We study maximal averages associated with singular measures on $\mathbb{R}$. Our main result is a construction of singular Cantor-type measures supported on sets of Hausdorff dimension $1-\epsilon$ with $0\le \epsilon <1/3$ for which the corresponding maximal operators are bounded on $L^p(\mathbb{R})$ for $p>(1+\epsilon )/(1-\epsilon )$. As a consequence, we are able to answer a question of Aversa and Preiss on density and differentiation theorems for singular measures in one dimension. Our proof combines probabilistic techniques with the methods developed in multidimensional Euclidean harmonic analysis; in particular, there are strong similarities to Bourgain's proof of the circular maximal theorem in two dimensions.

The purpose of this article is to clarify the Cauchy theory of the water-wave equations in terms of regularity indexes for the initial conditions, as well as for the smoothness of the bottom of the domain. (Namely, no regularity assumption is assumed on the bottom.) Our main result is that, after suitable paralinearization, the system can be arranged into an explicit symmetric system of Schrödinger type. We then show that the smoothing effect for the (one-dimensional) surface-tension water waves is in fact a rather direct consequence of this reduction, and following this approach, we are able to obtain a sharp result in terms of regularity of the indexes of the initial data and weights in the estimates.

Given any smooth circle diffeomorphism with irrational rotation number, we show that its invariant probability measure is the only invariant distribution (up to multiplication by a real constant). As a consequence of this, we show that the space of real $C^{\infty }$-coboundaries of such a diffeomorphism is closed in $C^{\infty }(\mathbb{T})$ if and only if its rotation number is Diophantine.

We show that any ancient solution to the Ricci flow which satisfies a suitable curvature pinching condition must have constant sectional curvature.