Makarov's principle for the Bloch unit ball

Makarov's principle relates three characteristics of Bloch functions that resemble the variance of a Gaussian: asymptotic variance, the constant in Makarov's law of iterated logarithm and the second derivative of the integral means spectrum at the origin. While these quantities need not be equal in general, we show that the universal bounds agree if we take the supremum over the Bloch unit ball. For the supremum (of either of these quantities), we give the estimate Σ^2_B < min(0.9, Σ^2), where Σ^2 is the analogous quantity associated to the unit ball in the L∞ norm on the Bloch space. This improves on the upper bound in Pommerenke's estimate 0.685^2 < Σ^2_B ⩽ 1

A smooth introduction to the wavefront set

The wavefront set provides a precise description of the singularities of a distribution. Because of its ability to control the product of distributions, the wavefront set was a key element of recent progress in renormalized quantum field theory in curved spacetime, quantum gravity, the discussion of time machines or quantum energy inequalitites. However, the wavefront set is a somewhat subtle concept whose standard definition is not easy to grasp. This paper is a step by step introduction to the wavefront set, with examples and motivation. Many different definitions and new interpretations of the wavefront set are presented. Some of them involve a Radon transform.Comment: 29 pages, 7 figure

On Gevrey singularities of microhyperbolic operators

We study the Gevrey singularities of solutions of microhyperbolic equations using exponential weighted estimates in the phase space. In particular, we recover some known results on the propagation of Gevrey regularity in an elementary way, using microlocal exponential estimates