can be illustrated by the fact that $\ln\ln 10^{100}\approx 5.4$ (quite small), $\ln\ln10^{10000}\approx 10$ (not so large), $\ln\ln10^{10^{12}}\approx 28$ (still rather small). I believe that $10^{10^{12}}$ is beyond imagination if you think that there are less than $10^{100}$ particles in the universe. Anyhow $\ln\ln$ is increasing very slowly and detecting numerically that function as not constant (!) would involve incredibly large numbers.