Download On Rough Frobenius-type Theorems and Their Hölder Estimates Book in PDF, Epub and Kindle
This thesis is dedicated to giving several results on Frobenius-type theorems in non-smooth settings, and giving H\"older regularity estimates for the respective coordinate systems. For the real Frobenius theorem, we extend the definition of involutivity to non-Lipschitz tangent subbundles using generalized functions. We prove the Frobenius Theorem with sharp regularity when the subbundle is log-Lipschitz: if $\mathcal V$ is a log-Lipschitz involutive subbundle of rank $r$, then for any $\varepsilon>0$, locally there is a homeomorphism $\Phi(u,v)$ such that $\Phi,\frac{\partial\Phi}{\partial u^1},\dots,\frac{\partial\Phi}{\partial u^r}\in\mathscr C^{1-\varepsilon}$, and $\mathcal V$ is spanned by the continuous vector fields $\Phi_*\frac\partial{\partial u^1},\dots,\Phi_*\frac\partial{\partial u^r}$. We also develop a singular version of the Frobenius theorem on log-Lipschitz vector fields. If $X_1,\dots,X_m$ are log-Lipschitz vector fields such that $[X_i,X_j]=\sum_{k=1}^mc_{ij}^kX_k$ for some generalized functions $c_{ij}^k$ that can be written as the derivatives of log-Lipschitz functions, then for any point $p$ there is a $C^1$-manifold $\Sf$ containing $p$ such that $X_1,\dots,X_m$ span the tangent space at every point in $\Sf$. Nirenberg's famous complex Frobenius theorem gives necessary and sufficient conditions on a locally integrable structure for when the manifold is locally diffeomorphic to $\mathbb R^r\times\mathbb C^m\times \mathbb R^{N-r-2m}$ through a coordinate chart $F$ in such a way that the structure is locally spanned by $F^*\frac\partial{\partial t^1},\dots,F^*\frac\partial{\partial t^r},F^*\frac\partial{\partial z^1},\dots,F^*\frac\partial{\partial z^m}$, where we have given $\mathbb R^r\times\mathbb C^m \times\mathbb R^{N-r-2m}$ coordinates $(t,z,s)$. When the structures are differentiable, we give the optimal H\"older-Zygmund regularity for the coordinate charts which achieve this realization. Namely, if the structure has H\"older-Zygmund regularity of order $\alpha>1$, then the coordinate chart $F$ that maps to $\mathbb R^r\times\mathbb C^m \times\mathbb R^{N-r-2m}$ may be taken to have H\"older-Zygmund regularity of order $\alpha$, and this is sharp. Furthermore, we can choose this $F$ in such a way that the vector fields $F^*\frac\partial{\partial t^1},\dots,F^*\frac\partial{\partial t^r},F^*\frac\partial{\partial z^1},\dots,F^*\frac\partial{\partial z^m}$ on the original manifold have H\"older-Zygmund regularity of order $\alpha-\varepsilon$ for every $\varepsilon>0$, and we give an example to show that the regularity for $F^*\frac\partial{\partial z}$ is optimal. Similarly we give a counterexample for the $C^k$-version of the Newlander-Nirenberg theorem: we give an example of $C^k$-integrable almost complex structure that does not admit a corresponding $C^{k+1}$-complex coordinate system. Combining the log-Lipschitz Frobenius theorem and the sharp Frobenius theorem, we show that if a complex Frobenius structure $\mathcal S$ is $\mathscr C^\alpha$ ($\frac12