We show that the Luzin area integral or the square function on the unit ball of ℂ^{$n$}, regarded as an operator in the weighted space$L$^{2}($w$) has a linear bound in terms of the invariant$A$_{2}characteristic of the weight. We show a dimension-free estimate for the “area-integral” associated with the weighted$L$^{2}($w$) norm of the square function. We prove the equivalence of the classical and the invariant$A$_{2}classes.

We classify the ordinary differential equations that correspond to elliptic CR-manifolds with maximal isotropy. It follows that the dimension of the isotropy group of an elliptic CR-manifold can only be 10 (for the quadric), 4 (for the listed examples) or less. This is in contrast with the situation of hyperbolic CR-manifolds, where the dimension can be 10 (for the quadric), 6 or 5 (for semi-quadrics) or less than 4. We also prove that, for all elliptic CR-manifolds with non-linearizable isotropy group, except for two special manifolds, the points with non-linearizable isotropy form exactly some complex curve on the manifold.

We obtain integral boundary decay estimates for solutions of fourth-order elliptic equations on a bounded domain with regular boundary. We apply these estimates to obtain stability bounds for the corresponding eigenvalues under small perturbations of the boundary.

The topological group $\mathcal{D}^k(\mathbb{S})$ of diffeomorphisms of the unit circle $\mathbb{S}$ of Sobolev class$H$^{$k$}, for$k$large enough, is a Banach manifold modeled on the Hilbert space $H^k(\mathbb{S})$ . In this paper we show that the$H$^{1}right-invariant metric obtained by right-translation of the$H$^{1}inner product on $T_{\rm id}\mathcal{D}^k(\mathbb{S})\simeq H^k(\mathbb{S})$ defines a smooth Riemannian metric on $\mathcal{D}^k(\mathbb{S})$ , and we explicitly construct a compatible smooth affine connection. Once this framework has been established results from the general theory of affine connections on Banach manifolds can be applied to study the exponential map, geodesic flow, parallel translation, curvature etc. The diffeomorphism group of the circle provides the natural geometric setting for the Camassa–Holm equation – a nonlinear wave equation that has attracted much attention in recent years – and in this context it has been remarked in various papers how to construct a smooth Riemannian structure compatible with the$H$^{1}right-invariant metric. We give a self-contained presentation that can serve as a detailed mathematical foundation for the future study of geometric aspects of the Camassa–Holm equation.

In this paper we consider the computational complexity of uniformizing a domain with a given computable boundary. We give nontrivial upper and lower bounds in two settings: when the approximation of the boundary is given either as a list of pixels, or by a Turing machine.