Lifting in Besov spaces (Q1985855)

The authors extend the results of \textit{J. Bourgain} et al. [J. Anal. Math. 80, 37--86 (2000; Zbl 0967.46026)] on the lifting problem for Sobolev spaces to Besov spaces. The lifting problem asks whether one can construct the phase of a complex valued function $u$, and if one can, what smoothness properties does it inherit from the function. Precisely, \(\Omega\) is a simply connected domain in \(\mathbb{R}^n\), and \(u:\Omega \mapsto S^1\) is a continuous function (or \(C^k$, $k \geq 1\)) with \(|u(x)| = 1\) for all \(x\). There is a continuous function \(\phi\) such that \(u = e^{\imath \phi}\), which is called a lifting of \(u\). The lifting is only unique mod \(2\pi\). When \(u \in W^{s,p}(\Omega; S^1)\), [loc.\,cit.]\ gave complete results for the smoothness and uniqueness of \(\phi\) for \(u\) in the Sobolev space \(W^{s,p}(\Omega; S^1)$, $s>0$, $1 \leq p < \infty\). The authors give comparable results for \(B^{s}_{p,q}\). The authors consider the case \(s>0\) in order to work with functions rather than distributions, \(1 \leq p < \infty\) and \( 1 \leq q \leq \infty\) in order to be able to compare with the results of [loc.\,cit.], although they say the results extend to \(0 < q <1\), and \(0 < p < 1\) if the functions in the space are locally integrable functions. To ease comparison, I give the main results below as the authors do, first for the Bourgain-Brezis-Mironescu case of Sobolev spaces and then for the results of this paper. \begin{itemize} \item[(a)] For \(n = 1\), if \(u \in W^{s,p}((0,1); S^1)\), lifting holds for every \(s, p\) as above. The lifting is unique if \(sp \geq 1\). \item[(b)] For \(n = 1\), if \(u \in B^s_{p, q}((0,1); S^1)\), lifting holds for every \(s, p, q\) except when \(1 \leq p < \infty$, $s = 1/p$, $q = \infty\). The lifting is unique iff \(sp > 1\), or \(sp = 1$, $q < \infty.\) \end{itemize} The range where uniqueness holds in higher dimensions is given by the same result: \(sp > 1\), or \(sp = 1$, $q < \infty\). For higher dimensions then, I will only discuss existence of liftings. \begin{itemize} \item[(c)] For \(n=2\), if \(u \in W^{s,p}( \Omega; S^1)\), lifting holds if \(sp <1\) or \(sp \geq 2\). Lifting does not hold if \(1 \leq sp <2\). \item[(d)] For \(n = 2\), if \(u \in B^s_{p, q}( \Omega; S^1)\), lifting holds if \(sp<1\) or if \(sp>2\) or if \([sp = 2$, $q < \infty]\) and does not hold if \(1 \leq sp <2\) or \([sp = 2$, $q = \infty]\). \item[(e)] For \(n \geq 3\), if \(u \in W^{s,p}( \Omega; S^1)\), lifting holds if \(sp<1\) or if [\(s \geq 1\) and \(sp \geq 2\)] or if \(sp \geq n\). Lifting does not hold if \(1 \leq sp <2\) or [\(0 < s < 1\) and \(2 \leq sp < n\)]. \item[(f)] For \(n\geq 3\), if \(u \in B^s_{p, q}( \Omega; S^1)\), lifting holds if \(sp <1\) or \([s > 1\) and \(sp > 2\)] or [\(s> 1\) and \(1 \leq q \leq p < \infty\) and \(sp = 2\)] or \(sp > n\) or [\(sp = n\) and \(q < \infty\)]. Lifting does not hold if \(1 \leq sp <2\) or [\(sp = 2\) and \(q = \infty\)] or [\(0 < s < 1\) and \(2 \leq sp < n \)]. \end{itemize} Notice that unlike in dimensions 1 and 2, in dimensions higher than 3 there are some values for which the lifting results are not known. These are collected in Section 6 along with an explanation of the reason for the difficulty. The results in one dimension follow the methods of [loc. cit.] for \(sp<1\) and \(sp>1\), but even there new results are required for \(sp =1\). One of the new approaches is to differentiate \[ u = e^{\imath \phi} \ \Longrightarrow \ \nabla u = \imath u \nabla \phi \ \Longrightarrow \ \nabla \phi = \frac{\nabla u}{ \imath u} = -\imath \overline{u} \nabla u = F \] and attempt to prove that \(F \in B^{s-1}_{p,q}\), but \(F\) is to be a gradient, leading to a need to check \(\mathrm{curl} \, F = 0\). This is more difficult than for functions in \(W^{s,p}\), and is explained by a disintegration argument for the Jacobian in {Lemma~3.11}. It is impossible to describe all of the methods in a short review; I refer the interested reader to the paper for details.