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Let $\psi=f+ig=\rho e^{i\theta}$ be a complex function on some open subset of $\mathbb{R}^n$, where $f,g,\rho$ and $\theta$ are real-valued. I happened to find that the identity of differentiation for the polar coordinate expression \begin{equation} (1)\qquad\nabla\psi=e^{i\theta}\nabla\rho + i\rho e^{i\theta}\nabla\theta \end{equation} is rather formal. In general, $f$ and $g$ has the same regularity as $\psi$: $\psi\in C^1$ iff $f,g\in C^1$; $\psi\in H^1$ iff $f,g\in H^1$, etc. But this is not the case for $\rho$ and $\theta$. For example, even $f,g\in C^{\infty}$, $\rho=\sqrt{f^2+g^2}$ may not be differentiable at points where $f=g=0$. One can see this by considering the one dimensional example $f(x)=x$, $g(x)=0$, and $\rho=|x|$ is not differentiable at $0$. Assuming $f,g\geq 0$ is a way to remedy this problem. In fact, $f,g\geq 0 $ and $C^1$ implies $\rho\in C^1$. A more natural way is to simply consider weak differentiability. For example if $f,g\in H^1$, I can show $\rho\in H^1$ (of course with weak derivatives in the same form as the usual ordinary ones. But although it seems to be very natural, I found this is a not so easy exercise in real analysis). I proved it by using approximation of $f,g$ by smooth functions. And I failed when using only the definition of weak derivatives. So, if only weak differentiability of $f,g$ is known and there is no approximation theorems as in Sobolev spaces, I don't know whether $\rho$ is also weakly differentiable. But anyway the result in the Sobolev setting may be sufficiently satisfactory.

The big problem comes when considering $\theta$. I have no idea about the regularity of $\theta$ at all. See the second term in the right hand side of identity (1), if $\psi\in H^1$, we should have $\rho\nabla\theta\in L^2$. But we know $\rho\in L^2$ (in fact $H^1$), so $\nabla\theta\in L^{\infty}$? $\theta\in W^{1,\infty}$? There is also a disturbing problem that on where $f$ and $g$ are both zero, $\theta$ can in fact be defined arbitrarily. So, a more correct question may be is there a natural choice of $\theta$ so that (1) is true in some sense?


Thanks to Denis Serre's references, I think my question may better be divided into two parts.

First, let $f+ig=\rho u$ with $\rho=\sqrt{f^2+g^2}$ and $u$ some complex function with $|u|=1$. Then, what is the best possible regularity of $u$ expected, in terms of $f$ and $g$?

And second, write $u=e^{iv}$, what is the best regularity of $v$ in terms of $u$?

The references deal with the second question. From their introductions it looks like the natural framework for this question are such spaces as VMO, BMO, etc. But does the first question lead to such answer as $u\in$VMO or $u\in$BMO, under for example the assumption $f,g\in H^1$? I have little knowledge about the spaces BMO and VMO. Is the answer to my last question naturally "yes"?

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The fact that $\rho\in H^1$ is a classical fact. Stampacchia proved that if $u\in H^1$ and $\phi$ is Lipschitz, then $\phi\circ u\in H^1$. This applies here,with $\phi$ the Euclidian norm. –  Denis Serre Nov 5 '10 at 9:09
    
Without some smoothness conditions, I don't think you can conclude that $\theta \in L^p$ for any $p$ ($\infty$ included). [Even when $\theta$ is smooth almost everywhere, you still can't conclude that. For example, pick a point $p$ in the open domain and construct $(n-1)$-spheres centered at $p$ with radius $1/k$ for positive integers $k$. Any $\theta$ can be modified to jump at the boundary of each sphere (that intersects the domain) by an arbitrary multiple of $2\pi$ and remain smooth almost everywhere. Obviously we can arrange such that $\theta$ is not $L^p$ for any $p$.] –  Tunococ Nov 5 '10 at 9:43
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1 Answer 1

If $f,g$ have weak derivatives, the vanishing of $\rho$ is not the only problem. Even if $\rho$ stays uniformly positive, estimating the regularity of $\theta$ in terms of that of $(f,g)$ is hard and does not always work. Ths has been studied by Bourgain and Brézis (Comm. Pure Appl. Math. 58 (2005) 529–551 ; Publ. Math. Inst. Hautes Etudes Sci. 99 (2004) 1–115), Mironescu (C. R. Math. Acad. Sci. Paris 346 (2008), no. 19-20, 1039–1044) and H.-M. Nguyen (C. R. Math. Acad. Sci. Paris 346 (2008), no. 17-18, 957–962).

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Thanks for the references. But the question they addressed is the relation between the regularity of $u$ and $v$ for $u=e^{iv}$. This seems not to answer my question directly. In fact it awares me that maybe my question should be divided into two parts. Please see my added note. –  Liren Lin Nov 6 '10 at 13:41
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